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ADVANCES IN

GEOPHYSICS VOLUME 4

This Page intentionally left blank

Advunces In

GEOPHYSICS by H. E. LANDSBERG Edited

U. S. Weother Bureou Woshingfon, D. C.

J. VAN MIEGHEM Royal Belgion Mefeorological Institute

Uccle, Belgium

Editorial Advisory Committee BERNHARD HAURWITZ WALTER D. LAMBERT

ROGER REVELLE R. STONELEY

VOLUME 4

ACADEMIC PRESS INC

PUBLISHERS

NEW Y O R K , 1 9 5 8

COPYRIGHT @ 1958, B Y ACADEMIC PRESS INC. 111 Fifth Avenue, New York 3, N. Y.

All Rights Reserved No part of this book may be reproduced in any form, by photostat, microfilm, or any other means, without written permission from the publishers.

Library of Congress Catalog Card No. 52-12266 PRINTED I N T H E UNITED STATES OF AMERICA

LIST OF CONTRIBUTORS JOSEPHW. CHAMBERLAIN, Yerkes Observatory, University of Chicago, Williams Bay, Wisconsin J. LEITH HOLLOWAY, JR., General Circulation Research, U. S. Weather Bureau, Suitland, Maryland CHRISTIAN E. JUNGE,A i r Force Research Center, Bedford, Massachusetts LINCOLNLAPAZ,Director, Institute of Meteoritics, University of New Mexico, Albuquerque, New Mexico PAUL MELCHIOR,Obsereatoire Royal de Belgique, Reporter General f o r earth tides of the International A ssociations o f Geodesy and Seismology, Uccle, Brussels, Belgium

V


FOREWORD As this fourth volume of Advances in Geophysics finds its way t o our colleagues, a fifth volume is being assembled and a sixth is through the planning stages. This reflects the rapid progress in our field. The International Geophysical Year-and the public interest in it-has also served to promote a once rather obscure area of science into the limelight. Also, more departments and institutes of geophysics or earth sciences are being established in universities the world over. Thus it has become one of our aims, among others, to bring in these volumes, to professors and advanced students alike, broad reviews of sectors of our science where new knowledge has accumulated fastest or where perhaps neglect has relegated an important facet into the background. I n all our contributions the authors not only report the solid progress but also point toward the great questions which remain open and where the avenues of future research lie. Wherever possible, we want also to show how other sciences impinge on the quest for knowledge about the physics of our planet. The meteoritical article in this volume reflects this concept. We are also always interested in discussions of analytical techniques for the rapidly accumulating data collections. I n the search for suitable topics the editor has had again the good advice of the editorial committee. This is gratefully acknowledged. Reviewers have voiced however, an important criticism of our past volumes. This is the preponderance of American authors for our articles. Although we are quite aware of the universality of geophysics the problem of language difficulties has loomed large. I n Volume IV we are trying our hand for the first time with a translated contribution. I n a limited way, this may provide an answer. We have been fortunate enough t o obtain a distinguished European colleague as associate in the editorship. Dr. J. Van Mieghem of Uccle, Belgium, is joining me in the editorial responsibility, and his active participation will be apparent beginning with Volume VI. A preview of Volume V indicates that the article on model experiments, already announced in the foreword to Volume 111, will be in it. Other titles will include cloud physics, atmospheric tides, photochemical problems of the high atmosphere, and the shape of the earth.

H. E. LANDSBERG August, 1957 vii


CONTENTS LIST OF CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . .

v

FOREWORD . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Atmospheric Chemistry

CHRISTIANE . JUNGE 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. TraceGases . . . . . . . . . . . . . . . . . . . . . . . . . . 44 71 4. Precipitation Chemistry . . . . . . . . . . . . . . . . . . . . . 5 . Air Pollution and Its Role in the Chemistry of Unpolluted Air . . . . 94 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Theories

of the Aurora

.

JOSEPH W CHAMBERLAIN 1 . Introduction . . . . . . . . . . . . . . . . . . . . 2. The Motions of Charged Particles in Magnetic Fields . 3 . Stormer’s Theory of Aurorae . . . . . . . . . . . . 4. Electric Currents bctween the Sun and Earth . . . . . 5. The Chapman-Ferraro Stream and Ring Current . . . 6 Other Electric-Field Theories of Aurorae . . . . . . . 7 Additional Mechanisms for the Production of Aurorae . 8. Theories of Auroral Excitation . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. .

. . . . . . 110 . . . . . . . 116 . . . . . . . 127 . . . . . . . 146 . . . . . . . 158 . . . . . . . 173 . . . . . . . 183 . . . . . . . 191 . . . . . . 206 . . . . . . 207 . . . . . . 210

The Effects of Meteorites upon the Earth (Including Its Inhabitants. Atmosphere. and Satellites)

LINCOLN LAPAZ 1. The Effects of Typical Meteorite Falls . . . . . . . . . . . . . . 218 2. Number. Classification. and Weights of Recovered Meteorites . . . . . 235 3 . Metcoritic Abundnnws and Terrcstrid Meteorit.ic Accretion . . . . . 240 4 . The Hyperbolic Meteorite Velocity Problem . I . . . . . . . . . . . 273 5 . The Hyperbolic Meteorite Velocity Problem . 11. . . . . . . . . . . 292 6. Crater-Producing Meteorite Falls . . . . . . . . . . . . . . . . . 307 Appendix I . Meteoritical Pictographs and the Veneration and Exploitation of Meteorites . . . . . . . . . . . . . . . . . . . . . . . . . . 329 ix

CONTENTS

X

Appendix I1. Basic Meteoritic Data and in Section 5 . . . . . . . . . . . List of Sym6oIs . . . . . . . . . . . References . . . . . . . . . . . . .

Classificational Criteria Employed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

336 339 341

Smoothing and Filtering of Time Series and Space Fields

J . LEITHHOLLOWAY. JR. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Time Smoothing and Filtering . . . . . . . . . . . . . . . . . . 3. Equalization, Pre-emphasis, and Inverse Smoothing . . . . . . . . 4. Smoothing and Filtering Functions . . . . . . . . . . . . . . . . 5. Frequency Response of Smoothing Functions and Other Filters . . . . 6. Design of Smoothing Functions and Filters with Specified Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. High-Pass and Band-Pass Filtering Functions . . . . . . . . . . . 8 . Elementary Smoothing and Filtering Functions . . . . . . . . . . 9 . Design of Inverse Smoothing Functions . . . . . . . . . . . . . 10. Design of Pre-emphasis Filters . . . . . . . . . . . . . . . . . 11. Filtering by Means of Derivatives of Time Series . . . . . . . . . 12. Space Smoothing and Filtering . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351 352 353 354 355 363 365 369 372 376 378 380 386 387 388

Earth Tides

PAULJ . MELCHIOR 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 392 2. Static Theory of the Tides . . . . . . . . . . . . . . . . . . . 394 3 . Definition of Love’s Numbers . . . . . . . . . . . . . . . . . . 400 4 Studs of the Amplitude of Oceanic Tides . . . . . . . . . . . . . 401 5 Periodical Deflections of the Vertical with Respect to the Crust . . . . 403 6 Measurement of Elastic Tensions and Cubic Dilatations Due to Deformations Produced by the Earth Tides . . . . . . . . . . . . . . . 418 7 . Deflections of the Vertical with Respect to the Axis of the Earth . . . 423 8 . Variations in the Intensity of Gravity . . . . . . . . . . . . . . . 426 9. The Role of the Geologic Structure of the Crust in the Indirect Effects 432 10. Theory of Elastic Deformations of the Earth . . . . . . . . . . . . 435 11 . Effect of Earth Tides on the Speed of Rotation of the Earth . . . . . 439 12. Program of the International Geophysical Year . . . . . . . . . . . 440 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

.

. . .

ATMOSPHERIC CHEMISTRY Christian E. Junge Geophysics Research Directorate, Air Force Cambridge Research Center, 1. G. Hanscom Field, Bedford, Massachusetts

Page 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 2.1. General Comments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. Size Distribution of Natural Aerosols.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Physical Constitution of Natural Aerosol Particles. . . . 2.4. Nature and Origin of Aitken Particles,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5. Sea-Salt Aerosols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6. Continental Aerosols. . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ 29 3. Trace Gases.. . . . . . . . . 3.1. General Comment 3.2. Carbon Dioxide.. . . . . . . . . . . 45 3.3. Ozone .............................. 3.4. Nitrous Oxide.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 59 3.5. Nitric Oxide, Nitrogen Dioxide, and Ammonia.. . . . . . . . . . . . . . . . . . . . . . 3.6. Sulphur Dioxide, Hydrogen Sulfide. ............................. 64 ............................. 66 3.7. Halogens ........................ 67 3.8. Methane, Carbon Monoxide, and Formaldehyde.. . . . . . . . . . . . . . . . . . . . . 3.9. Ratio of Gaseous to Particulate Matter for Some Chemical Components 68 4. Precipitation Chemistry.. . . . ............................. 71 4.1. The Removal of Trace Substances from the Atmosphere.. . . . . . 4.2. Physical and Chemical Processes of Wash-Out by Precipitation.. . . . . . . . 73 4.3. The Content of Rain Water with Respect to the Predominantly Continental Components NH4+, NO,-, NOz-, SOa-- and Ca++. . . . . . . . . . . . 81 4.4. The Content of Rain Water with Respect to the Predominantly Maritime Components C1-, Na+, Kf, Mg++ and the Role of Precipitation in the 88 Natural Cleansing Process of the Atmosphere.. . . . . . . . . . . . . . . . . . . . . . . 5. Air Pollution and Its Role in the Chemistry of Unpolluted Air. . . . . . . . . . . . . 94 .................... 94 5.1. The Main Components in Polluted Air. 5.2. The Area of Influence around Pollution Centers., . . . . . . . . . . . . . . . . . . . . 98 List of Symbols ................................................ 101 ...... ..... . 101

1. INTRODUCTION

Atmospheric chemistry, if it is to be comprehensive, must treat such varied problems as the origin of our atmosphere as a phenomenon of general geochemistry, the temporal changes and geochemical cycles of its components, the effects of biological processes at the earth’s surface and 1

2

CHRISTIAN E. JUNGE

in its oceans, and finally, the photochemistry of the upper atmosphere. Space limitations preclude any such comprehensive treatment here; our discussion will center, therefore, on one particular field of atmospheric chemistry, the chemistry of active trace substances in the troposphere. Active trace substances may be defined as those atmospheric constituents which form but an infinitesimal part of the total atmosphere, and which are continually in the process of being removed and added to the atmosphere by chemical or meteorological processes. As a result their concentration varies greatly with time and space, in contrast to the permanent gases, including the rare ones, which we will regard here as the given carrier gas. Trace substances comprise primarily compounds of nitrogen, sulphur, chlorine, carbon, and oxygen, as well as all substances present in the form of aerosols. We consider COz a borderline case because of its comparatively large concentration, small fluctuations, and relatively passive role in the atmosphere. Water, though it undergoes rapid changes, will not be treated here. We will restrict our discussion to the troposphere, since its chemistry is t o a large degree independent of, and different from, that of the stratosphere. A number of the trace substances to be treated here are formed at the earth’s surface and will be confined to the troposphere by the limiting action of the tropopause. On the other hand, a large number of the products formed photochemically at high altitudes will remain a t these levels, being unable to penetrate downwards through the tropopause. Our knowledge of tropospheric trace substances is quite fragmentary; there are, however, a number of vague hypotheses, most of which have been formulated on the basis of only a very few measurements. Therefore, our first concern will be to summarize as comprehensively as possible the most reliable factual material and to point out in what respect it is inadequate. Hypotheses and assumptions regarding the nature of the chemical decompositions and the cycle of the substances will be mentioned only insofar as they appear to be reasonably supported by observation, or possess a certain degree of probability on the basis of our present knowledge. In our description of the chemical components (Sections 2 and 3) we will distinguish them according to phase, i.e., whether they are present in particle form, as an aerosol, or as a gas. This appears to be the most reasonable division, and is of fundamental importance for those substances which may occur in both forms. Section 2 will therefore start with a detailed discussion of the physics and meteorology of natural aerosols, since it is only on this basis that we will be able to understand properly the behavior of certain chemical compounds and their role in cloud and precipitation chemistry.

3

ATMOSPHERIC CHEMISTRY

I n Section 4, cloud and precipitation chemistry will be discussed. These processes take on particular importance in a n age of increasing industrial pollution of the atmosphere, for they provide the major means whereby numerous chemical components are removed from the atmosphere. Again we feel the need for a careful discussion of the physicochemical processes involved in order to attempt a proper interpretation of the extensive observational data from rain-water analyses. Finally, in Section 5 we will discuss air pollution briefly in terms of its significance for the general chemistry of the troposphere. The role of these anthropogenic trace substances in the troposphere has occasionally been overestimated, though in certain cases their influence has been found to attain a continental scale. 2. AEROSOLS 2.i. General Comments I n view of their special importance for numerous branches of meteorology, we will begin our discussion of trace substances with the aerosol. T o provide the necessary basis for a proper understanding of the chemical composition of the aerosol particles we will first give a complete survey of their physical properties, size distribution, and concentration under various conditions. The concept of the aerosol is generally understood as the disperse state of matter in a gaseous medium, in our case the air. All sizes of particles, ranging from air molecules all the way to raindrops, appear in the atmosphere (see Fig. 1). However, our interest will be confined t o particles in the size range between about 5 X and 20 p. Particles in this size range a t relative humidities below water saturation are defined here as natural aerosols. I n meteorology they are labeled condensation nuclei, or just nuclei. This designation is fundamentally correct, since these particles are all capable of acting as condensation nuclei for water vapor. However, usually in the atmosphere only a small fraction of these potential nuclei (preferably the largest) actually take part in the formation of droplets. Thus i t is preferable t o speak of aerosol particles generally and refer t o them specifically as condensation nuclei only in those cases ere tJheyreally form the starting point for condensation or sublimation Certain regions of the aerosol spectrum, which extends over lmost four orders of magnitude, are well known from the investigations of special branches of meteorology. We might draw an analogy here between the spectrum of the aerosol particles and that of the electromagnetic waves. Similar t o the specific and quite different properties of the various wavelength regions, distinct size classes of particles are responsible for certain physical processes in the atmosphere (Fig. 1).

?

4

CHRISTIAN 1. JUNGE

For example, particles smaller than 0.1 p play a significant role with respect to the electrical conductivity of the air and thus the potential gradient. When charged these particles are called large ions. Particles in the range between about 0.08 p and 1.0 p are most important for light scattering. Beyond these limits the product of the particle concentration and the Mie scattering coefficient becomes so small under normal conditions in the atmosphere that their optical effect is practically zero. A third example is that of cloud formation. The main portion of true condensation nuclei, at least over land, consists of particles above approximately 0.1 p . Particles above about 5 p probably play an important role in the formation of rain. Particles below 0.1 1.1 may act as condensation nuclei only in areas with quite low aerosol concentrations, i.e., at high altitudes or over the oceans.

p radius FIa. 1. Nomenclature of natural aerosols and the importance of particle sizes for various fields of meteorology.

For convenient characterization, the over-all range of aerosol particles has been broken down into three parts, namely, the Aitken, the large, and the giant particles (see Fig. 1). Particles outside the range of the Aitken and the giant nuclei normally are not important carriers of chemical substance in the atmosphere. The small ions, which are very important for atmospheric electricity, have a different physical constitution than the Aitken nuclei and are separated from these by a distinct gap in the size distribution. They consist of about 10 to 30 molecules and have an average concentration of some hundreds per cm3. This is equivalent to a substance concentration of 10-7 ylmeter3, a value several orders of magnitude smaller than the corresponding value of all the other chemical components. On the other hand, particles larger than 20 p can remain airborne for only a limited time and their occurrence is therefore restricted to the vicinity of their source.


5

With rising humidity the aerosol particles gradually change, both in size and physical properties, into either cloud droplets, fog droplets or ice particles, and possibly even into raindrops. I n the process of growing, these particles may accumulate chemical materials in addition to the original condensation nuclei around which they formed, and their composition therefore differs from that of the aerosol particles. However, in view of the fact t ha t they cannot exist a t relative humidities below loo%, they will be treated in connection with precipitation chemistry. If not otherwise stated, we will use y/meter3 (i.e., grams per meter3) as the unit of concentration for both aerosols and gases. This amount of material is represented, e.g., by a water droplet of 0.12-mm diameter in a cubic meter of air. 2.2. Size Distribution of Natural Aerosols

For a better understanding of the results of the size-distribution measurements, we will briefly discuss the methods which are used for this investigation. The well-known Aitken nuclei counter, which operates on the principle of the Wilson cloud chamber, determines only the total number of particles. With proper adjustment of the expansion ratio (see, e.g., [l]) small ions are not counted. This total particle number is approximately identical t o that below 0.1 1.1 radius, since the number of the large and giant particles is comparatively low (hence the designation of the Aitken size range). Data on the size distribution below 0.1 p can be obtained by measuring the coefficient of diffusion due to Brownian motion [2], or the mobility of the charged particles in an electric field [3]. The latter method permits a quite accurate size resolution, which refers, of course, to thc charged Aitken nuclei (large ions) only. Findings indicate th a t the size distribution of large ions frequently resembles a “line spectrum”; i.e., the particles are concentrated around discrete sizes (Fig. 2). The reason for this phenomenon is unknown. To convert an “ion spectrum” into a n “aerosol spectrum,” it is necessary t o know the charge of the particles as well as the fraction of particles charged. Both quantities are functions of the particle size. These functions are approximately known for natural ionization conditions 141. Size determinations based on measurements of diffusion and ion mobility can be carried out only up to a radius of 0.08 to 0.1 p and are not, applicable for larger particles. I n fact no satisfactory method is available for the size range between 0.08 and 0.8 p. It is not difficult t o obtain somewhat representative samples of these particles with the Dessens spider thread method [ 5 ] , with an impactor [ 6 ] , or with thermal precipitation, the latter method being not very appropriate because of the low concen-

6

CHRISTIAN E. JUNGE

trations of aerosols in the atmosphere. The difficulty lies rather in the evaluation itself: the limit of measurement with the optical microscope is already reached a t 0.3 1.1. The application of the electron microscope, on the other hand, is unsatisfactory because the size and the shape of the particles change considerably with the evaporation of water and other volatile substances due to heating by the electron beam in the vacuum.

4o

t

p radiusFIG.2. Zon spectrum a t various places (solid columns) according to Israel and Schuls [3] in per cent of the total ion population. The total height of the columns indicates the calculated size distribution of the corresponding aerosol spectrum (ions plus uncharged particles) in arbitrary units.

The light microscope is an ideal instrument for evaluating giant nuclei, which can easily be collected on spider threads or with an impactor. For instance, Woodcock [7] collected giant particles on small glass plates, which he exposed to the wind or air stream from an aircraft. Also gravity deposits on horizontally exposed plates yield good samples of aerosol particles [7, 81. The advantage of this method lies essentially in the simplified evaluation. All particles appear in the samples in


7

about equal amounts due to the opposite trends exhibited by particle concentration and fall speed. We see from this brief survey that a determination of the entire size range requires the simultaneous use of several methods, but even then the range around 0.1 to 0.3 1-1 remains hard to cover. In view of the difficulties involved, very few measurements of the complete size distribution have been made t o date. On the other hand, extensive data have been obtained with single instruments, e.g., the Aitken nuclei counter, Owens dust counter, and other types of impactors. However, because these instruments cover only a narrow, poorly defined portion of the total size distribution, the measurements are only of relative value. As a consequence of the independent use of these various instruments, the view prevailed for a long while that the natural aerosols consisted of fairly independent groups of particles, e.g., the hygroscopic, in contrast to the dust particles counted by the impactor or the Owens “dust counter.” The realization that the natural aerosols form essentially a coherent size distribution is relatively new. A few words about the appropriate representation of the size distribution curves will be helpful here. A fundamental requirement of such representations, whether histogram or continuous distribution curve, is that the number of particles within a radius interval Ar or dr be represented not by the value of the abscissa, but by the surface area enclosed by the particular interval and the distribution curve. Unfortunately, this has not always been taken into account in the literature, making a comparison of values difficult or even impossible in some cases. Considering the wide range of particle sizes and concentrations, it seems advisable to use logarithmic scales. The log-radius distribution can thus be defined by (2.1)

n(r) = dN/d(log

r)

where N is the total concentration of aerosol particles (per cm3) from the lowest size limit up to size r. Therefore, dN = n(r)d(log r ) is the particle number per om3 found between the limits of the interval d(1og r ) . The log-radius distribution is not as important for atmospheric-chemical considerations as is the log-volume distribution dV/d(log r ) . Multir . r 3 we obtain plying the log-radius distribution by (2.2)

dV/d(log r)

=

?.$ ?r * r3 dN/d(log r )

Knowing the density, from the volume distribution we can immediately estimate the maximum amount of substance which may be present in any size range of the aerosol. This value is maximal because of the possible presence of other components in the particles.

8

CHRISTIAN E. JUNGE

104

103

102

f

10'

0 L Q)

0.

lo(

L

10-

lo-:

10-:

FIQ.3. Complete size distributions of natural aerosols, average da ta [4,81. Frankfurt/Main, curves 1, 2, and 5. Curve 1: ion counts converted to nuclei numbers. Curve 2: data from impactors. The point below 0.1 /I radius was obtained from the total Aitken-nuclei number under the assumption that the radius interval of the Aitken nuclei is A log r = 1.0. Curve 5: average sedimentation data over a period of I1 days. Curves 1, 2, and 5 are not simultaneous. Zugspitze, 3000 meters above m.s.1.: curves 3 and 4 correspond to curves 1 and 2 for Frankfurt and were obtained at approximately the same time. The figures in parentheses give the number of individual measurements. The dashed curves between 8 x 10-2 and 4 X 10-1 /I are interpolated.

Results of the measurements of the complete size distribution are shown in Fig. 3 for a densely populated area and for an elevation of 3000 meters [4,81. The curves were obtained by the combined use of ion counters, nuclei counters, impactors, and sedimentation chambers; they represent mean values from which the individual values can depart considerably. These curves exhibit the following main features: the lower


9

limit of the particle size, which may vary greatly from day to day, lies p and is separated by a break from the small ions. The a t about 4 X upper limit of 10 p for curve 2 is incorrect, because the observational procedure precluded detection of the few still larger particles. Curve 5 places the true value in the neighborhood of 20 1.1. The maximum of the logradius distribution lies between 0.01 and 0.1 p, usually a t 0.03 p , This has been repeatedly confirmed by Nolan and his co-workers [2] and also by other investigators. The most striking feature of the size distributions is their almost straight-line segment extending over two orders of magnitude from 0.1 to 10 p . It can be approximated by (2.3)

dN/d(log r )

=

const/ro

where p is equal to about 3. I n the log-volume representation this would correspond t o a constant value (Fig. 4).The interpretation important for atmospheric chemistry is that the large and giant particles represent approximately equal as well as the major amounts of substance, while the Aitken nuclei, despite their large number, are unimportant with respect to mass. The curve obtained on the Zugspitze shows that these features are valid for the 3000-meter level as well; however, the concentrations there are by one order of magnitude smaller and the upper radius limit is possibly somewhat lower. The contradiction between the continuous distribution curves in the Aitken nuclei range in Fig. 3 and the line spectrum in Fig. 2 is only apparent, being the result of a different method of representation. In Fig. 3 the ion numbers were evaluated from certain intervals of the Aitken particle range, so that the position and intensity of the ‘(lines” could not be detected. Moreover, the observational material seems to indicate that as the radius grows the line structure tends to disappear. It is hardly detectable in the region of the larger nuclei. The small number of direct measurements of the complete size distribution does not permit any conclusions about the general validity of these results. However, the optical behavior of atmospheric haze gives added confirmation that the exponential law is in any case valid within the optically effective p?rticle range between 0.08 to 0.8 p for a /3 approximately equal t o 3 [4]. Angstrom found that the pure haze extinction is almost inversely proportional to the wavelength of light. Theoretically it follows from our distribution law that with /3 = 3 the extinction is exactly inversely proportional to the wavelength. Numerous observers have confirmed h g s t r o m ’ s law [9] and their findings imply that the corresponding p values of the aerosol distribution must lie between 2.5 and 3.5. We might therefore infer that the aerosol distribution law for the

10

CHRISTIAN E. JUNGE

optically effective size range between 0.08 and 0.8 p has a wide validity. Observations of the scattered light around the sun lead t o a similar conclusion, as the detailed investigation by Volz [lo] has shown. All measurements of the entire size distribution as well as the optical observations cited above have thus far been made only over land. The

I0-

l o

FIG.4. Log-volume distribution curves for various aerosols [ll-131. Curve 3B is extended to a point determined by chemical analysis. The curve of Frankfurt is approximated by a “model” distribution, which is used for calculations in Fig. 5. (By courtesy of Tellus.)

conditions above the ocean appear to be fundamentally different [ll].Unfortunately, size-distribution measurements over the ocean have so far been made for giant particles only. Woodcock [la] has made the most extensive observations. The log-volume curves obtained by different observers above the ocean (Fig. 4) indicate that) with average wind forces, the giant nuclei exhibit nearly the same distribution as over land, i.e., approximately the same total concentration, identical upper limit, and a mean 0 of 3 (see also Fig. 10). However, all maritime curves show a defi-

11


nite decrease as the radius approaches 1 to 2 p , where the measuremeilts end. This is confirmed by data which have been obtained in chemical analyses and included in Fig. 4. The fundamental difference between continental and purely maritime aerosols is immediately evident. The sharp drop of the extrapolated curve reflects the small content of both the Aitken and the large particles above the ocean, a fact that was already well known. The extrapolated maritime distribution curve is valid for extremely pure sea air. A large variety of transitions between continental and maritime curves may therefore be expected over a wide area along the coast and seem to be implied in some observations obtained over the East Atlantic [13]. These general features in the size distribution of natural aerosols establish the framework within which the numerous individual observations of limited size ranges must be fitted. Table I contains a compilaTABLE I. Number concentration of Aitken particles per cm3 in different types of localities [I 1. Locality City Town Country (inland) Country (sea shore) Mountain 500-1000 meters 1000-2000 meters >2000 meters Islands Ocean

Number of Places Observations

Average

Average Max Min 379000 49100 1140OO 5900 66500 1050 33400 1560

28 15 25 21

2500 4700 3500 2700

147000 34300 9500 9500

13 25 16

870 1000 190 480 600

6000 2130 950 9200 940

7 21

36000 9830 5300 43600 4680

1390 450 160 460 840

tion of data on A i t k e n particles after Landsberg [l];it shows the considerable decrease in the concentrat,ions as one approaches the sea. In entirely undisturbed areas (i.e., in marine areas far from continents) particle numbers of only 100 to 200/cm3 have usually been observed. The particle numbers obtained by the various dust counters and impactors (conimeter) depend to a great degree on the lower precipitation limit (-0.1 p ) of these instruments, because of the steep slope of the size distribution curve in this range. Naturally, this limit fluctuates with instrument type and operating conditions; however, it generally lies around a few tenths of a micron. Moreover, when the particles are precipitated on normal glass surfaces or hygroscopic films rather than on hydrophobic surfaces, the true nature of atmospheric particles, as mixed particles, remains concealed; indeed, only a fraction of them may even

12

CHRISTIAN E. JUNOE

become visible. The numerous reported concentrations vary over a wide range between ten and a few hundreds of particles per cma (see, e.g., Effenberger [14]). Giant nuclei counts above the ocean 1121 lie between about 0.1 and l/cm3; similar concentrations in this size range were also found over land [8]. The question now arises as to the physical processes underlying the observed size distributions. There is a comparatively simple explanation for the lower and upper limits of the aerosol distributions. The lower limit results from coagulation (due to Brownian motion) of the small Aitken particles with larger particles. This process causes the number of the smallest particles to decrease very rapidly; it can be computed with good approximation 1151. Figure 5 shows how a given model size distribution, which follows closely the Frankfurt values (Fig. 4), changes with time. The decrease of the small Aitken nuclei number is rapid, compared t o meteorological processes, and results in a displacement of the size-distribution maximum toward larger particle radii. Because of the minute size of the small Aitken nuclei, this process brings about no noticeable change in the size and concentration of the particles larger than 0.1 p. From Fig. 5 it must be concluded that a decrease in the total particle concentration should be related to an increase in the average particle size. This can actually be verified statistically on the basis of numerous measurements made at various places throughout the world [4]. The corresponding representation of these results by a log-volume distribution (Fig. 5) shows that the decrease in the volume of the Aitken particles corresponds to an equal increase in volume above 0.1 p, which occurs preferably in the large-nuclei range, Hence, one should expect that the Aitken and the large nuclei have some chemical components in common, and that the individual particles consist of a mixture of various substances (mixed particles). The upper limit is established by the fall-out due t o gravity. Since the fall speed increases, according to Stokes’ law, with the square of the radius, we may expect this limit to be comparatively abrupt. The sources of aerosols are usually to be found right at the earth’s surface (above ocean and land), or a t any rate, quite near to it, as in the case of the various types of smoke and dust sources. The upper size limit is then determined by the equilibrium between the upward flux of particles (due to eddy diffusion) and the downward sedimentation flux. It can be computed under simplified conditions, as e.g., above the sea where the water surface is a uniform source of sea-salt particles [15]. If the eddy diffusion is represented by the wind speed a t a 5-meter height, the roughness parameter of the sea surface, and the constant austausch coefficient a t a relatively great height, the fraction E of the par-


13

ticles produced at the sea surface which can penetrate to various heights are given in Fig. 6 as a function of their radius. The variation of the individual parameters obviously exerts little influence. Above about 20 p radius, E drops rapidly to zero, indicating that this size is virtually the

FIG. 5. Calculated change in the size and volume distribution of natural aerosols, due to coagulation resulting from Brownian motion. The size distribution at the beginning (Oh) is the model distribution of Fig. 4. ( h = hours, d = days.)

upper limit for aerosols present for any considerable length of time in the atmosphere under normal conditions. The conditions over land are more complex, but estimations show that the same, or a somewhat higher upper limit, may be expected. Although it is possible to explain the limits of the size distribution on the basis of well-known physical laws, there is no satisfactory explana-

14

CHRISTIAN E. JUNGE

tion of the distribution law between these limits. As becomes evident from Figs. 5 and 6, the effects of coagulation and sedimentation do not extend far enough into the range between 0.08 and 10 p t o exert any great influence on shaping the size distribution in this range. Figure 4 seems to point to the nearly constant log-volume distribution as the basic phenomenon requiring clari$cation. As far as we can see, there are two processes that could be responsible. First, the large-scale mixing of innumerable small aerosol sources everywhere over the continents may result statistically RADl US I .o

I

I

I

I

I I

l0P l l l

1

I

I

I l

'OP l l l

I

I

I

I

10P

I , I ,

1

I

1

,,,,,

w 0.4 0.2

b =4crn

RADIUS-

b =4cm u5 = 5 rn/sec

FIG.6. Calculated concentration t of sea-spray particles as a function of the radius a t altitudes of 10, 50, and 250 meters if the concentration at the surface of the sea is 1.0. In the upper part of the figure the wind speed, u6,a t 5 meters altitude is varied, while the austausch coefficient for high altitudes ( A , ) and the roughness parameter of the sea surface ( b ) remain constant. I n the lower part of the figure A , and b are varied.

in a constant log-volume distribution. This could easily explain the relatively large deviations in time and place. Second, the formation, coagulation, and re-evaporation of cloud and raindrops in the atmosphere might exert an effect which tends to re-establish a constant log-volume distribution. Nevertheless, our present knowledge is still too inadequate for anything beyond a suggestion. 2.8. Physical Constitution of Natural Aerosol Particles

The physical properties of the particles are just as important relative to their role in air chemistry as is their size distribution. The particles in


15

natural aerosols can consist of solid material or droplets, or of a mixture of both. It is well known that a considerable portion of the natural aerosols are droplets of a solution of hygroscopic matter which grow with increasing humidity [16]. On the other hand, insoluble particles are very widespread, e.g., the soot particles in smoke, or the mineral dust over large land masses. A great variety of mixtures is encountered, depending upon the geographical location and the history of the air masses. The factors determining the physical characteristics of the particles are their content of soluble substances, the relative humidity at which these substances form a saturated solution (usually somewhat unclearly termed hygroscopicity) and the relative humidity of the air itself. The vapor pressure above a solution droplet is reduced by the dissolved substances and increased by the curvature of the surface. For particles with radii above 0.1 p, the latter effect is smaller than 1% relative humidity and may thus be disregarded here. Aerosol particles of salt solution then assume such a volume that the concentration of the solution is in a vaporpressure equilibrium with its surroundings. The corresponding growth of the particles with rising humidity depends only slightly on the temperature and the type of the dissolved substance. I n computing such growth curves one should use the measured data of the water-vapor partial pressure for different concentrations of solution, and not Raoult’s law, which holds true for dilute solutions only. Figure 7 gives some examples of growth curves for various substances. An NaCl crystal becomes a droplet at, or a little below, 75% relative humidity, which is the relative humidity of a saturated solution. However, crystallization with decreaszng humidity does not occur until 40 %50% relative humidity is reached, i.e., not until the solution becomes considerably supersaturated. The same holds true for other salts as well. The growth curves of mixed particles show all transitions between a pure droplet of solution and the straight line of a completely dry particle. The computed growth curves have by and large been confirmed by measurements [17]. The measured mean growth curves for natural aerosols over land [Fig. 7(d)] are not much different for giant and Aitken particles [17, 181. We see immediately that they agree well with those of mixed particles which contain a noticeable amount of insoluble substance. Growth curves of indiuidual giant aerosol particles show great variety, indicating fairly comulex chemical compositions 1171. The predominance of mixed particles over land is confirmed by direct observations. Figure 8 shows electron microscope pictures of large and giant particles, a considerable number of which were obviously droplets before they evaporated in the vacuum [19]. Impaction of particles on dif-

16

CHRISTIAN E. JUNGE

2

3

4

6

0

1

0

2

3

4

6 0 1 0 0

radius 4 FIG.7. Growth of aerosol particles with relative humidity. The curves are valid for the size range of the large particles, but are not very different for other sizes. With the logarithmic radius abscissa, the curves are almost independent of the absolute size of the particles. (a) NaC1. The solid curve is calculated. At 75% the crystal goes into solution, increasing the radius by a factor of about 2. The dashed lines represent observations made in the size range of the giant particles. With increasing humidity the crystal goes into solution somewhat earlier than the calculated value; with decreasing humidity a considerable salt supersaturation in the droplet delays the crystallization. (b) Calculated growth curves for various hygroscopic substances. 1 = HNOa, 2 = CaCL (the arrow indicates the expected crystallization point), 3 = H2S04. The differences are small. (0) Calculated growth curves for mixed particles composed of a solid spherical core (indicated by the hatched areas) and of a liquid layer of HtSOn of varying thickness (indicated by the length of the solid line). (d) Measured average growth curves of continental giant (1) and Aitken (2) particles. The hatched area indicates the size of the solid core, which agrees best with curves in (c).

ferent type surfaces has provided additional confirmation. Using alternately clean dry glass plates (which catch only droplets) and plates covered with a viscous film (which catch both droplets and dry particles) in an impactor has shown that below 70% relative humidity a noticeable fraction of the particles are dry, but that above 70% almost all of the particles have assumed the properties of a droplet 1171. The sea-salt nuclei in pure maritime air contain practically no insoIuble components. If the relative humidity rises to more than 70%, the crystals go into solution, as may be expected from their chemical composition [20]. Figure 7 shows that mixed nuclei and pure solution droplets grow with increasing rapidity and gradually become fog and cloud droplets when water saturation is approached. But even dry dust particles will be covered by several layers of water vapor below water saturation, due

.ITMOSPHERIC CHEMISTRY

17

to physical adsorptioii. In the presence of soluble gas traces the water coiitrwt of aemhol particles may be important for proiiiotiiig chemicd reactions betwer.11 such gas traces aiid the inaterial of the particles t heriiselves. S..4. NatiirP a d Origin of ,titlietL Particles

The history of twearvh iiito the nature of iiatural aerosols is esseiitially the history of the study of the Xitkeii nuclei, iiiitiated by the famous iiivestigatioiis of Aitken. These particles attracted particular attention because of the high values and pronounced fluctuations of their number (ionrelitrations wider various geographical and atmospheric conditions, hecause of their significance for air electricity, and last but not least, herause the iiuclei counter provided such a handy iiistrumeiit to count them. ,Is a result of these investigations the natural aerosol was for a loiig time considered to be identical with the Aitken iiuclei. The importance of the less numerous, but much bigger particles above 0.1 p for such fields as air chemistry, cloud physivs, and atmospheric optics

18

CHRIRTIIN E. J U N O E

was for a long time completely overlooked, or at, least underestimated. To a certain extent this resulted in a vonsiderahle delay in a proper approach t o the problems of the natural aerosol. ,111 this research into Aitken nuclei, however, was far more concerned with the origin and the geographical distribution rather than with the chemical composition of the particles, the importance of which was stressed almost exclusively by bioclimatologists, as indicated by the work of Cauer [2l, 221. Since the extremely small amounts of material represented by the Aitken nuclei excluded and still exclude today any direct analytical method, indirect methods mere tried to obtain information on the nature of these particles. I t was, for instance, attempted to determine to what degree the nuclei counter responded to soluble, hygroscopic substances only, and not to dry and insoluble ones. But this approach failed after it became evident that all substances can act as condensation nuclei provided that they are of the proper size. Another approach was to investigate the most efficient processes of Aitken-nuclei formation. A considerable amount of effort was devoted t o these studies, the results of which are summarized in excellent# monographs [l, 231. The large number of such processes can be divided into three major groups: (1) Condensation and sublimation of substaiices in rapidly cooled airgas mixtures. This group includes all smokes produced b y heat and combustion. Typical substances formed in large quantities in this way are ashes, soot, tar products, as well as sulphuric acid and sulfates in cases where the fuel contains sulphur. The variety of particles which can be formed in this way by industrial installations is, of course, enormous and of considerable importance for the production of aerosols in continental atmospheres. The sizes of these particles cover a wide range, but are primarily within that of the Aitken nuclei. ( 2 ) Reactions between trace gases through the action of heat, radiation, or humidity. Examples are the formation of NH&1 in the presence of NH, and HC1 vapors, the oxidation of SOt to SO3, and the oxidation processes yielding higher iiitrogeii oxides by the action of heat, ozone or short -wave radiation. ,111 these processes are efficient sources of Aitken nuclei. Because of the small size of the particles formed in most of these reactions, even large particle concentrations represent only minute fractions of the reacting gases. This is best illustrated by the careful investigatioii of SO? ouidation. Gerhard [24] found that SO2is directly oxidized to so3 by solar radiation at a rate of 0.1% to 0.2% per hour. This SO3 immediately forms droplets of H2S04in the presence of watcr vapor in normal air. An SO2 concentration of 10 y 'meter3, which can be considered average for coun-


19

try air, would yield about 0.03 y/meter3 H2SO4per hour. This value corresponds approximately to the following particle concentrations: 106/cm3 with a radius of 5 X p, 104/cm3with p, and 3 X 102/cm3with 3 X p. These concentrations are quite high, though we should bear in mind that they would actually be reduced considerably due to rapid coagulation with each other, or with larger particles already present. Though this photochemical process is apparently a very efficient Aitkennuclei producer, it is still of minor importance compared with the sulphate formation resulting from oxidation of SO2 in cloud and fog droplets (see Section 3.9). (3) Dispersion of material a t the earth’s surface, either a s sea spray over the oceans, or as mineral dust over the continents. I n contrast t o processes 1 and 2, which generally yield particles below 1 1.1 radius, the mechanical and chemical processes considered here usually produce much coarser particles, of which only those with radii below 10 to 20 p remain as aerosols for any appreciable length of time. While the sea-salt mechanism has been fairly well investigated, very little is known about the role of mineral dust and soil particles in the aerosols. This is partly due t o the fact that dust and soil particles consist for the most part of insoluble matter, which makes quantitative analyses in the microgram range very difficult. It becomes quite obvious from this brief survey that the numerous possible sources of natural aerosols do not permit us to draw any conclusions as t o the composition of aerosols in general and of the Aitken particles in particular. A third indirect method to obtain information on the composition of the Aitken nuclei was by statistical correlations between the concentrations of Aitken nuclei and the geographical location, season, wind direction, and general weather situation. The large number of observations have been carefully examined by Flohn, Burckhard, and Landsberg [ l , 251. All the interrelationships found thus far point t o continental areas with dense populations and much industry as the major sources of Aitken nuclei. The Aitken nuclei turn out to be a very sensitive index of air pollution, though they in no way represent an important pollub ant themselves. But again the results of these investigations fail to answer the question of their general composition. Despite the development of the electron microscope and of advanced microchemical techniques, this situation has not changed in recent years. It is true that some of these methods are well suited for the analysis of particles above 0.1 p radius; however, identification of particles below this size by examination of electron micrographs becomes nearly impossible. Moreover, the amount of material in the Aitken particle range is

20

CHRISTIAN E. JUNGE

so small and its precipitation and separation from the much larger amount of aerosol material above 0.1 p radius so difficult, that the chemical composition of the Aitken nuclei in our atmosphere is still an open question. Two facts may be of some help in this situation: the similarity in the respective growth curves of the Aitken and the giant particles in Fig. 7 and the mass exchange by coaguIation between the Aitken and the large particles according to Fig. 5. From these we may infer that there probably is not very much difference between the composition of particles with radii smaller and larger than 0.1 p. This conclusion is supported by the observation that a large number of aerosol sources produce particles over a wide size range from about 0.01 p up to 1.0 p . However, for particles above 0.1 p some results on their chemical composition are available, which will be discussed in the next sections.

2.6.Sea-Salt Aerosols Until recently it was generally assumed that the foam from breaking waves, as wind-carried spray, provided the main source of sea-salt nuclei.

I

,. 9

Water Surface

FIG.9. The formation of sea-salt particles from the bursting of bubbles. The large droplets ( W ) originate upon disintegration of the jet; they have been investigated thoroughly by Woodcock 1261. More numerous, though much smaller, particles (M)formed from the bursting bubble jilm have been assumed by Mason [27].

However, apart from those that originate in surf, such particles are not very numerous and are also too large to remain in the air for any considerable length of time. Woodcock and his co-workers were able to show that the numerous small air bubbles bursting in the foam of waves represent a much more effective source of sea-salt nuclei [26].Figure 9 shows schematically how the bursting of a bubble creates a small jet, which shoots high into the air and breaks up into about 5 t o 10 droplets of nearly the same size. The droplet size is about %,J that of the air bubbles,


21

so that the size distribution of the particles is determined t o a large extent by that of the bubbles. Woodcock suggests that the lower size limit of the bursting bubbles is determined by the solution of the smallest bubbles if the sea water is not saturated with atmospheric gases. So far it has not been determined whether this, or a purely mechanical process, limits the size of the droplets. After their formation the droplets of sea water shrink to about one-third their original size, due to evaporation, until they are in equilibrium with the humidity of the surrounding air. Besides the particles formed in the jet of bursting bubbles, Mason [27] suggests that about 100 much smaller particles are formed from the remnants of the thin liquid film which covers the bubble prior to breaking. Researchers who have supported the thesis that the sea-salt particles play a n important role not only with respect to the amount of substance but also in terms of particle number have always been disturbed by the fact that considerably more Aitken nuclei are found over land than above the ocean. They therefore welcomed observations which indicated tJhe possible subsequent occurrence of an increase in the sea-salt nuclei. For instance, some have claimed that when a sea-salt particle dries and crystallizes very rapidly, it shatters into several fragments; more recently it has been stated that the drying of sea-salt particles which were attached t o spider threads produced a large number of very small particles [28]. The first process was not verified by a careful check [29]; nor did unpublished investigations of the author substantiate the second process when spider threads were not used. Rau’s measurement [30] in Central Europe certainly prove that the number concentration of chloride particles is of no importance in the interior of a continent. It might very well be th a t a fraction of the chloride particles which Rau counted in the range of the large nuclei are still of continental origin (see Section 2.6). The question now arises a,s to whether the sea-salt particles produced in this manner havc and maintain the same composition as sea water. Our information is quite poor on this question. Neither Facy’s notion [31] that there is an increase of Mg in the particles produced by bursting of bubbles, nor Cauer’s view [21] that C1 is released from the sea-salt partoiclesthrough the action of ozone, are anything more than hypotheses. More impressive is the finding made by Swedish researchers and others [32] that in rain water the ratio of C1 to Na over land deviates considerably from that of sea water. But we will see in Section 4 that this is probably due t o additional Na and C1 sources over land rather than t o a decomposition. The few direct analyses of marine aerosols so far do not indicate a decomposition. Twomey [20] found that sea-spray particles undergo a phase change from crystal to droplet a t the expected relative humidity and concluded that their composition must be the same as sea

22

CHRISTIAN E. JUNGE

water. Junge [ l l ] analyzed sea-salt nuclei in Florida for C1, Na, and in some cases, also for Mg, and found agreement with the composition of sea water within the limits of accuracy of his analytical method. Very good agreement of the C1, Na, and Mg ratio with that of sea water was found in rain water by Gorham in North England [33]. Since the air masses which arrive there from the Atlantic are influenced very little by the European continent, these data are of special significance. It appears from these measurements that any decomposition of sea-salt particles, if it occurs a t all, can only take place very slowly. Additional studies along this line are necessary. Of basic importance for the role which the sea-salt particles play within the natural aerosols is their size distribution. Data above or near the ocean for different wind forces and at various heights have been obtained by Woodcock, Moore, Fournier D’Albe and Lodge [12, 34-36]. All these researchers were primarily interested in the giant nuclei because of their importance for cloud physics. Woodcock and Moore collected particles on small plates which they exposed to the wind or the air stream from an aircraft. Aerodynamic conditions restricted this collection to particles above 1 p radius. Fournier d’Albe used a cascade impactor, but counted only particles above 2 p radius. The particles counted in these investigations were assumed to consist of sea salt. The circumstances of the investigations as well as the physical behavior of the particles certainly justify this assumption. Lodge employed a microchemical method specific for chloride. The particles were collected on the surface of fine membrane filters and identified by their reaction with mercurous fluosilicate. The size of the resulting color spots can be calibrated in terms of the original particle size. The results of the various sea-salt-particle counts agree quite satisfactorily (see Fig. 4).Some of Woodcock’s measurements, which he made a t a height of about 600 meters and primarily in the region of the trade winds of Hawaii, are also plotted in Fig. 10. With increasing wind speed the increase in the concentration of the largest particles is somewhat more pronounced than that of the smaller ones; however, on the whole, the character of the size distribution changes only slightly. Unfortunately, a t present there are no measurements of marine aerosols below a radius of 1 to 2 p. Based on chemical analyses, Junge [37] found that in the pure sea air of Hawaii the C1 concentration of nuclei with radii between 0.08 to 0.8 p amounts to only 1.501, of that of the nuclei with radii between 0.8 to 8 p. This result was used in Fig. 4 and Fig. 10 for extrapolation of the curve for Beaufort 3. According to this tentative extrapolation, the most frequent size of the sea-salt particles should lie around a radius of about 0.5 p. This seems t o agree with Isono’s

23


L

=T I -B 0

1.0-2

10-1 1,oo 10' .lo2 PI radius at 993 relativle humidikv 10-1"

1043

10-10

10-7

I

Iwelghf of ska,sal/ inlpo,rtir+e gr FIG.10. Average size distribution of sea-salt particles according to Woodcock [12] for wind forces 1, 3, 5, and 7 Beaufort. The dotted line, a, gives the size distribution at Frankfurt (Fig. 3) for comparison. The dashed line, b, is the extrapolated size distribution of marine particles according to Fig. 4. The various scales a t the bottom allow a comparison of the different units used in the literature to indicate the size of sea-spray particles.

24

CHRISTIAN E. JUNGE

observations [38] that the residues of evaporated snow crystals contain many NaCl crystals of this size. The total number of sea-salt nuclei estimated from Fig. 10 is 2 per cms for Beaufort 3 and is not expected t o exceed 10 to 20 per cms a t higher wind speeds. The data on total sea-salt nuclei are compiled in Table 11. There is strong evidence that the majorTABLE 11. Data on the total concentration of sea-salt particles.

0b ser v er Woodcock

Fournier d’Albe

Moore

Rau

Junge Moore

Remarks

Total number concentration/cms

Beaufort 1 0.05 3 0.15 5 0.30 7 0.36 12 20 All particles larger than I p radius a t 80% 2 X 10-4 to 0.3 relative humidity in Pakistan a t various distances from the coast. The particles probably predominantly of marine origin. All particles larger than 2 p a t 80% relative humidity over the Atlantic a t 10.5 meter/sec wind 0 . 3 15 meter /sec wind 0 . 8 Maximum numbers during 0 months of Max. . . . . . . . . 18 observation in Central Europe of all 96% of all chloride particles larger than 0.25 p observations, . f 1 radius a t 80 %, relative humidity. Estimated total number of sea-salt nuclei 2-10 for average wind speeds (Fig, 10). Lowest Aitken counts over the Atlantic 77 during 3 weeks of observation Average 703 All particles larger than 1.5 p radius a t 80 % relative humidity for various wind speeds.

..

ity of the Aitken nuclei, even over the ocean where they reach their minimum concentrations, is not of maritime origin. This is confirmed by Moore [34] who found no relationship between the Aitken nuclei and the wind force or wave height over the Atlantic Ocean, quite in contrast to the giant particles. It is probable that these Aitken nuclei are remnants of continental aerosols. I n this connection, a few words should be said about the controversy between Wright [39] and Simpson [40] over the role of sea-salt nuclei with respect t o visibility in marine air. Simpson claims that the constancy of visibility with humidity below 70% precludes any change in particle size in the optically important size range. If the particles were composed of sea salt, the process of crystallization, even if distributed over the humid-


25

ity range between 40% and 70% due to supersaturation of the solution [Fig. 7(a)] would cause a significant variation in visibility. This variation, however, is not observed. It must therefore be concluded, in agreement with Simpson, that the major part of the large particles in Valentia even with west winds are not of maritime origin [41]. This appears t o be probable, because, according t o Fig. 10, the concentration of sea-salt particles below 1 p is small compared to that of continental aerosols and because continental aerosols penetrate far over the oceans as we shall see later.

W I N D FORCE

FIG.11. Sea-salt concentration over the ocean (Woodcock [12],Moore [34]) or a t the coast (Fournier d’Albe, cited in [34], Junge [49]) as a function of the wind force.

On the basis of the data in Fig. 10 we may infer a close relationship between the total content of sea salt in the air and the wind force. Woodcock [12] showed that all his measured values fall between the two dashed lines in Fig. 11. His observations were usually made slightly below the base of the clouds (at a height of approximately 600 meters) in the tradewind regions of Florida, Hawaii, and Southern Australia. We have supplemented his data with the values of Fournier d’Albe, Moore and Junge [42], obtained at sea level. Considering the different conditions under which these observations were obtained, the agreement is quite surprising and indicates a generally valid relationship. As this relationship permits quite reliable estimates of the amount of airborne sea salt under

26

CHRISTIAN E. JUNQE

varying conditions, an attempt will be made t o explain this phenomenon. To do this we must examine more closely the vertical distribution of the sea-salt nuclei above the ocean. Figure 12 represents some examples of the vertical distributions under trade-wind conditions, which Lodge [36] obtained near Puerto Rico. We see that the number of particles decreases exponentially with height, independent of particle size. The slope

2000

* .-c d 1000

E

lo-’

part i d e s per cm3

-

10-3

FIG. 12. Average vertical distribution (based on data from 5 flights) of sea-salt particles of various sizes on the windward sidc of Puerto Rico, according to Lodge [36]. These observations are compared with calculated distributions under the assumption of a production of 5 X loT3 cm-2 sec-1 particles at the surface of the sea, a n average austausch coefficient of 255 gm om-2 sec-1, and an inversion layer a t 2000 meters. At time 1 = 0 the particle concentration was assumed to be zero a t all altitudes.

of the curves varies with the individual measurenients and is sometimes only weakly indicated. Lodge’s measurements are the only ones available which cover the space below the cloud base (which is particularly important here). On the basis of the available observational material (including Figs. 10, 11, and 12) the following main features become evident for the height distribution of sea-salt nuclei and their dependence on the wind: (1) A fairly good relationship between wind force and total salt content in the mixing layer over various parts of the oceans. (2) A more or less well-pronounced exponential decrease with height of the particle numbers with varying slope. (3) Independence of the vertical distribution of particle sizes.


27

These features, established predominantly on data from regions of the trade winds, can be adequately explained by the following model. The surface of the ocean produces a constant amount of particles per unit surface and unit time for each wind force. If these particles are smaller than the critical fall-out radius, a large percentage of them penet.rates into higher layers. Given the vertical distribution of the eddy diffusion, the nuclei concentration can be computed as a function of height and time [42]. Such computations show that the time-dependent vertical particle distribution is little influenced by the assumed vertical distribution of eddy diffusion but depends almost entirely on the average austausch coefficient. The calculations in Fig. 12 are based on an eddy diffusion (represented by the austausch coefficient) which a t first increases linearly with height and later approaches a convection-controlled constant value, thus approximating the actual conditions quite closely. We see how the concentration increases rapidly at first and then more slowly with time and how the vertical gradient is approximately exponential, becoming increasingly steeper with time. Naturally, below the size limit of 10 to 20 p this process must be independent of the particle size. The calculation was based on the assumption that the trade-wind inversion, which stops further vertical mixing, was situated a t an altitude of 2000 meters. However, the result would not be essentially different without this assumption. After a certain initial time has elapsed, the concentration in Fig. 12 increases only slowly with time and height and then becomes essentially proportional to the wind-force-dependent particle production rate. Generally we should thus find a relationship between the salt content and the wind force as indicated in Fig. 11. Naturally, in the atmosphere there are further complicating factors such as time and space variations of the wind force, cloud formation, and precipitation. Hence, we can expect this model to conform only approximately to natural conditions. Observations show that sea-salt particles can penetrate far into the interior of the continents. On a flight over the USA at a height of 3000 meters, Seely and his colleagues [43], employing a microchemical method, counted chloride particles larger than about 10-13 gm (= 0.5 p radius at 80 % relative humidity). The highest number concentration of 0.46/cm3 was encountered in an air mass of marine origin. On a 600-mile flight over the southeastern corner of Australia at a height of 700 to 2700 meters, Twomey [44] counted hygroscopic salt particles (probably all sea-salt nuclei) which were larger than 10-lo gm (= 4 p at 80% relative humidity). He found little correlation between the particle number and the watervapor mixing ratio, the distance of the trajectory of the air mass over land, and the length of time since the air mass had crossed the coast.

28

CHRISTIAN E. JUNGE

However, his values show that convection over land rapidly works toward the creation of a uniform vertical distribution of sea-salt nuclei. The basic difference in the vertical distribution of sea-salt particles over land as a result of mixing is best demonstrated by Fig. 13. On several flights in Illinois and southward toward the Gulf of Mexico, Byers and his group [45] measured the size distribution of salt particles with Lodge’s microchemical filter methods. Figure 13 gives their mean concentrations for particles with a dry radius larger than 3 p. For comparison

FIG. 13. Vertical distribution of the number of sea-salt particles having a dry radius 23 p (equivalent to a radius of 7 p at 80% relative humidity and a mass of 5X gm). The curves marked W are measured by Woodcock [12] in regions of the trade winds; the portion of the curve beween the ocean surface and the measured values around 0.5 km is assumed. The values show the rapid decrease a t the level of the trade-wind inversion. The curves marked B are given by Byers [45]. B 1is a n average of three soundings made in Illinois. Bz are average concentrations for four overland flights southward from Chicago. The lengtth of the bar indicates the altitude range during each flight.

we have also plotted Woodcock’s figures for the corresponding particle size [12], which he obtained in the trade winds, and which are probably also representative of the Gulf of Mexico. Lodge’s absoEute numbers obtained in Puerto Rico are not reliable enough for comparison here. The main finding of Byers’ flights is that in general the particles are fairly uniformly distributed horizontally and vertically, with a mean value of approximately 300/meters. The sharp drop in the lowest 200 meters apparently caused by fall-out and impaction of particles on trees, etc., is an interesting phenomenon. It shows that the earth’s surface can act as


29

a sink for aerosols, a t least for giant particles, so that surface measurements must be interpreted with caution. The total number of sea-salt particles above 1 meter2 over land and ocean can be estimated on the basis of Fig. 13. From Woodcock’s data we obtain a value of 1.8 X lo8per meter2 and from Byers’ data, 2.3 X lo6 per meter2.’ The approximate agreement of the two figures indicates that the loss of sea-salt nuclei due to precipitation is small, even in air masses which are 1000 miles from the coast. We will return to this question in Section 4.4. A marked increase in sea-salt nuclei has been observed in the immediate vicinity of the coast, a phenomenon which can lead to chloride values of 50 to 1000/metera [46]. Due to fall-out, impaction, and vertical mixing the concentration of these local and shallow sea-spray clouds from surf areas rapidly decreases inland by approximately 1 to 2 orders of magnitude in about 10 km. 2.6. Continental Aerosols The invention of the electron microscope presented new possibilities for obtaining information on the nature of the natural aerosol particles. Such investigations were made primarily in Japan. Kuroiwa [47] obtained evidence on the origin of particles by observing the changes of their structure after repeated exposures to humidity; Yamamoto and Ohtake [48] obtained similar information from observing evaporation of nuclei substance in connection with an intensity increase of the electron beam; Isono [38] made electron diffraction diagrams of individual aerosol particles. These researchers investigated exclusively particles found in fog or cloud elements, i.e., true condensation nuclei. Among these particles, which were in the main larger than 0.1 p, they found many minerals from the soil, combustion products, as well as sea salt. Jacobi and Lippert [8] determined the presence of ammonium sulfate in continental aerosols by electron diffraction. The electron microscope, however, as an instrument for investigating aerosols has its limitations. The shape of the particles and their physical behavior under electron bombardment preclude any definite conclusions about their chemical composition. Even the electron diffraction method can detect, of course, only those components which are present in a crystallized form, but even for these no quantitative data can be obtained on the actual amount of material. On the basis of these shortcomings of electron-microscope techniques, it proved advisable to employ more direct microchemical methods. TWO 1 Here, we assume a uniform mixing over an altitude range of about N of the atmosphere.

30

CHRIBTIAN E. JUNQE

particles size ranges, 0.08 S r 4 0.8 1.1 and 0.8 5 T 5 8 p, which represent approximately large and giant particles, respectively, were separately collected in a cascade impactor [8]. I n accordance with Fig. 4 both size ranges should yield about an equal mass in continental aerosols. The aerosol samples were precipitated on Plexiglas plates, dissolved with a drop of distilled water, and then analyzed for NH4+, Na+, Mg++, Sod--, C1-, NO3-, and NOz- by spot test, turbidity, or in case of Na, by polarographic methods [49, 111. The selection of these chemical components for test was partly determined by the sensitivity of available methods, so as to make possible the detection and determination of a few tenths of a y of material with a reasonable accuracy of +20%. The first measurements were carried out in densely populated areas of West Germany. They confirmed the electron diffraction pattern with respect to the ammonium sulfate content [8]. In the large particles NH4 and SO4 were apparently essential constituents of the soluble material. The ratio of these components was approximately that of (NH4)zS04. On the other hand, in the giant-nuclei range only a small amount of NH4 was found and it must be assumed that a considerable fraction of the SO4 was bound to another cation. In Frankfurt no analysis was made for NO,, and the analyses for NO2 invariably gave negative results in both size ranges. The C1 analyses confirmed its presence in both size ranges. With advection of maritime air the C1 concentration increased in the giant particles and decreased in the large particles. This indicates that the seaspray component of the natural aerosol is limited to particles larger than 0.8 p, and that smaller C1 particles are of continental origin. These findings have been completely confirmed by measurements made at Round Hill on the east coast of the USA [49]. I n contrast to the site in Germany this coastal setting is entirely rural, although air masses in this area are to a certain degree still influenced by the industrial and more densely populated regions of the northeastern part of the United States. Analyses were made for NH4+, Na+, Mg++, SO4--, C1-, NO3-, and NOz-. Figures 14 and 15 give the daily values of the analyses. Here, NH4 and SO4 again predominate in the large nuclei and show about the same mass ratio as in Frankfurt. In Fig. 16 the NHa values are plotted against the SO4. It appears that they are present as (NH4),S04 and NH4HSOI (with the exception of one value in sea fog). Here, C1 and Na were found almost exclusively in the giant nuclei, even when the wind came directly from the ocean, which was not more than 100 meters away. The large nuclei contained C1 and Na in amounts above the limit of detection in only a few cases. This shows that the sea-salt component was quantita-

-

m

x

v)

IY I* 0

f

J

3

?

P

'+,

E

n

C w / r e +01

-5 W

8 3 J I I#

8

I*

c 0

9

c

-

t


t n * N

E

n

ii

6 0

L

3 : J o

o

$ 0 " Y

a 0

2 0

-

N 0 0 0

0

o

0

i

0

n

2

+

o

-

-€-

o

o

0

.

0

2 s O

0

L

t

0

n

0

0

n

.

0

-

0

0

0

-

zs oo + o Q :

t n * N

O

I

t O I

GIbNT NUCLEI 0 8 C -8”

6 4

2

6 4

2 n

E

>

12

00

‘0

I0

z

X

8

M

6

aZ

4

z

2

4

2

4 2

t

SEA FOG

JUNE

t

t

SEA FOG SEAFOG

JUNE

t

JULY

SEA FOG

FIQ.15. The same as Fig. 14, but for giant particles. (By courtesy of the Journal of Meteorologg.)


33

I

I

I

I

I1

0

E

‘ f a

0

.I ’ In E

5

4

3

2

I

NH.‘

10-8~r/m’

FIQ.16. Relation between SO1-- and NH4+ content for large particles a t Round Hill. Each set of values is represented by a line indicating the limits of accuracy. The line corresponds to (NH4)zSOn. (By courtesy of the J O U T n d of Meteorology.)

tively confined t o giant nuclei, a fact that has been confirmed in all subsequent measurements. Again, no NO2 was found; however, the results of the NO3 analyses were surprising. Similarly t o C1, NO3 was limited almost exclusively to giant nuclei and a relationship between the C1 and NO3 contents seemed indicated. These findings at Round Hill were later confirmed in the trade winds

34

CHRISTIAN E. JUNGE

of southeastern Florida and also in Hawaii [ll].The data obtained in Florida are plotted in Fig. 17, together with the values of gas traces, which will be discussed later. With the exception of sea salt, all components show essentially smaller concentrations. We note that, when obtained under like geographical conditions, the C1 and Na analyses agree well with those of Woodcock and others (Fig. 11). All results for large and giant particles are summarized in Figs. 18 and 19 and are arranged by sampling sites in order of increasing maritime influence with arbitrary equal spacing. The figures contain mean concentrations and, in the case of the maritime sites with their low values, also the average detection limits of the various components. For the sake of clarity, the difference between the two is indicated by hatching. An NH4 value for large nuclei, obtained above the trade-wind inversion in Hawaii (Mauna Kea 3200 meter), is included. Figure 18 shows a decrease of all values with increasing maritime influence. Here, SO4 and NH4 follow nearly a parallel course up to the Florida notation; from there on the determination of SO4 was no longer possible, due to the comparatively high detection limit. The line “b-b” represents the sulfate content, which corresponds to the observed NH4 content and the formula (NN4)zS04.The actual SO4 values are a little higher; however, the major portions of NH4 and SO4 are reciprocally bound. Very likely a corresponding SO4 content would have been found on Hawaii and on Mauna Kea also, had a more sensitive method been employed. Chlorine decreases up to the Florida line and then starts to rise. The line “u-a” corresponds to 1.5% of the C1 content of the giant nuclei of Fig. 19 and thus gives the average maritime C1 component in the large nuclei range. The difference between the actual C1 values and the line “ a-u” indicates the continental C1 component in the large-particle size range. The nitrate content of the large particles is small and there are no figures available for Frankfurt. The giant-nuclei composition curves shown in Fig. 19 present a similar picture for NH4 and sod, the only difference being that the SO4 surplus is essentially more marked at all sites. I n Hawaii the SO4 content increases again due to its presence in sea water. This maritime SO, content is computed on the basis of the C1 values and represented by line “a-a.” The trend of the C1 values is completely different from that of all the other components, indicating at once its marine origin. The NOs exhibits a surprising feature: the results from Round Hill point to a marine origin, but the curve in Fig. 19 does not. Summarizing, we can state that the sea-spray component can clearly be distinguished from that of continental origin, and that the injluence

'C

It

LARGE PARTICLES

GIANT PARTICLES

GAS

10

LARGE PARTICLES I

GIANT PARTICLES I

IC

LARGE PARTICLES

.

n

€

5

GIANT PARTICLES

k .

ul

t 3 z

w 2 4

0

1

cn

GAS

5

I LARGE PARTICLES

I GIANT PARTICLES I

10

DUST

RAIN

rzm==--

JULY I AUG FIG.17. The chemical composition of large and giant particles in Florida [ll]and the concentration of the gases NH,, NOZ,Clt, (?) and SOZ, measured simultaneously and plotted as NH,, NOI, C1, and SO1 for better comparison. 8 and L indicate sea and land breeze, respectively. Note the different scales for gases and aerosols! (By courtesy of Tellus.)

35

36

CHRISTIAN E. JUNOE

of the latter can be traced as f a r as the middle of large oceans. This is the

counterpart to the penetration of the sea-salt particles into the interior of the continents. On Mauna Kea, above the trade-wind inversion, ammonia was detected in the large particles, but no C1. This might well indicate a continental rather than a marine history for these air masses.

n

E

\

h

0.01

CONTINENTAL

c-----)MARITIME

sites are arranged with respect to their continental and maritime character. The shaded areas indicate the difference between tho actual values and the corresponding limits of detection, Lines "a-a" and "b-b" are explained in the text. (By courtesy of Tellus.)

However, more observations are necessary to confirm this interesting observation. Figures 18 and 19 give a clear answer to the old controversy over the role of continental and marine aerosols. It is emphasized that the components quantitatively determined may not represent the total soluble substance of the aerosols, For instance, rain-water analyses show that con-

37


siderable amounts of Ca are encountered over land, and that additional amounts of Na, C1, and Mg also occur. However, NH4 and SO4 certainly appear among the prevailing components in the large-particle range. As for NO3,its continental origin is not as evident as, e.g., for NH, and Sod. Additional investigations in the vicinity of Boston show that during advection of air from the interior of the continent the NO8 content

B

GIANT PARTICLES

SO;-

+

0 NU4

A

-!-I

O.O1

t

I

t

WINTER SUMMER FRANKFURT

t

ROUNDHILL

CO NTINE NTALC------,

NO; I

CL-

t

FLORIDA

,-T

t

HAWAII

MARITIME

FIQ.19. The same as Fig. 18, but for giant particles. (By courtesy of Tellus.)

decreases to values comparable to those in Fig. 19 for Hawaii; and that with the influx of marine air the NO3values become comparable to those from Round Hill. Furthermore, parallel measurements of the NO3 content in the center of Boston and in the suburban area show that urban atmospheres are not an important source of NO3. We can summarize the facts on the NOS content of the aerosols as follows: (1) With a few exceptions in or near Boston, NO3 was always confined t o the giant nuclei.

38

CHRISTIAN E. JUNGE

(2) The smallest concentrations were found at the center of the ocean and in pure continental air masses. (3) The largest concentrations were found in marine air masses on the northeast coast of the USA. (4) The increase of the NO3 content in the polluted atmosphere of Boston compared to the level of the unpolluted atmosphere in its suburbs was much less than, say, that of NH4. These observations suggest that NOs possibly forms in sea-salt particles in coastal areas when they come in contact with the higher NO2 concentrations of continental air. Woodcock has personally suggested to the author that nitrate may be formed in sea-spray droplets by decomposition of marine micro-organisms. At certain seasons the marine areas along the east coast of the USA are rich in these micro-organisms. However, the available observational material is still too sparse for such speculations; besides, the rain-water analyses, which we discuss below, present a different picture. A comprehensive study of the composition of aerosols in polluted atmospheres was recently undertaken by the U.S. Public Health Service [50]. This study is important for the problems under discussion, since a considerable fraction of the continental aerosols come from these sources. Under this program, in thirty cities for a little more than a year, aerosol particles larger than about 0.3 p were collected on glass filters. The numerous chemical components determined were selected with regard for their importance to air hygiene. Tables I11 and IV give the average values for medium-sized cities and nonurban areas. These nonurban areas are mostly suburbs and, therefore, may not be quite representative of the undisturbed background of country air. The only components which were also investigated by Junge are SO4 and NO3. The reported concentrations agree satisfactorily with the corresponding values of Figs. 18 and 19. So far as can be concluded from the few NO3values, the cities are less efficient sources for NO3 than for so4 and NH3, a conclusion which is in agreement with the observations in Boston. There are two interesting exceptions with respect to the NO3 content in the Public Health network; the high values in San Francisco (3.4?/meter3) and in Los Angeles (14.4 ?/meter3). Both are coastal cities and it is very tempting to regard these data as confirmation of the abovementioned conditions for the formation of nitrate in areas near the coast. The other components show the strong influence of certain industries, (e.g., the relative high F1 values) and of automobile traffic (Pb). The high percentage of acetone-soluble organic components shows that the

39


TABLE111. Particulate analyses from cities having populations between 500,000 and 2,000,000. (Average values in ?/meters.) Cineinnati Total load Acetone soluble Fe Pb

F-

176

PortKansas land City (Oregon) Atlanta 146

143

137

Mn

cu V Ti Sn As Be

sod-N01-

129

24.2 18.4 S9.1 6.1 4.1 3.3 1.2 1.o 1.8 0.01 Nil 0.05 0.23 0.12 0.94 0.08 0.18 0.04 0.05 0.01 0.09 0.002 0.009 0.024 0.06 0.21 0.24 0.12 0.0s 0.0s 0.01 0.0s 0.02 0.09 0.09 0 ) , the magnetic field tends to decelerate the forward motion of the particle. But as we have shown in equation (2.2), the magnetic field alone cannot change the total speed of a particle. Hence, it is clear that the loss of velocity along the field must reappear as an increase in the absolute value of the v+ component. We shall investigate this point further. It is convenient to write the equation of motion (2.16) in terms of the magnetic moment of the particle, p, which is defined as the product of the current produced by the particle times the area encircled by this current. Thus (2.17) where the second equality follows directly from equation (2.3). Multiplying the equation of motion (2.16) by v2 and substituting from equation (2.17), we obtain

p

(2.18) Here d/dt indicates the substantial derivative, which is taken along the path of the particle. (For a stationary observer, we would have aB/at = 0.) Another relation between v, and p (or vd) can be found from energy considerations. Since the total kinetic and potential energy of the system is a constant, (2.19)

=

qE,

dz

=

qE,v,

Using equation (2.17) we then obtain (2.20)

z”(’ t z mvZ2)= qE,v2 - d

(pB)

A comparison of equations (2.18) and (2.20) illustrates that, in the limit of our approximation of a slowly converging field, p is a constant. Let x be the angle between the total velocity vector v and the magnetic field. Then v+ = v sin x. Further, let E = Xrn(uz2 v + ~ )= %mu2, the kinetic energy of the particle.

+

122

JOSEPH W. CHAMBERLAIN

The constancy of the magnetic moment may then be expressed as (2.21)

If there is no electric field acting on the particle, so that B = constant, and “initially” the field and angle of pitch are B1 and X I , respectively, then the particle will be magnetically reflected when the field seen by the particle increases to the value B - 1 (2.22) B1 sin2xl At this point all the kinetic energy has been transformed into the gyration of the particle. But it is clear from equation (2.16) that so long as aB/& > 0, there will be a force on the particle in the -2 direction so that the particle recedes, gaining speed parallel to the field as 1 0 + 1 decreases. If there is an electric field involved (which is possibly the case for extraterrestrial auroral particles accelerated into the atmosphere) the last equation in the set (2.21) should be used, rather than the more familiar relation (2.22). Again we must caution that these relations are not exact and do not strictly apply, for example, to the motion of particles over large distances in the field of a dipole. Nevertheless, Alfv6n [27] has successfully applied equation (2.21), along with equation (2.1 1) for the perpendicular drift of the guiding center, to an approximate treatment of the Stormer problem-the motion of a charged particle in the field of a dipole. Also in Section 8.2.3 we shall use equation (2.21) for an approximate calculation on the angular distribution of protons to be expected in the production of a quiet-arc aurora.

2.2. Motion of a Particle in the Field of a Monopole The solution of the problem of a charged particle moving in the field of a single magnetic pole was obtained by Poincar6 [Ill, soon after Birkeland announced his first experiments. Birkeland had suggested auroral particles from the sun were pulled toward the earth by the dipole magnetic field. The monopole problem is, of course, unrealistic, in that single poles are merely a mathematical construction. It was felt, however, that the solution to this problem would give a qualitative description of the auroral trajectories, as the geomagnetic field near one of the poles crudely approximates the monopole field. Since the speed of a particle in a magnetic field remains constant, it is clear that a particle is not strictly “pulled” toward the pole. The trajectory will, however, circle about a line of force and if there is a

THEORIES OF THE AURORA

123

component of motion toward the pole, the guiding center of the orbit will move straight toward the pole along a line of force. There is no drift arising from the inhomogeneity of the field, since the lines of force are straight [cf. equation (2.11)]. In the case of a dipole field, the motion is not so simple; and, indeed, unless a particle coming from an infinite distance is initially on the dipole axis, it must cut across closed lines of force in order to penetrate close to the dipole. Thus the dipole field can be a barrier as well as a trap, depending on the initial conditions for any particular particle. So in this sense, also, the monopole problem is not too descriptive of the dipole phenomena. We shall work through its solution, however, as an introduction to Stormer’s theory, since the problem does illustrate fundamental concepts and is rather basic to an understanding of the more difficult dipole problem. The monopole field follows an inverse square law: B = Zmr/ra, where ill?is the strength of the pole. The lines of force thus radiate in all directions. For zero electric field, the equation of motion (2.1) for a positively charged ion is then dv - = - -qZmvXr (2.23) dt m r3

It is convenient to use ds, the element of length along the trajectory, as the independent variable in place of dt. Since v = ds/dt, we have dv/dt = vdv/ds; and if we choose our unit of length as (q%Jl/mv) cm, equation (2.23) becomes d v-- -v X r ds r3 This unit of length, rather than centimeters, will be used throughout the remainder of this section; since v is constant, the unit of length is a constant for any one particle. Further, we have v = dr/dt = v drlds, and

-=-+ dv vdzr ds

as2

(:;)- :; -

The last term vanishes, however, as the scalar speed remains constant. Thus the equation of motion is written finally as (2.25)

If we take the vector product of r times equation (2.25) and expand the resulting triple vector product on the right, we find (2.26)

124


In deriving equation (2.26) we have used the identity r * dr/ds We may write equation (2.26) as (2.27)

L (r x ds

g)

=

=

r dr/ds.B

4 (I) ds r

which is easily proved by expanding the derivatives in equation (2.27).

This equation may be integrated directly: (2.28)

where h is an integration constant. If we take the scalar product of r times equation (2.28), the left side vanishes, and if we let a be the angle between h z and r we have h COB a = 1 (2.29) Since h is a constant, a is also a constant. Thus all the vectors r that are crossed by the trajectory make a constant angle a with the vector h; that is, the trajectory lies on the surface of a cone and h is along the axis of the cone or along the guiding center of the orbit; h is along the z axis in Fig. 2. To investigate further the nature of the orbit, let x be the angle between the unit vector dr/ds (along the path of the particle) and r. Then taking the absolute value of equation (2.28), we have, since the unit vector r/r is at a constant angle FIQ.2. The mag- with h, netic monopole prob- (2.30) r sin x = \he\ = tan a

lem. Lines of force lie along the vectors r. The orbit of a particle in the neighborhood of the pole is shown.

where he is the component of h perpendicular to the cone. If the cone is unrolled into a plane sheet, the trajectory as given by equation (2.30) is obviously a straight line; hence the orbit is a geodesic on the cone.' The distance of closest approach to the origin is r = tan a;after passing this point the particle again recedes to infinity. 'The reader should bear in mind that d r l d s indicates dlrllds and not Idrldsl. This point may be demonstrated somewhat more elegantly from equation (2.25). The acceleration given by the left term must be along the principal normal to the trajectory since D remains constant. From the right side it appears that this normal is also normal to the surface (r) on which the trajectory lies. By definition, the path is therefore a geodesic. 7

125

T H E O R I E S OF T H E AURORA

Since the magnetic field follows the inverse square law, equation (2.30) is equivalent to sin2 - x - constant (2.31) B This is the relation (2.21) governing magnetic reflection that was previously shown to hold approximately in the general case of a slowly converging field. Here we have demonstrated that for the particular case of a monopole field, the equation is exact. Figure 2 illustrates the trajectory near its closest approach to the origin. 2.3. Currents and Magnetic Fields

If the number of charged particles moving in a magnetic field is large, it may be necessary to consider the magnetic field arising from the current that is produced by these particles. In this case the equations governing the motion of a single particle in an external magnetic field no longer apply, and we must seek a more general treatment. We now make a distinction between the mean velocity of a set of positive ions and electrons and the current, which is a result of their differential velocity. The mean m a s s velocity is (2.32) where n is the number density of the particles and m is their mass; subscripts i and e refer to positive ions and electrons, respectively. Similarly, when the positive ions are singly charged the current density is

J

(2.33)

=

q(nivi - we)

Neglecting any external gravitational field and assuming that a scalar pressure p (rather than a stress tensor) is appropriate, we have for the equation of motion for singly charged positive ions (cf. Spitzer [26]) (2.34)

dv dt

nimi 2 = niq(E

+

Vi

X B ) - Vpi

+ 'pie

and for electrons, (2.35)

nCme5 dt

=

-n,q(E

4- V ,

X B) - v p s

+ '$hi

Here pie and ' p s i represent the rate of transfer of momentum to protons from electrons and vice versa.

126


The total derivative on the left sides of these equations is

av---av + v * v v at at

(2.36)

But in order to proceed with the analysis, we neglect the second order term in v and its derivatives in equation (2.36). Actually this term may be appreciable for solar ion streams, but the following equations may be applied where the stream is assumed to be in a steady state. Specifically, the gradient term vanishes if the velocities are axially symmetric (&/a+ = 0) and are constant along the length of the stream (av/az = 0 ) , and if the stream is not rapidly expanding or contracting (VR= 0). We shall neglect m,/mi compared with unity and assume electrical neutrality: Ini - n81/n, 10 amp. But if the cone is as wide as a = 12O, the current must be I 2 lo6 amp. All this assumes that mutual electrostatic repulsion of the ions is halted by an influx of interplanetary electrons (see Section 4.2.1). If the current is not strong enough to produce the pinch, the stream merely continues t o accelerate outward (in the R direction), since the magnetic field is not strong enough to overpower the pressure gradient in equation (2.37). But would this really make so much difference as far as an auroral theory is concerned? Perhaps yes, if the solar streams are initially ejected with a wide angle. But in Section 4.3 we shall review Ferraro’s criticism, which maintains that currents of sufficient intensity to focus wide-angle streams cannot exist. 42.4. The Radial Electric Field. To illustrate how the pinch effect operates in constricting fast protons and slow electrons, we shall derive a steady-state expression for ER, the radial electric field that would be necessary t o balance exactly the Lorentz and diffusive forces of the stream. (It should not be concluded th at in a real stream this equilibrium would be achieved; the problem considered here is merely illustrative.)

154

J O S E P H W. CHAMBERLAIN

From equation (2.40), for an infinite conductivity (q

=

O), we have

(4.8)

Since the charge separation must be small, we may substitute in the diffusion term the approximation 1 ap, = -1-ap -(4.9) piaR pan

+

where p is the total pressure ( = p , pi). Also from equation (2.39) we have _ -- -J,B4 (4.10)

zL

Using equations (4.9) and (4.10) to eliminate the pressure gradient from equation (4.8), we find for the electric field measured by a “stationary” observer (4.11)

According to the Lorents transformation (2.6), the apparent field ER’ seen by an observer moving with velocity w Zis (4.12)

ER’ = ER - wZB4

Hence an observer moving with velocity W, = ER/B+will detect no electric field (ER‘ = 0). And a particle traveling with velocity (4.13)

will be affected by an inwardly directed field ( E R ’ < 0). I n the simple case where p , = pi (or T, = Ti)this condition for ER’ < Ois

+

(4.14) wz > v z - S ( V-~v e ) = % ( ~ iv,) where we have used equation (2.33) to eliminate J, and have set vi = v,, the mean mass velocity, according to equation (2.32). The interpretation of equation (4.14) is that a particle traveling with a velocity greater than the mean of the ionic and electronic velocities will see an electric field directed toward the axis. For slower particles the field is directed outward. Hence our previous statement that fast protons and slow electrons are constrained toward the axis. In the above derivation all the simplifying assumptions have been introduced for the purpose of clarifying the basic physics involved. I n a real stream the apparent field seen by a particle is likely to be more complicated than is given in equation (4.14).

155


4.8. Ferraro’s Criticism Ferraro [43] discussed the possibility of the emission of electronic currents from the sun. Recently [44] he has republished these same arguments to call attention to difficulties inherent in such modifications t o Stormer’s theory as have been proposed by Vegard and by Bennett and Hulburt. If a current flows along the z axis it will have a magnetic field associated with it, according to AmpBre’s law (2.41), with the lines of force being circles with R = constant. And if the current is increased or decreased there will be a corresponding change in the magnetic field. But this changing magnetic field produces an electric field by Faraday’s law (2.42), which tends to oppose the original current. Ferraro concludes that this induced field is so strong as to make it unlikely that there could be appreciable change in the current. We shall discuss this further after presenting the mathematics. The discussion presented here is a slight modification of that given by Ferraro. The magnetic vector potential A is related to the field by i

B = V X A

(4.15)

Faraday’s law (2.42) may then be written (4.16)

where @ is the electrostatic potential. We shall neglect V@, which is equivalent to postulating no net separation of charge. At any instant the relation between the current and its magnetic field is given by equation (2.41), which may, in terms of A, be written (4.17)

V X (V X A )

=

-V2A

=

4aJ

where the first equality is true since V * A = 0 when there is no space charge. The general solution to equation (4.17) is

A

(4.18)

=

/’& r

where r is the distance from the point where the potential is being evaluated to the volume element dr, and where the integration is over the volume of the stream. Except near the ends of the stream, we obtain approximately (-1.19)

A

=

2nR12JIn

L RI

156


where we take J to be uniform across a cylindrical stream which has a radius R1 and a length L. Near one end of the stream the value of A drops to approximately one-half the value of equation (4.19). Putting equation (4.19) into equation (4.16) we may compute the electric field set up by a change in the current density: (4.20)

L aJ E = -2?rRI2 In R~ a t

We may consider d J l d t to be composed of two parts: The primary change, a1J/6t, arises from some unspecified cause, such as differential slowing of electrons and protons by collisions. The secondary change, 62J/6t arises because of the induced field E. To find &J/St we appeal t o the generalized Ohm’s law (2.38). For negligible resistivity ( r ] = 0)’ a constant pressure in the direction of the current flow, and negligible net motions perpendicular to &J/&t, we have (4.21)

Combining (4.20) and (4.21), we find (4.22)

where (4.23)

neq2 me

I, RI

A = 27rR12__ ln-

Equation (4.22) may be written (4.24)

For A >> 1 we see that the induced current is practically equal and opposite to the primary component of the current; in this case the net change in current density will be very small, because of the large self-inductance of the stream. Alternatively, we may express equation (4.22) in the form (4.25)

which gives the final current in terms of the initial displacement of charges. Suppose that initially the electrons and protons have equa.1 velocities: v(0) = v i ( 0 ) = ~ ~ ( 0In ) . accordance with Bennett and Hulburt’s assumptions, suppose further that the primary change in current results from the electrons being completely stopped by collisions but the protons suffer no change in velocity. The electrons are then re-accelerated


157

by the induced electric field and from equation (4.25) we may obtain the final net difference in proton and electron velocities: (4.26) (We neglect the presence of coronal or interplanetary electrons, which might be accelerated by the electric field at the expense of some of the original stream electrons.) Rough estimates suggest that A is indeed much greater than unity (-loll) and therefore the differential velocities of protons and electrons must remain a small fraction of the mean velocity. Ferraro makes an estimate of the maximum total current likely to flow in the solar stream, with differential velocities given by equation (4.26). Using equation (2.33) we have (4.27) Inserting the expression for A given by equation (4.23), with the logarithmic factor set equal to unity, we find approximately (4.28) For ~ ( 0 = ) lo9 cm/sec, which is probably an upper limit for auroral ion streams, I = 30 emu = 300 amp, which would have negligible importance as a current interacting with the geomagnetic field. On the basis of Bennett's calculations with similar parameters, it appears that a stream with this current would have to be ejected with a half-angle of less than one degree for the pinch effect to be appreciable. But with such narrow cones, magnetic constriction is not necessary to make the beams sufficiently small to satisfy the various criteria listed in Section 4.2.1. However, Ferraro's criticism as outlined above is not strictly valid when the stream moves through an ionized medium. First of all, when the electrons tend to lag behind, the induction process will act on all electrons in the region, not just on the original stream electrons. Even if the total current density J remained negligible in the stream, as in equation (4.25), Ferraro's conclusion that the protons in the stream have the same velocity as the electrons is not necessarily true. Also, Bennett, in a paper now in press, justly points out that the vector potential A immediately outside the stream decreases very gradually (as log LIR). Hence, the induced return current is not confined to the outward moving stream, but is distributed over a much larger volume of space, The solar stream itself may thus possess a finite current density,

158


although the total net current vanishes when integrated over a very large surface. An ionized interplanetary medium from the sun to the earth requires a complete revision of several concepts and ideas that have been cherished for many years. Needless to add, however, it has not yet been convincingly demonstrated that sun-earth currents do exist or that they could cause aurorae if they did. 5. THE CHAPMAN-FERRARO STREAM AND RING CURRENT

Chapman and Ferraro have adopted, in their theories of a neutral, ionized stream and a ring current, certain simplifying assumptions. But then, like Stormer, they have attempted to develop their theory along precise analytical grounds. I n order to accomplish this mathematical development, it was necessary occasionally to make assumptions of a physical nature that depart significantly from real conditions. They have maintained, however, that the precise treatment of certain idealized problems would give added insight to the real problem. Hence an extrapolation of the Chapman-Ferraro theory to an explanation of aurora, for example, involves considerable speculation and cannot be said to rest on a very firm foundation. Nevertheless, the mathematics has suggested one or two ways in which aurorae could be produced by solar particles, and it is not likely that these ideas would have originated from purely speculative and qualitative reasoning alone. The division of the theory of Chapman and Ferraro into two parts, the solar stream and the ring current, is a natural one for several reasons. According to their ideas, the stream itself is responsible for the initial phase of a magnetic storm, whereas the ring current is associated with main phase. Also the theoretical developments are quite distinct: Both the ring current and the stream are hypothesized to exist; it has not been demonstrated, for example, how the ring would be formed from the stream. And, finally, it seems that an aurora might appear either directly from the stream particles or through the intermediary of the ring, but these two methods of auroral formation are quite distinct. 6.1. Theory of a Neutral, Ionized Stream

The basic assumptions of Chapman’and Ferraro’8[45-50b]-theory’ofa solar corpuscular stream are that the stream is composed of an equal number of electrons and singly charged, positive ions, all of which are moving with the same velocity toward the earth. Before arriving in the neighborhood of the earth, the stream is unaffected by any magnetic or electric fields and moves through a vacuum. We may wish to question the validity of some of these basic assump-


159

tions. As we pointed out at the end of Section 4.3, the assumption of equal electron and proton velocities is questionable, even if we accept the conditions of zero current and electrostatic neutrality. Further, the recent work of Behr, Siedentopf, and Elsasser [51-531 and of Blackwell [53a] on the zodiacal light indicates an electron density of the order of 600 cm-a at the earth’s distance from the sun. If this continually increases toward the corona, as the observations indicate, then the interplanetary density of ionized matter is probably considerably greater than that of the solar stream throughout its journey from the sun to the earth. Also the research of Storey [54] and others on radio whistlers seems to confirm the high electron density around the earth. Alfven [27] has also objected to the neglect of the general magnetic field of the sun, which he feels would have an important effect on the stream even at one astronomical unit. But we shall defer discussion of this matter to a later section. A somewhat different objection was first raised by Hoyle [55], who points out that a cloud of ionized material that breaks away from the sun would have to carry some of the solar magnetic field with it. The electrical conductivity is so high that the lines of force would move with the gas. The importance of these considerations cannot be decided until more is known about the sun’s field, both in the neighborhood of the region where matter is ejected and the general (dipole) magnetic field. More recently Hoyle [56] has suggested that the interstellar medium accompanied by an interstellar magnetic field of the order of gauss, penetrates the solar system. At a distance of about 10 earth radii, the terrestrial field is closed and beyond that distance the interstellar field prevails. These considerations could certainly make a profound difference in the behavior of a stream near the earth. Discussion of this matter will be deferred to Section 6.4. Various assumptions are made by Chapman and Ferraro on the form of the stream and in some cases simplifying assumptions are made for the magnetic field of the earth. We shall mention these approximations in the subsections below, as they differ for the different problems. Their several idealized problems proceed from simple to the more difficult; as we progress through these cases, different aspects of a gas stream are illustrated. At the end of each subsection we shall attempt to summarize the “moral” of that particular problem. Here we shall only sketch the fundamental aspects of the problems; but Chapman and Ferraro have in some cases carried the discussion of the pressures, temperatures, and other characteristics of the stream to an astounding degree of completeness. Additional investigations of the physics of an ionized stream have been published by Landseer-Jones [57, 611.

160


The basic idea of a neutral, ionized stream without a current was proposed first by Lindemann 1121, in a criticism of an earlier theory of Chapman’s (see Section 4.1). At first, Chapman [58] insisted that the conveyance of a current was an essential feature of the solar stream, but he has since rejected this idea and adopted Lindemann’s hypothesis. 5.1.1. Motion of a Plane Slab in a Uniform Magnetic Field. Chapman and Ferraro [45, p. 831 first consider an infinite slab with faces parallel (away from the to the 2, z plane. The slab is moving with velocity origin) and the field is in the + x direction. If the magnetic field were applied all of a sudden, the particles would be deflected in the direction pv X B; hence positive particles would go in the -y direction and electrons would go toward +y. This drift would continue until the electrostatic polarization field balanced the Lorentz force. From equation (2,1), when there is no acceleration, we have (5.1)

EU(O)= V,(O)&

inside the slab. On the periphery, however, where a charged layer exists, the net relativistic field will not be zero. If the thickness of a charged layer is d, then at a distance 8 d (where 0 < 8 < 1) measured from the inside edge of a charged layer, the electric field will be (5.2)

E ( e J = E(o)(1- 8)

where E(O)is given by equation (5.1). By equation (2.1) the equation of motion of a particle at position 8 d is then

From the discussion in Section (2.1). we see that the motion may be represented as a drift velocity (5.3)

and a circular velocity (5.4)

y’

= OV,(O)

The radius of the gyration would be (5.5)

And from the known expression for the field EU(O)inside an infinite condensor of surface charge uo, we find, for the thickness of the charged


161

layer (or displacement of the charges from their neutral position), (5.6)

These equations show the following properties of the stream: If the magnetic field is suddenly impressed on the stream so that all particles start moving in phase, the surface layers will pulsate with a frequency w/2a = qB,/2~mand with an amplitude depending on position and given by equation (5.5). There will also be a shear between layers of different 8. Outside the slab (0 = 1) the electric field and drift velocity are zero; inside the slab (e = 0) the electric field is balanced exactly by the Lorentz force and all the motion is a uniform drift. The importance of these pulsations in a real stream might be questioned. Putting numerical values in equations (5.5) and (5.6) (e.g., B, = 100 gammas, u , ( O ) = lo8 cm/sec, we) find p >> d. In . , n, = lo2 ~ m - ~ this case, any divergence of the stream 2 from plane-parallel motion as well as the random thermal motions would tend to smear out the systematic pulsations. These deviations from the ideal situation would cause a single particle to move through varying values of 8 ;its radius of gyration would change over a single revolution. At the surfaces of the slab there are currents resulting from the motion of charges of only FIG. 12. Cylindrical stream one sign and the magnetic field associated in a uniform magnetic field. with these currents has been neglected in The diagram shows the polarcomparison with the external field. ization of the stream and the Thus an accurate discussion of even a direction of the electric field outsimple problem becomes an exceedingly side the cylinder. complicated task, except when quite idealized conditions are postulated. 5.1.2. Cylindrical Stream in a Uniform Field. Consider the cylindrical stream of radius Rl illustrated in Fig. 12. The displacement is now twice as much as in equation (5.6) (for the plane slab) : (5.7)

where (TO is the surface charge per unit area perpendicular to the y axis. At a point on the periphery ( y = R1cos 4; z = R1sin b), the “algebraic” surface charge per unit area perpendicular to the radius vector is (5.8)

u = -uo cos

4

162


where the minus sign enters because negative charges tend toward the y direction. Outside the cylinder the field does not vanish, as in the case of the plane-parallel condenser, and the potential will be

+

=

- ~ T u ~ c ~ ( Rcos ~ ~4/ R )

The potential inside the cylinder is Gin = -2moc2R cos 4. It may readily be shown that the displacement quoted in equation (5.7) is consistent with this internal potential when the Lorentz force balances the electrostatic field. Outside the cylinder the field E = -V@ is (5.9)

and

(5.10) The component Ev is crossed with B, and merely gives rise to oscillations and shearing motions as in the plane slab. The component E, causes an acceleration of particles away from the plane z = 0, as shown in Fig. 12. Hence, in the general problem of an ionized stream in a magnetic field, the surface charges are not stable (as they are in the special case of a plane slab), but are accelerated away from the stream. The electrons escape with higher velocities than do the heavier particles, and the electric potential and resulting distribution of surface charge become horribly complicated. I n any event the field inside must remain E, = v , ( ~ ) B , . Chapman and Ferraro ([45], p. 91) estimated that protons accelerated by this mechanism along the lines of force might reach terminal velocities of the order of los cm/sec. However, precise calculations in the field of a dipole are rather difficult, so they were reluctant to claim too much importance for this mechanism of accelerating auroral particles until a more detailed analysis became possible. 5.13. Advance of a Stream into a Magnetic Field. To illustrate some of the phenomena accompanying the advance into a region of increasing field strength, Chapman and Ferraro [46] considered an infinite plane sheet moving perpendicular to itself and to the magnetic field. The sheet is assumed to be rigid (so that distortions in the surface that would be produced by the interaction of the sheet with the field can be neglected) and perfectly conducting. Let the ionized sheet be centered at the origin; the velocity of the sheet is +v, (toward the dipole), the lines of force in

163

THEORIES O F T H E AURORA

the equatorial plane extend toward +z, and the dipole moment M oextends toward - 2 (see Fig. 13). (Alternatively, we can think of the sheet at rest in the dipole moving toward it with velocity -uz.) This problem was considered by Maxwell [59] in terms of the magnetic scalar potential. We suppose that outside the sheet there is no ionized matter, so that V X B = 0. Then the scalar potential il can be defined for every point except in the sheet so that B = - V Q . Let Oo be the potential due to the permanent dipole with moment Mo. Maxwell showed that the field as modified by the induced currents in the sheet can be described by the superposition of the permanent-dipole field and an image-dipole field, but we must use different image dipoles for the field in front of and behind the sheet. 2

REGION II

t

REGION I

n, + n;

FIG.13. The magnetic field produced by an infinite sheet of ionized matter, which moves toward a permanent dipole, Mo.The induced dipoles are M Iand MI';actually MI'is coincident with MO, and is separated in the figure only for clarity. The scalar magnetic potential for the regions in front of and behind the sheet are given in terms of the potentials of the permanent and induced dipoles.

The situation is illustrated in Fig. 13. For region I (on the same side of the sheet as the permanent dipole) the image is MI, which is located behind the sheet. In region I the total scalar potential is Q O %. For region I1 the image dipole is M1', which is superimposed on Mo (in the figure they are separated slightly for clarity) and is in the opposite direction to Mo. Let the electrical resistance of the currents in the sheet be 2nb. Maxwell showed that -MI' = M i = Mo- 02 (5.11)

+

b

+

02.

For infinite conductivity we have b = 0; the field behind the sheet vanishes, whereas that in front of the sheet is increased in the equatorial

164


plane.130 I n the terminology ofihydromagnetics we may alternatively think of this change in the magnetic field as resulting from a compression of the lines of force by the ionized gas. For the ideal case of infinite conductivity, it may be shown that the lines of force move with the fluid and cannot therefore penetrate the sheet. The currents induced in the sheet are illustrated in Fig. 14. The current lines are intersections with the sheet of the equipotential surfaces of the dipole. These currents move in the eastward direction in the equatorial plane (counterclockwise as viewed from the north) , which, of course, is the proper direction to increase the magnetic field a t the earth. This increase of the terrestrial magnetic field has been identified by Chapman and Ferraro with the Jirst or initial phase of a magnetic storm. It will be noted in Fig. 14 that there are two foci where the currents vanish. These points are of some further interest in the following discussion of the deformation of the stream. Although we are considering a rigid sheet, let us examine the interaction of the current in this sheet with the magnetic field and the resulting retardation in the sheet’s advance toward the dipole. If the pressure (which would have a physically implausible discontinuity in the sheet) is neglected, the acceleration on the gas is, by equation (2.37) av, 1 (5.12) - = __ (J,B, - J,B,) 5 0 at nmi The deceleration vanishes a t the foci (where both the current is zero and the field lines are perpendicular to the sheet). Whereas an accurate discussion of the deformation of such a sheet would be exceedingly difficult and has not been done, a qualitative picture of the profile of the stream is shown in Fig. 15. The so-called horns of material that protrude toward the earth seem to be a possible route of entry into the atmosphere for auroral particles. Chapman [GO] has discarded this possibility, on the grounds that the particles are not likely to gain energy and speed and should therefore not have the penetrating power of auroral protons. To summarize, the rigid, plane sheet acquires a current system over its surface as it moves toward the dipole. These currents modify the magnetic field, and through interactions between the field and the currents, the stream is distorted. 185 I n some regions off the equatorial plane the field lines from the image dipole will be in the opposite direction to those from the permanent dipole. Hence the field is in some places decreased rather than increased. However, the total magnetic energy of the field in region I is increased. The deceleration of the stream is essentially a result of the conversion of kinetic energy into magnetic energy.

165


This analysis, however, postulates a scalar magnetic potential everywhere except in the current-bearing sheet itself. Suppose, as we have good reason to believe, that the interplanetary medium has a density of ionized matter that is a t least comparable to that in the stream. The N I

\

W

S FIG. 14. Currents induced in an infinite, plane, rigid sheet by the motion of the sheet into a dipole magnetic field. The points A and B are null points, where the current vanishes. The stream is viewed from the magnetic equator of the earth as the stream approaches the observer. After Chapman [50a]; courtesy Geophysics Research Directorate, A. F. Cambridge Research Center.

changing magnetic field will induce currents in this medium so that the condition V X B = 0 will no longer be strictly true and the scalar potential is no longer defined. Although this idealized problem may be descriptive of real streams to some extent, the adoption of such an approximation to discuss fine

166


details, such as the formation of horns and the penetration of auroral particles to the earth, would seem to require some further demonstration of applicability. 6.1.4. The Cylindrical-Sheet Problem. The shape of the stream of ionized material treated in this problem differs radically from any real solar stream, but the geometrical simplification is necessary if an analytical solution is to be obtained. Chapman and Ferraro [50] wished to consider in detail a current system, such as that described in the previous

OF f ’/,
a is

+

+

+

(3.11) Consequently, if a < b 6 A < B , then the ratio q of the total amount of meteoritic iron lying in the strata between depths of B and A , and b and a, respectively, is given by the formula

(3.12) On the basis of the observed penetration of iron meteorites for which the radius of the meteorite, or of its spherical-equivalent, was r cm, we are led to take d(r) = 5r. It is then found that the inner limits of integration of the double integrals in (3.12) have the values x / 6 , x / 4 , respectively. On computing the value of q with A = b = 25.4 and with values of No(r) consonant with observational data, the writer found (see [58], pp. 111112) that something of the order of 100,000 times as much sideritic material lies buried below maximum plow depth (25.4 cm) as occurs above this depth. Even though the per cent of the total number of meteorites found by the plowman is large, the actual number of meteorites found in this manner is surprisingly small if account is taken of the vast tracts of land which are under cultivation. Since the areal extent of systematic excavations carried to depths exceeding 25.4 cm is, and probably will long remain, vanishingly small in comparison with the extensive acreages plowed up, it becomes clear that there is most urgent need for instruments designed to detect the presence of deeply buried masses of meteoritic iron.

258

LINCOLN LAPAZ

3.4.3. Meteorite Detectors. An iron meteorite buried in the ground is a n example of a n object hidden in an opaque medium. Clearly, detection of such a n object is dependent on the condition that some of its physical properties differ from those of the medium in which it is embedded. In the case of a buried siderite, the magnetic permeability of this body is one such property. I n comparison with that of rock or soil, the magnetic permeability of a n iron meteorite is quite large. For example, unoxidized meteoritic material from the Canyon Diablo, Arizona, area has been found to have a magnetic permeability over a million times greater than that of the sedimentary rocks in which the Barringer meteorite crater was formed [65]. The unusually large magnetic permeability of a buried iron meteorite produces changes in the natural magnetic field of the earth in the vicinity of the meteorite. These changes, if sufficiently pronounced a t the surface of the earth, can be measured by means of devices known as “magnetic balances,” of the type employed in Sweden and Norway in prospecting for buried iron ores. The ordinary dip-needle is the oldest and simplest such balance [SS]. I n meteorite hunting, this relatively crude device seems t o have been the first instrument used in field investigations; and, as late as 1932-1933, in spite of unsatisfactory performance, ordinary dip-needles were still employed at Canyon Diablo [67]. Long before this date, however, such refined equipment as the Hotchkiss Superdip [68] and various forms of stable, portable magnetometer had become available. Results of field search employing these more modern instruments have been quite satisfactory, and the devices in question would seem well adapted for such definitely localized investigations as the search for subsidiary meteorite craters surmised to lie buried in the immediate neighborhood of known meteorite craters. Buried iron meteorites, in addition to magnetic effects, may, under suitable conditions, produce measurable electric effects a t points on the earth’s surface above them. It is easy to enumerate such possible electric effects in terms of the following fundamental electric phenomena:

(1) The field of dielectric force set u p b y an electric charge. (2) The current (and associated magnetic field) set up by such a charge in motion. (3) The electromotive force (and associated notion of potential) appearing when a current passes through a resistance, as it must in any physical situation. 4 In 1940, a small buried crater, approximately circular in outline with a diameter of 70 f t and a depth of 17 ft, was located near the main Odessa crater by such a magnetometric survey conducted by the Humble Oil and Refining Company, in collaboration with the University of Texas.

EFFECTS OF METEORITES ON THE EARTH

259

(4) The electric-cell effect, arising when two dissimilar conductors are so arranged as t o produce an electromotive force by electrochemical or thermoelectric action. The first and fourth of these effects offer little promise in connection with the detection of meteorites; and the potential-drop-distribution method based on the third effect is applicable only to objects of considerable size (and even for such objects only in case the specific resistance of the object differs greatly from that of the surrounding medium). Consequently, the effects in the second category remain as the only source of practical meteorite detection methods. Such so-called electromagnetic methods may be subdivided into those which are concerned with the differential of flux density and those which depend on magnetic field distortion. The flux-density method rests on the fundamental fact that if the flux originating from a coil is caused to change slightly, then the coil will show a n increment of inductance directly proportional t o the change in flux. One of the modifications of the Wheatstone bridge may be employed to measure this increment of inductance. This scheme was utilized in certain of the “dud-detectors” designed in Europe to locate unexploded shells buried in fields slated for recultivation following World War I. Since i t is impossible to build a search-coil with pure inductance, the performance of these devices was not satisfactory. The search-coils used on these instruments always have associated with them an internal resistance and a distributed capacity. Consequently, the bridge employed can be balanced with a given setting for one frequency only. For all other frequencies, the bridge is out of balance, and the instrument will react in the same manner as if a metallic object had been brought near it. A more promising flux-density method would seem to be the beatfrequency-oscillator system. I n this method, the inductance of the searchcoil is used t o control the frequency of a vacuum-tube oscillator. Unfortunately, many and serious difficulties are encountered in the development of instruments of this type which possess the stability essential for satisfactory performance under the conditions enountered in a field search for meteorites. Finally, we come to instruments designed to measure the magneticfield distortion produced by proximity to metallic masses. It is a wellknown fact that when a ferromagnetic mass is brought into the magnetic field surrounding an energized coil, the flux from the coil is attracted and reinforced by the mass in question. Consequently, the coil giving rise t o the field will exhibit a positive increment of inductance. Foucault currents will be induced in the meteorite if the field intensity oscillates, and the

260

LINCOLN LAPAZ

existence of these currents will set up a counter-magnetomotive force in such a direction as to oppose the impressed field and, hence, to neutralize a portion of it. This “eddy-current effect,” as it is called, repels and diminishes the impressed field. As a result, among other things, the coil exhibits a negative increment of inductance. The positive effect is independent of the frequency of the energizing current, but the negative effect is proportional to this frequency. For a meteorite of given size, therefore, a frequency of operation exists at which the net distortion and, with it, all possibilities of detection vanish. This is only one of a number of facts which must be considered in deciding at what frequency a magnetic-fielddistortion instrument should be driven. Under the assumption that a frequency of operation is employed at which the net distortion does not vanish, we proceed to an examination of the two chief methods by means of which this distortion can be measured. For the most part, commercial instrument-makers have specialized in the construction of what may be designated as perpendicular-coil detectors. Such instruments consist of a horizontal (vertical) “power” coil or transmitter rigidly joined to a vertical (horizontal) pickup coil by handles having a length of several coil diameters. In the absence of meteoritic material, these instruments may be brought into a condition of balance in which a vanishingly small portion of the flux emanating from the power coil is linked by the pickup coil. The mutual inductance is zero in this condition, and no signal is induced in the pickup system. If the instrument so adjusted is carried over a meteorite, the resultant field distortion results in linkage of power-coil flux by the pickup coil. A voltage is thus induced in the pickup system and, boosted by suitable amplification, this voltage reveals itself either as a signal tone or as the deflection of an indicator needle, or both. The following difficulties were encountered in the use of perpendicularcoil instruments as meteorite detectors on the First and Second Ohio State University Meteorite Expeditions in 1939 and 1941: A weight excessive in view of the long hours required to search extensive meteoritestrewn areas; sizes and shapes precluding convenient use on steep, brushed slopes like those on which the lost Port Orford, Oregon, meteorite is probably located; the necessity of frequent awkward adjustments; excessive battery drain; sacrifice of rigidity for the sake of portability; spurious signals, for example, from lightning; and loss of effectiveness over moist ground or over soil irregularly impregnated with comminuted ferromagnetic material, Some of these difficulties have since been eliminated by the trend toward miniaturization and, particularly, by the introduction of transistorized equipment; but certain difficulties, e.g., the last one mentioned, seem inherent in the perpendicular-coil type of instrument.

261


Instruments of quite different design from the perpendicular-coil detectors have been used with considerable success in several investigations where search has been made primarily for ferromagnetic masses such as unexploded airplane bombs and iron meteorites. The forerunner of all such instruments appears to be the “N.A.C.A. Bomb Detector,’’ designed in 1930 by Theodorsen [69], “For the immediate purpose of locating unexploded bombs which were known to have been dropped from airplanes a t targets in close proximity to the site of the new Seaplane Towing Channel at Langley Field, Virginia.” I n the original Theodorsen bomb-detector, three coils were wound symmetrically on a hollow, cylindrical, wooden frame three inches thick, three feet in diameter, and 1% ft high (Fig. 2). A special 110-volt, 6-amp \ \ \ I /I,’ generator, furnishing 500 cps current, ener\ I 1 1 1 \ \ gized the central or power coil, the strong \ \ 1 1 l ’ / \.\ I I I / alternating field supplied by the power coil :’ 1\ I I( II ’l I’ // : inducing electromotive forces in the nearby pickup coils, which were so connected th at \II ‘ only the difference of the emf’s appeared I+$\\ across a telephone receiver connected in i/ I I 1 I I \. /+I1 series with the pickup coils. When the / / ’ 1 1 \ \ instrument was in balance, one pickup coil I \\ \ \ was an “image” of the other with respect FIG. 2. Theodorsen coil. to the central power coil, and no current flowed through the receiver. If the device were carried over a buried bomb, the resulting distortion of the magnetic field destroyed this condition of balance and a signal was heard in the phones. Because of the weight of the cylinder on which the coils were wound, two men were required t o carry the instrument, which was suspended from a ladder-like frame by means of ropes in order to minimize flexure. A large powersupply truck was necessary in field work in view of the great weight of the generator used. A consideration of the cost and the weight of the Theodorsen apparatus will lead t o the conclusion th at no matter how satisfactorily it performed, this type of device would not be suitable for use by the meteorite hunter. I n this age of electronic gadgeteering, however, many possible modifications of the N.A.C.A. instrument speedily suggested themselves; and several of these were developed into first-rate meteorite detectors by the staff of the Meteorite Bureau at The Ohio State University in the 1930’s. Firstly, large scale, three-coil instruments were constructed which could be energized by standard, portable, gas-engine-driven 110-volt generators, of a size suitable for mounting in the luggage compartment of an automobile. Secondly, portable, three-coil instruments energized by

+-+ i$

\‘;

;/q;\

262

LINCOLN LAPAZ

vacuum-tube oscillators and small enough to be carried in a knapsack were built. Finally, a variation of the vacuum-tube meteorite detector, employing only one pickup coil and one power coil, was designed and constructed by Wisman and the writer in 1940 [70]. Even before miniaturization, this two-coil instrument had an exceedingly small battery drain and could be used anywhere a man was able t o walk or climb. With a search-coil only 10 in. in diameter, the two-coil instrument had a performance comparable to that of Theodorsen’s bulky, three-coil detector and, furthermore, would clearly pick up a 15-lb Odessa iron buried a t a depth of 12 in. in a water-filled hole in muddy ground-a crucial test none of the many perpendicular-coil instruments studied a t The Ohio State University would pass satisfactorily. Since World War 11, numerous attempts have been made t o employ land-mine detectors in the search for meteorites. Many instruments in this category have become available as war-surplus items. All were designed for a specific purpose, namely, the detection of military mines buried a t shallow depth. If this design limitation is kept in mind by the user, late model land-mine detectors like the SCR-625, in perfect adjustment, give satisfactory performance. Theodore Johnson, formerly custodian of the Barringer Meteorite Crater Museum, once informed the writer that more than 10,000 small Canyon Diablo meteorites-none buried t o depths in excess of a few inches-had been recovered by use of landmine detectors in the years between 1946 and 1950. W. A. Cassidy and H. L. Baldwin, Research Assistants of the Institute of Meteoritics, have had equal success with such instruments in shallow searches conducted about the Odessa, Texas meteorite crater. To the writer’s knowledge, however, no large meteorites buried at great depth, like the 130 kg “accordion meteorite’’ found near the Odessa meteorite crater, embedded in limestone a t a depth of over a meter, by the First Ohio State University (OSU) Meteorite Expedition in 1939 [71] have been recovered by the use of land-mine detectors. Before concluding this subsection, evidence must be presented testifying a t once to the large proportion of the iron meteorites buried at considerable depth and t o the efficiency of well-designed meteorite detectors in locating such deeply buried masses. Understandably, such evidence has been collected almost exclusively about meteorite craters, for almost all really systematic instrumental search has been confined to the meteorite-enriched areas surrounding such craters. It should be noted that the sample secured by search of these areas is biased in favor of shallow burial, for most of the meteorites recovered in such regions are “fragments” thrown out at the instant of the crater-forming explosion. These fragments strike the earth a t speeds very low in comparison with

EFFECTS O F METEORITES ON THE EARTH

263

those possessed by the “outriders,” i.e., smaller, companion meteorites that accompanied the main crater-forming mass in its orbital motion. The “outriders” show a depth of burial which increases roughly as the cube root of the mass, and hence for large meteorites may amount to several feet. Equally heavy “fragments” may be buried only slightly below plow depth.5 Although more meteorite hunts have been carried out around t,he Barringer meteorite crater than in any other locality, nevertheless, the data t o be presented below do not relate t o finds made at the Arizona crater. The reasons for this are as follows: On the one hand, recoveries made during systematic steam-plowings t o depths far in excess of normal plow-depth had depleted large areas about the Barringer meteorite crater years before meteorite detectors appeared on the scene at all. On the other hand, prior t o the Ohio State University Meteorite Expeditions of 1939 and 1941 t o the Barringer crater, a great many commercial meteorite hunters and curiosity seekers had worked and reworked (often surreptitiously) the entire environs of the crater with meteorite detectors of one kind or another, without publishing any information concerning either the capabilities of the instruments used or the weights of the finds made a t depth. Consequently, no amount of care on the part of OSU personnel could guarantee collection of dependable observational datla with respect t o the amount of Canyon Diablo meteorites originally buried in strata a t various depths. In contrast t o the confused situation just described, the field work carried out by the First Ohio State Meteorite Expedition a t the Odessa meteorite crater between September 15 and 17, 1939, was conducted on what, a t least from the viewpoint of instrumental search, was essentially virgin ground. Prior to August, 1939, occasional brief visual searches for meteorites had been carried out-chiefly a t the Odessa crater itself-and a number of small surface finds had been made, the largest specimen recovered having a reported weight of about 8 lb. During the three weeks prior t o arrival of The Ohio State University party a t the crater, a group of approximately thirty WP.4 workers, employed on the recently initiated WPA-University of Texas Odessa Crater Project, had searched for meteorites diligently, but without instruments, not only a t the crater, but over a region of several square miles surrounding it. This search was an intensive one since the very continuance of the WPA-University of Texas Project depended on discovery of enough meteoritic material t o prove tha t the Odessa crater had originated in large-scale meteoritic impact, 6 The terminology employed and results stated stem from observations made during instrumental surveys of the Odessa and Canyon Diablo meteorite craters in 1939 and 1941 [72].

264

LINCOLN LAPAZ

thereby justifying proposed elaborate and expensive excavations in and about the crater. In spite of this compulsion, the widespread visual search had resulted in the discovery of a single meteorite weighing less than a pound at a distance of over half a mile from the crater rim. Because of high winds, it was possible t o employ the meteorite detectors effectively only during the early morning and the late afternoon. In spite of this fact, in less than 12 hr work during the interval September 15-17 over 400 lb of Odessa octahedrites were recovered from depths between 8 and 40 in. This instrumental search was not confined to areas contiguous to the crater, but rather covered a number of small subregions distributed in various directions and at various distances from the crater. Since the area swept over by the instruments was certainly much less than one-thousandth of that covered by the earlier WPA search, the ratio, q, of deeply buried iron to superficially embedded iron may conservatively be set at 400,000. It is of interest to note that a value of Q of the same astonishing magnitude is clearly indicated by the weight-depth of burial data now becoming available as a result of the exhaustive explorations carried out by several successive meteorite expeditions sent by the Meteorite Committee of the U.S.S.R. to investigate the strewn field of the massive granular-hexahedrite shower of February 12, 1947-a witnessed crater-producing fall in contrast to the Odessa fall, which occurred in remote prehistoric times. 3.5. The Rate of Accretion of Meteoritic Dust

I n earlier sections of this review, it has been-pointed out that no dependable appraisal of the rate of accretion of sideritic materials is possible until exhaustive instrumental surveys have made trustworthy information available concerning the distribution of meteoritic iron in that considerable three-dimensional volume accessible to deeply-penetrating siderites of high sectional density. Again, dependable appraisal of the rate of aerolitic accretion will become possible only when meteoritic strewn fields are subjected to such systematic searches as those basic to application of population criteria of the sort described earlier in this chapter. Although the determination of what might be termed “ordinary” meteoritic accretion thus seems to be a problem for future investigation, the opinion has been widely held for some time that reasonably dependable estimates of the rate of accretion of meteoritic or, as it is commonly called, “ cosmic ” dust are possible. The excellent historical review presented by Buddhue [73], together with the comprehensive annotated bibliography on meteoritic dust later published by Hoffleit [74], will serve to give the reader a complete picture of the exceedingly diverse


265

collection techniques employed prior to 1951 at stations scattered all over the globe, and of the manner in which the crude data of observation have been used to obtain rate-of-accretion estimates. Among the investigations in this field carried out since 1951, special mention should be made of the systematic and carefully controlled research on the abundance and rate of infall of meteoritic dust carried out under an ONR-supported Airborne Particle Study by the group at New Mexico Institute of Mining and Technology supervised by W. D. Crozier. Many interesting conclusions have been reached by the NMIMT group, three of which merit special emphasis: first, that a global rate of infall of magnetic spherules of 230,000-265,000 metric tons per year is indicated by NMIMT data; second, that the maximum contribution to this total is made by spherules with diameters in the neighborhood of 15 microns; and third, that magnetic spherules like those obtained from the atmosphere have been found not only in lake sediments in New Mexico but also in clay from Georgia and in water-deposited shale of upper Cretaceous or lower Tertiary age collected near Datil, New Mexico. Because, whatever their particular origin, cosmic dust particles are quite small (ranging in size from a few microns to a maximum of at most 250 p ) , their rate of descent through the atmosphere is always very slow. Having regard to the diverse motions of the medium through which the dust particles fall, their long-continued interaction with it apparently has been considered to guarantee widespread uniformity in the distribution of meteoritic dust, from whatever source, over the face of the earth. Therefore, the simplest, single-station collection techniques and identification procedures seem to have been regarded as adequate for determination of the world-wide rate of accretion of such dust. This view receives no support, however, from such extremely divergent estimates of the accretion rate in question as those to be presented later in this section. In place of the penetration and population difficulties earlier encountered in connection with appraisal of the contribution of the siderites and aerolites, respectively, a new difficulty-that of identification as cosmic dusl-arises to prevent realization of the optimistic forecast given above. Because of the complexity of the identification problem, a controversy dating back a t least to the contributions of Nordenskiold [75] and von Lasaulx [76] has remained unresolved for nearly a century; and even today it calls forth such flatly contradictory opinions as those held by Buddhue [73] and Krinov [77]. Vincent J. Schaefer’s development in 1946 of an etching technique applicable to minute particles under inspection through the electron microscope may eventually provide a thoroughly dependable criterion

266

LINCOLN LAPAZ

which will permit the investigator t o distinguish between the various cosmic and terrestrial magnetic dusts. Since the equipment necessary for application of Schaefer’s tests is at once expensive and quite difficult t o obtain, there is need for an inexpensive alternative test. Because of the past history of dusts of extra-terrestrial origin, arriving as these do from lengthy sojourns in space and in the higher layers of the earth’s atmosphere, i t would seem certain that meteoritic dusts would exhibit cosmic radiation effects not present in earth-bound terrestrial dusts. Conceivably, well-known and economical nuclear-emulsion techniques would permit detection of the anticipated differences in cosmic radiation effects. The writer suggested such a n autoradiographic test t o V. H. Regener in 1948. TABLE VIII. Estimates of the rate of accretion of meteoritic dust. (Based on tables earlier prepared by J. D. Buddhue and P. M. Millman.) Annual deposit (in metric tons) 8

26 56 100 332-3320 810-129,600 332,000-3,320,000

Source Black spheres from abyssal red clay deposits only From meteors only From meteors only From meteors only From fireballs; visual meteors; faint radio meteors; telescopic meteors Meteoritic dust brought down in rainfall From faintest telescopic meteors; micrometeorites; zodiacal dust

References Buddhue [73], p. 50 Watson [80] Watson-Buddhue [all, p. 115, Table 19 Watson [Sl] Millman [82] Buddhue [73], p. 54 Millman [82],p. 81

Pending development of dependable identification criteria based on electroumicrography, autoradiography, or on some or all of the tests employing chemical, microchemical, and physical means well summarized by Buddhue in 1950 (see [73], pp. 44-49), there would seem to be no alternative but t o present all of the recently published estimates of the rate of accretion, grossly discordant as these estimates may be. The decision to keep a n open mind in this question finds support in the fact th a t recent accretion estimates of the order of several thousand tons per day for the whole earth have won strong approval from Opik [78], but have been criticized by Whipple [79]. Inspection of the trend in Table VIII will bring conviction of the need t o set a n upper bound on the annual increment of cosmic dust received by the earth, preferably by an argument divorced as completely


267

as possible from dust collection and identification difficulties. We shall conclude this section with a calculation directed toward this end. Accumulation of meteoritic dust on the surface of the earth, by increasing the moment of inertia of our globe, must decrease the speed of its rotation and hence increase the length of the day. Since an incomparably more potent agency, that of tidal friction, also acts in the same direction, the total observed increase in the length of the day, say AT, must greatly exceed that attributable solely to the accretion of cosmic dust. To calculate the amount of dust that would be necessary to account for the whole of AT would, of course, give an upper bound on the annual meteoritic increment to the earth, but a bound of such magnitude as to be without interest. We therefore shall suppose that only a minute fraction, e * AT(0 < 0 ,respectively, a t least for sufficiently short intervals of time, can be calculated. In line with fundamental results obtained by Epstein [ 1291 in investigations of an analogous two-dimensional shock-wave problem, the first relation between 5 and .w‘ may be taken in the form

-

(4.5)

-

6 = hLi)

where h is a constant. The second desired relation between Z and 5 is secured, after suitable choice of a wv-coordinate system, by a limiting process applied t o the equation v 2 = cw of the approximating parabola (see [29], p. 372). The final result obtained can be summarized as follows. For given h, the mean values of w and ir within sufficiently short intervals of time can be calculated as soon as the value of the parameter c has been obtained from a study of the parabola approximating the vertex portion of the projection of the coma on the wv-plane. In the application to the Kybunga photograph, h was taken equal to one-third, in line with opik’s evaluation of this parameter for Epstein’s I( ideal case.” Because, as opik has stressed, in the actual case the speed of escape sidewise is smaller than in Epstein’s ideal case, it can be shown that use of h = will lead to a lower bound for the speed of the Kybunga meteorite a t the time a t which it was photographed. Although the Kybunga fireball was deep in the atmosphere a t this time (in fact, its height was only 31.3 km), nevertheless, the lower bound obtained by the coma method was found to lead to a value in excess of 45.8 km/sec for the heliocentric velocity of the Kybunga meteorite. This meteorite was therefore certainly moving in a strongly hyperbolic orbit before it became entrapped in the earth’s atmosphere (see [29], p. 374). 4.6.3. A Method Which M a y Be Applicable to High-SpeedrCarbonBearing Meteorites. All ballisticians are familiar with the fact that the appearance of a cloud of black smoke after the explosion of a charge of T N T is a certain indication that the charge actually detonated and did not merely burn a t low pressure. The extremely high pressure consequent on genuine detonation acts t o produce solid (uncombined) carbon. It is also well known that for many organic powders the conditions of pressure and temperature under which carbon, that is to say, black smoke, will be produced have been calculated [130]. Visual observations of meteorite falls show that certain types of meteorites produce spectacular clouds of black smoke when so-called “explosions” occur prior to or a t arrival of the meteorite at the Hemmungspunkt (point of retardation). Since this “ black-smoke” phenomenon has been chiefly remarked in connection with the carbon-bearing carbona-


291

ceous chondrites (Kc), it is a t least conceivable that an observed cloud of black smoke from such a meteorite is an indication of the very high pressure t o which the meteoritic vapor in the cloud is subjected. The carbonaceous chondrites are notoriously fragile and it is therefore reasonable to expect that as the head resistance on such a meteorite peaks up near the point of retardation, the whole (or a large portion) of the stone may quite suddenly be comminuted into powdery form. The result is a shower of very tiny meteorites moving through relatively dense air not a t the moderate speed that each meteoritelet would have attained had it penetrated as an individual t o such low levels in the atmosphere (see [29], p. 380), but, rather, endowed with the very great kinetic energy corresponding to the high velocity possessed by the main mass of the meteorite just before its disruption. The subsequent almost instantaneous transformation of the large kinetic energy carried by each tiny particle must lead to what could more aptly be called a “meteoritic detonation” than the characteristic meteoritic noises for which this term already has been preempted, If the pressure developed in this “detonation” of a carbon-bearing meteorite is in excess of what the ballistician calls the “smoke-formation pressure,” then it may be anticipated that a cloud of black smoke will be produced. This conjecture has been tested on one of the most recent wellobserved carbonaceous chondrite falls-that near Murray, Kentucky in 1950-in connection with which a single but notable cloud of black smoke was reported, The Murray meteorite was chosen for the test in question not only because of its recency, but also because Olivier [131], by use of the classical methods, and the writer (see [132], p. 115, footnote (*)), by use of the inverse acceleration method, had found strongly hyperbolic velocities for the Murray fall. The Murray meteorite was found by Horan [132] t o contain 2% of carbon. Available data on the smoke formation pressures for low carboncontent high explosives indicate that, since the temperature in the Murray fireball was certainly of the order of 3000”K, the smoke formation pressure p for the material in the Murray meteorite would fall in the high pressure range 6.9 x 10” dynes/cm2 < p < 6.9 X 1 O I 2 dynes/cm2. If these limiting values for the pressure p are introduced into a retardation formula of the type rpVn = p , where y is taken as 1 (see [124], p. 252) and p is the atmospheric density in gm/cc a t an altitude of 46 km (the height a t which the Murray black smoke cloud appeared), it is found that n = 2 leads to values for V so far in excess of even the largest hyperbolic velocities reported for telescopic meteors by Boothroyd [114] as to merit no consideration. For n = 2.5 and n = 3, the values of V in km/sec are given in Table XI.

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LINCOLN LAPAZ

I n view of the wealth of concordant real path data collected in the case of the Murray fireball, the very low V values given by the choice n = 3 are clearly inadmissible. The smaller value, 88 km/sec given by the choice n = 2.5 is near the first velocity value (77 km/sec) found by Olivier for the Murray meteorite; as well as near the value 70 km/sec given by the inverse acceleration method as a lower bound on the speed of the Kentucky meteorite. TABLE XI. Velocities computed from the retardation formula p = 6.915 X 10"

V(2.5) V(3)

8.812 X 10 krn/sec 6.131 km/sec

ypVn =

p.

p = 6.915 X 10**

2.214 X lo2km/sec 1.321 X 10 km/sec

Since the meteoritical importance of the particular velocity-exponent n = 2.5 has already been signalized in another connection [133] it would seem that the velocity method of this section, with n chosen as 2.5, although obviously speculative, merits consideration. 5. THE HYPERBOLIC METEORITE VELOCITYPROBLEM. I1

Two quite incompatible points of view are revealed by the account of the meteorite velocity controversy given in Section 4. One holds that meteoritic material is observed to move with high (hyperbolic) as well as with low (elliptic) velocities, while the other, in effect, denies that a hyperbolic component exists a t all. Since there seems little possibility of securing agreement as to the relative importance of either instrumental peculiarities and limitations or of the role of the personal element, it is natural t o cast about for some crucial test capable of disclosing in an unambiguous way whether the meteoritic universe displays the comparative uniformity that would characterize a system of bodies all moving with low (elliptic) velocity about the sun, or rather is comprised of two categories of objects as diverse dynamically as would be a low-speed solar system component and a high-speed interstellar component. Obviously, such a crucial test, to be of any value in resolving the heated controversy, must be completely divorced from the many and varied velocity methods already devised, each with its own partisan champions. I n view of the well-established uniformity not only of the material basis of the known universe, but also of the physical processes operative within this universe, it seems clear that the most significant observable difference between the interstellar and the solar system meteorites would be in their different mean velocities a t the earth's distance from the sun.


293

Since the wide variety of observational techniques already applied t o detect such velocity differences has led only to the controversy briefly reviewed in the previous section, it may seem hopeless to hunt for a method that will successfully reveal the existence of these differences. A clue, however, is to be found in a remarkable paper of Lowell’s [134]. His highly interesting dynamical analysis clearly shows not only that the velocity due to the sun’s attraction and that due to the earth’s upon a particle falling to the latter under the action of both are not simply additive, but also that the increment in meteoritic speed produced by the action of the terrestrial gravitational field is very much greater for slow meteorites than for fast ones, the ratio of the speed increments in the particular case treated by Lowell having the value 5 . If changes in the directions rather than in the magnitudes of the meteoritic velocity vectors are considered, the same differential effect is present. These facts suggest that if we replace the three-body problem considered by Lowell (involving the sun, the earth, and a meteorite) by an analogous threebody problem (involving the earth, the moon, and a meteorite), the differential in the disturbing effects exercised by the moon on fast and slow meteorites may lead to observable differences in behavior between these two categories of bodies. For example, one would anticipate that the moon would have little or no effect on the fall to earth of swift hyperbolic meteorites passing through the earth-moon system; whereas, our widely roving satellite might be capable of diverting numbers of slow-moving solar system meteorites into the more or less void torus which, as opik [135] has pointed out in an important paper, is essentially the nature of the track now swept out by the earth. If the above reasoning is valid, evidence of a moon effect should be most notable for the very slowest meteorites intercepted by the earth. Consequently, one should seek such evidence first in the case of processional fireball falls like the famous Canadian incident of February 9, 1913,’O for which an exceedingly small initial velocity relative to the earth has been found by most investigators of the phenomenon. Influenced by these considerations, the writer made a search of the literature which disclosed that as long ago as 1923, Pickering [138]had already noted with some surprise that both the Canadian procession and a later occurrence quite similar in character but involving much fainter meteors had developed when the moon was in essentially the same position on the celestial The vaIidity of Chant’s [I361 satellitory interpretation of the Canadian fireball procession recently has been questioned by Wylie [137]. The procedures adopted and the conclusions reached in Wylie’s papers on the Canadian incident are discredited in two papers (one by A. D. Mebane and the other by the present writer) which have appeared in issue No. 4 of the journal Meteoritics, pp. 402-421.

2 94

LINCOLN LAPAZ

sphere. In fact, on the basis of the near identity of the lunar positions a t the two times of fall, Pickering was led to hazard the conjecture that the moon had played an important role in bringing both processions down into the earth’s atmosphere. Since it is, of course, possible that the near identity in lunar position which attracted Pickering’s attention was simply a fortuitous coincidence, an extensive statistical investigation is necessary to establish whether or not the moon is actually capable of increasing the probability of fall of meteorites or at least of certain categories of meteorites. In the present section, the results of such a statistical inquiry, based on examination of the times of fall of 317 meteorites are presented. In the writer’s opinion, these results leave no doubt that the frequency of fall of one subclass containing 165 meteorites is strongly dependent on the moon’s nearness in direction to that of the line of apsides of Jupiter’s orbit. On the contrary, the 152 members of the remaining group of meteorites constitute a quite distinct subclass of meteorites, the frequency of fall of which is affected but little or not at all by the position of the moon in the sky. It is the writer’s belief that the first category comprises the solar system meteorites and the second consists of the much faster interstellar meteorites. 5.1. The Frequency Distribution of Meteorite Falls as a Function of the Right Ascension of the Moon at the T i m e of Fall

The time interval selected for the statistical test suggested in the previous paragraph was a half-century 1896-1945, inclusive. Choice of this interval was dictated by the fact that an examination of the literature of meteoritics disclosed that a larger percentage of dependable meteoritic classifications and times of fall were available for the more recent falls; and, furthermore, that the half-century in question contained almost as many critically evaluated meteorite orbits as the preceding 500 years 11391. Within the half-century chosen, 269 meteorites fell [140]. Of these, 265, or 98.5%, are included in the totality of 317 meteorites to be reported on in this section. Times of fall or classifications, or both, of the necessary accuracy were lacking for the remaining 1.5% of the meteorites in question. In addition t o the 265 meteorites just referred to, 17 meteorites that fell after January 1, 1945, and 36 meteorites that fell before January 1, 1896, are included in the group of 317 meteorites treated in this section. Details on the manner in which these additional 53 meteorites came into consideration (given in Appendix 11) will make clear that they stand on exactly the same footing as those meteorites that fell in the half-century to which the investigation originally was limited. The conclusions t o be

295


announced below would not be significantly altered if the 53 meteorites in question had been excluded from consideration. In Table XII,the distribution of the totality of 317 meteorite falls with respect to the right ascension of the moon a t the time of fall is given, together with other relevant data such as the year, month, day, and hour of fall, the name, and the meteoritic classification. It is necessary to specify that the time “Noon?” has been used in every case where the day but not the hour of fall was the only information that could be found. In every such case, the tabulated a may, of course, be in error, but by no more than one-quarter of the moon’s daily motion for the date in question. The effect of errors of this magnitude is of no importance in the present connection. In view of the manner in which the tabular data in Table XI1 are grouped, it is easy to verify that the number of meteorites in the four quadrants, Q1, Q 2 , Q 3 , and Q4, centering in sequence on a = 0”, a = 90”, a = 180”, and a = 270”, is, respectively, n(Ql) = 98, n(Q2) = 60, n(Qa)= 88, and n(Q4)= 71. The remarkable nonuniformity of this distribution with respect to a certainly would not seem to be the result of chance. To verify this common-sense judgment mathematically, we may proceed as follows: The probability p that a meteorite falling at random would belong in (Q1 Q3)is p = 48,as is the counter probability q = 1 - p . One would therefore expect 3 1 7 4 of the meteorites in Table XI1 to fall in (Q1 Q3). The discrepancy d between this number and the number actually observed has the large value d = 27.5. To ascertain the probability of occurrence of a discrepancy of this magnitude, we shall employ Laplace’s theorem to the effect that if in a single trial the probability for success is p and that for failure is q, then the probability, P, that in n trials the discrepancy will not exceed numerically the number d is very nearly equal to

+

+

(5.1)

In our case calculation gives d / d = = 2.184, from which it follows that the corresponding value of P = 0.998. One can therefore infer that there are less than two chances in a thousand for occurrence of the observed discrepancy, d = 27.5. It should be emphasized that this significantly nonuniform distribution has been deduced from observational data affected by all of the potent and highly variable terrestrial factors that preclude recovery of more than a small fraction of the meteorites actually seen to fall [26, 1411. For further progress, it is essential that standard smoothing techniques be applied to such rough observational data as those in Table XII. Application

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LINCOLN LAPAZ

TABLE XII. The right ascension

a of the moon at the time of fall of each of 317 meteorites.

No.

(a)

Year

Mo. Day

Hour

*I *2 3 4 5 *6 7 *8 *9 *lo 11

316% 31834 319 319% 319% 320 320M 323 324 32434 32534

1917 1905 1903 1937 1911 1922 1929 1932 1908 1911 1868

Feb. Apr . July Sept. Sept. Apr. Feb . Nov. Apr . June Mar.

20 8:30 A l:oo P 27 12 1O:OO A 17 11:45 P 6 3:30 P 20 7:45 P 9 12:45 A 5 1o:oo P 25 2:OO A 16 5:OO P 20 Noon?

12 13 *14 15 16 17

326 32654 326% 32735 329t.4 330

1914 1930 1926 1931 1938 1903

July July Dec. Aug. Feb. Apr.

10 Noon? 13 Noon? 10 9:30 A 27 3:OOP 1 5:OO P 22 11:30A

18 19 *20 *21 22

33035 33235 335% 336 33734

1932 1933 1898 1927 1903

May Apr. Jan. Apr. Jan.

5:30 P 26 19 8:30 P 24 2:30 P 27 1:00 A 3 11:OO P

23 *24 *25 *26 27 28 *29 30

339 339 34134 344 34435 345 345t.i 34534

1915 1927 1869 1882 1920 1869 1933 1933

Jan. June May Aug. Aug. Sept. July Mar.

19 Noon 21 6:OO A 5 6:30 P 29 Noon? 30 11:15 A 19 9:oo P 11 9:30 A 24 5:OO A

31 *32 33 34 *35 *36 *37 38 39

3463i 346% 347 34834 348t.i 34934 350% 351 35135

1897 1896 1931 1933 1889 1913 1905 1882 1927

June Apr . Apr. Oct. June Jan. May Aug. Apr.

20 8:30 P 9 6:15 P 14 11:53 A 2 6:OO A 18 8:30 A 12 6:OO P 27 10:45 A 2 5:OO P 28 9:00 A

Name Ranchapur, India Karkh, India Valdinizza, Italy Mabwe-Khoywa, Burma Demina, Siberia Hedeskoga, Sweden Padvarninkai, Lithuania Prambachkirchen, Austria Novy-Projekt, Lithuania Kilbourn, Wisconsin, USA Daniel’s Kuil, South Africa Saint-Sauveur, France Miller, Arkansas, USA Ojueltos Altos, Spain Yukan, China Aztec, New Mexico, USA Jackalsfontein, South Africa Kuenetzovo, Siberia Brient, USSR Mjelleim, Norway Sopot, Romania St. Mark’s Mission, S. Africa Visuni, India Trysil, Norway Krahenberg, Bavaria Pirgunje, India Merua, Iridia TjabB, Java Athens, Alabama, USA Pasamonte, New Mexico, USA Lanpon, France Ottawa, Kansas, USA Pontlyfni, Wales Pesyanoe, USSR Mighei, Ukraine Banswal, India Minnichhof, Hungary Pavlovka, Russia Aba, Japan

Classification cg cg Ci AE Chyi ChYW Aleu sh Chy Cg Cg Cenk Cenk Ceng Chy Civ AE Ci Ci Aleu AE C cc Cck cw Chyib (Cho) cwv Cbrg Chyk Cgb Aiho Civ Cgb(Cho) AE Aiho Cr cg Cw Aiho AE

297


TABLE XII. The right ascension 01 of the moon a t the time of fall of each of 317 meteorites. (Continued) ~

No.

(a)

Year

Mo. Day

Hour

*40 *41 42

3523.6 1882 Mar. 353 1933 Aua. 3533.i 1933 Aug.

19 1:00 P 8 8:OO P 8 10:30 A

43

3533.6 1919 June

20

*44 45

356% 1929 Oct. 35755 1927 Dec.

15 11:30 P 30 11:30A

46 47 48 49 50 *51 52 53 54 55 *56 57 58 *59 60 *61 62 *63 64 *65

3583.6 3583.6 358% 3593i 35955 1 3 33 i 3 36 3% 43.6 5 35 6 3.i 6% 7 3.6 7%

June May Sept. Sept. Mar. Mar. Nov. Apr. Mar. June Oct. Dec. July Sept. Oct. June Aug. June Aug. June

10 11:oo P 26 3:OO P 19 l:oo P 24 Noon? 31 Noon? 31 8:45 A 12 1:oo P 14 11:25 A 12 10:30 P 30 1o:oo P 1 2:oo P 22 5:OO P 11 7:15 P 29 5:OO P 9 2:OO A 1 1o:oo P 5 7:30 A 15 Noon 22 8:OO P 11 8:56 P

66 67

1901 1916 1910 1942 1938 1908 1902 1923 1899 1918 1868 1868 1868 1928 1938 1902 83.5 1898 9 1906 9% 1902 11 1939

6:OO P

1917 July 1865 Mar.

11 26

Noon 9:OOA

68 69 *70

11% 1910 Oct. 13 1907 May 13 1933 Jan.

17 9 31

Noon? 1:30 P 4:30 P

*71 *72 *73 *74 *75 *76 77

133.6 143.6 15% 213.i 213.4 22 223.5

Sept. 10 Feb. 2 Jan. 30 Apr. 3 Apr. 13 Oct. 30 Aug. 6

2:15 P Noon? 7:OOP 3:30P 7:30 A 5:OO P 9:00 P

11 11

1930 1922 1868 1916 1896 1944 1939

Name

~

Classification

Fukutomi, Japan cgv Repeev Khutor, Russia 0 Sioux County, Nebraska, Alho USA St. Mary’s County, Md., Ceng USA Beardsley, Kansas, USA Cg Oesede, Hanover, Cbrc Germany Sindhri, Bombay, India Cc Calivo, Philippine Islands A Khohar, U.P., India ccg Maziba, Uganda Ci Kasamatsu, Japan AE Avce, Italy H Kamsagar, India Ci Holetta, Abyssinia AE Bjurbiile, Finland Chyc Richardton, N.D., USA Cbrc Lodran, India S Moti-ka-nagla, India Ck Ornans, France Chyc Naoki, India cg Civ Zhovtnevyi, USSR Marjalahti, Finland LPa Andover, Maine, USA Cc Kijima, Japan C Caratash, Smyrna AIam Dresden, Ontario, cg Canada Cc Nan Yang Pao, China Vernon County, Cbr Wisconsin, USA Palahatchie, Miss., USA Cc Chainpur, India cc S Dyarrl Island, New Guinea Oldenburg, Germany Cw Baldwyn, Miss., USA Cwv Pultusk, Poiand Cbrgv Treysa, Germany Om Lesves, Belgium Chyg Valdavur, India cg Andura, India Ci

298

LINCOLN LAPAZ

TABLE XII. The right ascension CY of the moon at the time of fall of each of 317 meteorites. (Continued) No.

(01)

78 *79 80 *81 *82 83 84 85 86

2635 26% 28% 29% 32 33 33 36% 37

87 88 *89 *90 91 *92 *93 *94 295 96 *97 98

37 37% 39 39% 4034 41 41 35 42 43 43 ?4

99 100 101 102 103

Year 1925 1921 1944 1921 1939 1933 1865 1932 1921

1897 1916 1868 1932 1865 1930 1938 1933 1947 1914 43f4 1944 44% 1924 46)i 47 48)i 48% 49%

1941 1868 1943 1899 1938

Mo. Day Sept. June Mar. Jan. Sept. Dec. Aug. Aug. Oct.

Hour

6 Noon? 30 3:OO P ,26 10:15 A 17 9:oo P 3 Noon 26 6:OO P 12 7:OO P 22 3:OO P 17 11:oo P

Sept. 15 Noon? 5 2:30 A Apr. Nov. 27 5:OO P Apr. 8 Noon? May 23 6:OO P May 27 Noon Jan. 11 2:30 P 2 8:33 P Feb. Feb. 26 3:45 P July 17 Noon? Feb. 1 2:oo P June 27 3 3 0 P July Feb. July Sept. June

18 4:OO P 29 11:OO A 25 5:35 P 23 3:OO P 24 6:30 P

*lo4 105 * 106 107

51fi 1914 May 5236 1919 May 54% 1906 Mar. 553i 1902 Nov.

24 1 29 15

Noon? Noon Noon? 6:45 P

108 109 *110 *111 112

57% 6l)i 65% 67 68%

1934 1920 1939 1936 1948

Mar. Dec. Dec. July Feb.

20 23 24 15 18

6:30P 5:30 P 7:40 P 4:45 P 4:56 P

*113 *114 *115 *116

70)i 7154 7235 76%

1926 1905 1937 1868

Apr. Jan. May May

16 Noon? 17 9:30 P 12 8:45 P 22 10:30A

Name

Classification

Numakai, Japan C Tuan Tuc, China Chyg Tulung Dzong, Tibet AE Haripura, India Cr Santa Cruz, Mexico Cr Pervomaisky, USSR Cenivk Dundrum, Ireland Cbrk Douar Mghila, Morocco Chyi Rose City, Michigan, Cbrsb USA Gambat, India Civ Ekh Khera, India Civ Danville, Alabama, USA Chygbv Khor Temiki, Sudan cw Gopalpur, India Cbrc Kurumi, Japan C Lavrentievka, USSR cw Zemaitkiemis, Lithuania Chywv Seldebourak, North Africa Cg Nyirabrhny, Hungary AE Hallingeberg, Sweden Cg BBrBba, French West Aleu Africa Phuoc-Binh, Annam Cbrg Motta di Conti, Italy Cc Benoni, South Africa Cbr Donga Kohrod, India Ci Chicora, Pennsylvania, Chy USA Kisvarshy, Hungary AE Adhi Kot, India cs Kulp, Caucasus C Bath Furnace, Ci Kentucky, USA Tirupati, India Ci Atarra, India cs Nyaung, Upper Burma 0 Nassirah, New Caledonia Cbrgv Norton County, Kansas, Alno USA Urasaki, Japan AE Tomakovka, Ukraine Chyg Kaptal-Aryk, USSR cgv Slavetic, Yugoslavia Cgb

299


TABLE XII. The right ascension (I of the moon at the time of fall of each of 317 meteorites. (Continued) Year

Mo. Day

No.

((I)

117 *118 119

78 1923 Oct. 78% 1935 Mar. 7936 1942 Aug.

120 *121 *122 123 124 *125

79% 81% 8335 89ji 89% 95%

*126

963/4 1930 Apr.

127 '128 129 130

96>/4 96% 98>/4 983/,

1899 1904 1917 1907

Oct. Oct. Apr. Nov.

*I31 *132 133 *134 *I35 *136 *137 *138

99)/4 99j6 lOlj4 10334 104 10535 105% log>/,

1916 1913 1803 1918 1947 1935 1916 1899

Jan. Apr. Apr. Jan. Dec. Apr. Oct. Jan.

139

109% 1094/4 llO!i 11255 115 116 117 120 12136

1942 1900 1897 1950 1935 1803 1908 1925 1925

Apr. July Oct. Oct. July Oct. June July Apr.

140 *141 142 *143 *144 *145 146 *147

1933 1946 1925 1910 1940 1925

Jan. May June Jan. Dec. Dec.

148 12:3j/4 1911 June *149 12634 1919 Oct.

*150 *I51 152 *IC3

127 12755 129>q 13054

1944 1920 1917 1914

June Aug. hc. Oct.

Hour

Classification

Name

1 11:OOA Serra de Mag6, Brazil 12 12:52 A Lowicz, Poland 7 3:OO P Forest Vale, New South Wales 9 4:30 P Phum Sambo, Cambodia 4 Noon? Krasnyi Klyuch, USSR 20 3:oo P Renca, Argentina 22 9:30 P Vigarano, Italy 15 2:30 P Ramnagar, India 29 5:OO P Newtown, Connecticut, USA 6 2:OO P Gundaring, Western Australia 24 7:OO A Peramiho, East Africa 29 4:OO P Pfullingen, Germany 26 1O:OO A Troup, Texas, USA 22 10:30 A Bali Mission Station, W. Afr. 18 9:OOA Baxter, Missouri, USA 13 Noon? Sakouchi, Japan 26 1:00 P L'Aigle, France 25 2:30 P T a d , Japan 28 8:OO A Reliegos, Spain 10 7:30A Sungach, USSR 18 11:47A Boguslavka, Siberia 25 7:45 A Zomba, British Central Africa 22 7:OO P Kapoeta, Sudan 25 Noon? 0-Feherto, Hungary 18 Noon? Delhi, India 5 4:lO A Monse, Africa 29 2:20 P Patwar, India 8 10:30 A Apt, France 30 7:OO A Kagarlyk, USSR 20 7:OO P Villarrica, Paraguay 30 8:OO P Queens Mercy, South Africa 28 9:00 A Nakhla, Egypt, 16 8:OO A Bur-Gheluai, Ital. Somaliland 23 Noon? Fort Flattcrs, Algeria 13 Noon? Nanseiki, China 3 1:15 P Strathmore, Scotland 13 8:45 P Appley Bridge, England

Aleu S C Cbri C ChY ccs AE D Og Aleu 0 Ci

cs

C 0 Chyih cw cg Cga H Chyw Ci C cw ChY S cgv cw AE Cbrg Alna Cg AE LPa Chyi Chyg

300

LINCOLN LAPAZ

TABLE XII. The right ascension (Y of the moon a t the time of fall of each of 317 meteorites. (Continued) No.

((Y)

Year

Mo. Day

Name

Hour

Classification

.-

2:OO P Cherveltaz, Switzerland 8:30 P Hainaut, France 9:30 A Santa Isabel, Argentina

Nov. Nov. Nov. Feh. July Aug. Sept. May Apr. Apr. Feb. Jan. Apr. Apr.

30 26 18 26 10 23 19 23 6 2 3 9 13 !I

168 143$/a 1869 Jan. 169 146 1868 Dee. 170 147j/2 1949 Jan.

1 5 16

1230 P

26 3 1 li

Noon? Noon? Noon? 7:42 P

*175 15274 1929 July

9

3:00 P

June May June Dec.

30 22 29 26

11:30 P 1O:OO A 11:58 A

180 16036 1949 Sept. 21 6 181 160% 1924 July

1:45 A 4:20 P

154 155 *156 *157 158 *159 *160 161 *162 163 164 *165 *166 167

$171 172 *173 *174

176 177 178 *179

*182

183 *184 I85 186 187 *188 *189 190 *191

13034 13014 131 13235 133j.i 135 1363/4 13615 137 138M 13955 141 14235 143

14934 1493.5 150 1503/4

1553i 15636 15735 159

1613/4 162 163>/4 164 165 165 16536 167 167 170>/4

1901 1934 1924 1896 3899 1900 1949 1950 1914 1936 1882 1947 1916 1919

1940 1926 1873 1936

1903 1904 1903 1934

1932 1928 1897 1910 1926 1869 1949 1907 1924 1944

Oct.

May June Aug.

July Aug. Aug. Nov. Dec. Jan. Sept. Feb. Mar. May

Noon? 8:OO A 9:OO P 9:00 A 2:OO P 7:OO A 1:OO A 4:OO P 7:OO A Noon? Noon

3:oo P 4 3 0 I’

Noon?

8 1:00 P 16 7:OO P 1 11:30A 24 6:OO P 25 6:50 A 30 5:OO A 21 Noon? 1 Noon? 19 11:30 P 3 Noon?

Atemajac, Mexico Allegan, Michigan, USA Leonovka, Ukraine Karewar, Nigeria Madhipura, India Kuttippuram, India Yurtuk, USSR Mocs, Transylvania Git Git, Nigeria Tomita, Japan Cumberland Falls, Ky., USA Hessle, Sweden Frankfort, Alabama, USA Benton, New Brrinswick, Canada Semarkona, India Chaves, Portugal Virba, Bulgaria Crescent, Oklahoma, USA Bald Mount,ain, N. Car., USA Kerrnichel, Francc Altai (Barnaul), Siheria Uberaba, Brazil Fayet,teville, Arkansas, USA Beddgelert, North Wales Johnstown, Colorado, USA Khanpur, India Utzenstorf, Switzerland Zavid, Yugoslavia Lakangaon, India Ulmiz, Switzerland Angra dos Reis, Brazil Akaba, Transjordan Domauitch, Asia Minor La Colina, Argentina Mike, Hungary

Cck Cib Chyw ChYg Cbrc Cenw cw ChY cwv Atam Chywv cw C Wht, Cbrc Aiho C AE A,am cwv Crs

cfv Ck AE ccv AE Cks A d Cwb Chyi Chygb Aleu Chykc Alan cw AE Cbrg C

30 1

EFFECTS O F METEORITES O N THE EARTH

TABLE X I I . The right ascension

Q of the moon a t the time of fall of each of 317 meteorites. (Continued)

NO.

(a)

Year

192 193 194 195 *196 *I97 198

171 171 172 17235 172% 173>/4 1743/4

1946 1939 1931 1936 1935 1908 1903

Mo. Day

Hour

Jan. June June May July Dec. June

21 23 22 29 7 15 30

Noon? 1:OO P 3:OO A 7:34 P Noon*? Noon? 2:oo P

1912 June

21

2:oo P

*200 1763/4 1904 Aug.

13

8:OO P

201 *202 203 *204 205 206 *207 *208 209 210 *211 *212 213 214

27 12:45 P 9 150 P 21 9:30 B 12 7:25 P 12 3 : 3 0 P 14 11:OO P 20 9:00 A 25 Noon? 26 Noon? 19 7:15 P 24 5:44 A 23 5:00 A 2 5:OO P 1 9:42 A

199 176!/4

17835 182)/4 1823/4 183 183% 18634 18635 187 191% 192!/4 193>/4 193% 194j/4 195>/4

1918 1914 1916 1910 1906 1935 1921 1928 1929 1912 1909 1873 1945 1933

Feb. Apr. Nov. July Nov. May Apr. June Feb. July July Sept. Feb. July

6:49 P Noon

215 1953i 1950 Oct. *216 198 1865 Jan. 217 198$/4 1946 Aug.

11 19 2

3:oo P

218 20055 1865 Sept. *219 20094 1904 Apr. *220 2013/4 1930 Feb.

21 28 17

7:OO A 6:20 P 4:50 A

*221 *222 223 224 225 226 227 *228 229

202?4 203>/4 204ji 20436 205>/4 205j/4 20536 206>/, 206j/,

1865 1938 1865 1909 1930 1939 1869 1921 1939

Aug. Dec. Aug. May Mar. May May Aug. Apr.

25 11:30 9 16 5:30 P 25 9:00 A 30 10:30 P 17 Noon? 2 7:25 P 22 1o:oo P 9 9:OO A 5 6:OO P

Name

Classification

Krymka, Ukraine C Chervony Kut , Ukraine Aleu Malotas, Argentina Cbr Ichkala, Siberia ccg Sakurayama, Japan O AE Sete Lagbas, Brazil Rich Mountain, N. Car., Chyiv USA Leeuwfontein, South Chyi Africa Shelburne, Ontario, Chygvb Canada Civ Glasatovo, Russia Ryechki, Ukraine Chyg C Rampurhat, India St. Michel, Finland Chyw ‘42 Kirbyville, Texas, USA Perpeti, India Cen 0 Pitts, Georgia, USA AE Yoshiki, Japan Chyvb Olmedilla, Spain Holbrook, Arizona, USA Chykc CW Gifu, Japan Cenk Khairpur, India Cih Meru, Kenya Ci Cherokee Springs, S. Car., USA Vengerovo, Siberia c: Cgh Supuhee, India Aran Pena Blanca Spring, Texas, USA Chyr Muddoor, India Gurnoschnik, Bulgaria ChY R Paragould, Arkansas, cg USA CVW Aumale, Algeria Cr Ivuna, Tanganyika. Aish Shergotty, India C Blanket, Texas, USA C Zindoo, Korea Kendleton, Texas, USA CdJ Cbrkv KernouvB, France cwv Shikarpur, India Ekeby, Sweden cg

302

LINCOLN LAPAZ

TABLEXII. The right ascension CY of the moon at the time of fall of each of 317 meteorites. (Continued) ~~

No.

(01)

230 231 *232 *233 *234 *235 *236 237 238 239 *240 *241 242 243

207% 2093i 209% 209% 210% 21135 21336 21436 215 218 221 22 1 22 1 2213i

244 221% 245 222 *246 222%

Year

Mo. Day

1930 1869 1943 1905 1787 1911 1924 1920 1916 1942 1921 1947 1929 1913

May Oct. Nov. Sept. Oct. Jan. Aug. Sept. July Aug. May Feb. Mar. Apr.

Hour

11 4:OO P Lillaverke, Sweden 6 11:45 A Lumpkin; Georgia, USA 7:OO P Leedey, Oklahoma, USA 25 9:30 P Modoc, Kansas, USA 2 12 3:OO P Kharkov, Ukraine 22 3:55 P Tonk, India 7 2:30 P Muraid, India 16 Noon? Kushiike, Japan 10 11:OO A Sultanpur, India 18 Noon? Kamalpur, India 5:30 P Samelia, India 20 12 10:38 A Sikhote-Alin, USSR 1 5:24 A Khmelevka, USSR 21 5:OO P Moore County,

4 15 16

1934 Dec. 1928 Oct. 1898 Oct.

247 *248 249 250 *251

225% 22835 2303/4 231x 23136

1905 1928 1903 1900 1928

Oct. 29 Nov. 12 Oct. 22 8 July 8 Apr.

*252 *253 254 *255 256

232>/4 23236 234 235 235

1940 1922 1910 1873 1920

Mar. Jan. Jan. Sept. Jan.

*257 258 *259 *260 *261 '262

235% 235% 239 241% 245% 248%

1937 1934 1932 1938 1900 1918

Dec. Mar. Aug. June May May

'

27 21 7 26 15 29 7 10 11

15 26

283 249% 1944 Sept. 23 264 251 265 251%

1803 Dec. 1902 July

13 17

266 252%

1906 Dec.

15

Classification

Name

N. Car., USA Noon? Farmville, N. Car., USA 3:OO P Oter~jy,Norway mdt. Mariaville, Nebraska, USA 8:30 A Bholghati, India 7:30 A Isthilart, Argentina 7:OO P Dokachi, India 4:OO P Alexandrovsky, Ukraine 7:15 P Narellan, New South Wales Noon? Bhola, Pakistan 8:OO P Florence, Texas, USA 11:30 A Mirzapur, India Noon? Santa Barbara, Brazil 8:00 P Aguila Blanca, Argentina 1O:OO A Rangala, India 12:45 P Mangwendi, Rhodesia 4:30 P Archie, Missouri, USA 2:OO P Kukschin, Ukraine 11:30 A Felix, Alabama, USA 9:40 A Witklip Farm, South Africa 12:30 P Torrington, Wyoming, USA 10:30 A Mtissing, Germany 9:30 A Mount Browne, New South Wales 9:30 A Vishnupur, India

cv Chyck cw

Chywv cwv Cr cw C cs AE og H Ck Aleu AE Ci 0 Aiho ChY Chyiv C Chyw AE Chygb Cibv Cho Chyi Chywv Chyib cg

cw

Chycr cg C

Atho Cbrc Cib

303


TABLE XII. The right ascension of the moon at the time of fall of each of 317 meteorites. (Continued) No.

(a)

Year

Mo. Day

Hour

12 2:oo P Forsbach, Prussia 29 9:00 A Benld, Illinois, USA 9 l:oo P Ashdon, England 23 9:30 P Bahjoi, India 3 12:45 P Changanorein, India 28 11:30 A Ellemeet, Holland 23 Noon? Po-Wang Chbn, China 23 8:00 A Macibini, South Africa 8 10:30 P Nio, Japan 7 Noon? Unkoku, Korea 28 7:25 P Lanzenkirchen, Lower Austria 278 2663i 1937 Sept. 13 2:15 P Kainsaz, USSR 279 2673/a 1949 Jan. 25 7:56 P Mezel, France *280 270 1908 Nov. 26 12:30 P Mokoia, New Zealand 2 5:30 P Jajh deh Kot Lalu, India 281 270% 1926 May 282 27336 1931 July 27 1:30 A Tatahouine, South Tunisia *283 275% 1907 Jan. 12 8:OO P Leighton, Alabama, USA 7 6:35 A Okano, Japan $284 2773i 1904 Apr. *285 27836 1927 July 13 l:oo P Tilden, Illinois, USA 1 9:15 A Simmern, Prussia *286 279% 1920 July 5 6:22 P Fenghsien-Ku, China *287 280)i 1924 Oct. 1926 June 26 4:30 P Lua, India *288 283 4 6:30 P Colby, Wisconsin, USA 1917 July 289 285 290 2853i 1897 May 19 7:45 P Meuselbach, Germany *291 285% 1896 Feb. 10 9:30 A Madrid, Spain "292 286 1790 July 24 9:oo P Barbotan, France 1938 June 16 8:45 A Pantar, Philippine 293 293 Islands 9 3:OO P Tromoy, Norway 1950 Apr. 294 293 2 7:35 A Washougal, Washington, 295 294 1939 July USA 1:30 P Boriskino, USSR $296 297 1930 Apr. 20 *297 29736 1945 Sept. 17 Noon? Soroti, Uganda 1 Noon? Sharps, Virginia, USA 298 297% 1921 Apr. *299 29835 1900 June 15 Noon? N'Goureyma, French W. Africa 300 299 1924 June 19 8:00 A Olivenza, Spain *301 29936 1902 Sept. 13 10:30 A Crumlin, Ireland 1930 Nov. 25 10:53 P Karoonda, South 302 300 Australia *303 301 1934 June 28 8:OO P Saeovice, Czechoslovakia 304 30191 1924 July 16 5:45 P Forksville, Virginia, USA 267 *268 *269 *270 *271 272 $273 274 275 *276 *277

25236 25436 25435 259 25934 260ji 261% 261% 26235 264M 265

1900 1938 1923 1934 1917 1925 1933 1936 1897 1924 1925

June Sept. Mar. July July Aug. Oct. Sept. Aug. Sept. Aug.

Classification

Name

Ci ChYW Chyw og cg Azan AE Aleu cc C ChY

cs Chyiv Cr Ckv A d

Cr 0 Chykc Ob Chygc Chyg Cc(Asiderite) ChYW Chyc

304

LINCOLN LAPAZ

TABLE XII. The right ascension a! of the moon a t the time of fall of each of 317 meteorites. iContinued) No.

(a)

*305 302 *306 *307 *308 *309 310 311 *312 313 314

Year

1932 D e r .

1897 1940 303)/4 1910 30636 1794 30635 1901 306)5 1922 307 1898 309>/4 1915 31055 1941 30255 303

*315 311 316 312% *317 3133i

Mo. Day 1

Hour

Name

5:OO P

Witsand Farm, S.W. Africa Higashi-KGen, Japan Erakot, India Baroti, India Siena, Italy Hvittis, Finland Tjerebon, Java Quesa, Spain Meestcr-Cornelia, Java Black Moshannon Park, Pa., USA Plantersville, Texas, USA Beyrout, Syria Majorca, Balearic Isles

Aug. 11 Noon? June 22 5:OO 1' Sept. 15 1O:OO A June 16 7:OO P Oct. 21 Noon July 10 10:30 P Aug. 1 9:oo P June 2 6:OO A July 10 6 3 0 A

1930 Sept. 1921 Dec. 1935 July

4 31 17

4:OO P 3:45 P 11:35 A

Classification Chyw Cg Cr Chyw Cho Ccnk ChY Of Cbr Cck Chywv Chyb 0

of such techniques not only provides data showing discrepancies d much less probable than the d = 27.5 of the unsmoothed data; but also discloses a new feature of great significance. I n Fig. 3 (111), the upper (dotted) and the lower (dashed) curves exhibit the smoothed distributions of the group, CrIr,of 317 meteorites in 5' and 30" sectors, respectively. The sinusoidal dependence on the moon's nearness in direction to that of Jupiter's line of apsides shown in the lower curve in Fig. 3 (111) is so pronounced that, in view of the wellknown concentration of the perihelia of asteroidal bodies near Jupiter's perihelion, i t might be concluded that all of our 317 meteorites belong t o a slow, solar system category of asteroidal meteorites. There is, however, a possibility that must not be overlooked. While one might expect asteroidal type meteorites to show such a dependence as that exhibited in the lower curve in Fig. 3 (111),it would be anticipated t,hat the infall of swift interstellar meteorites, being unaffected by the moon, would show a nearly constant frequency for all values of a. Clearly, a subgroup, Cr, of meteorites possessing a frequency of fall independent of a, if present in our totality of 317 meteorites would not mask the sinusoidal distribution curve which it is natural to attribute to the subgroup, CII, of solar system meteorites present in CIrr. Furthermore, if C, were removed from CIII,the remaining subgroup, CII, should exhibit the sinusoidal dependence t o be expected in the case of a group of solar system meteorites There thus arises the question as to whether any appreciable subgroup,


305

7

6 5 4

3 2

1 0

30 20

10

0

z ui

%

z

4

2

3

E

-

n $

2

1

s o z 20 10

0

3 2

1 0

20 10

0 0

30 60

90 120 150 180 210 240 270 300 330 01,

0

30 60

in degrees

FIG.3. Distribution of 152 interstellar meteorites (I),of 105 solar system meteorites (11),and of the totality of these 317 meteorites (111) with respect to the right ascension, 01, of the moon a t the time of fall.

306

LINCOLN LAPAZ

Cr, exhibiting a frequency of distribution essentially independent of a, can be removed from the group, CIII,without destroying such sinusoidal type dependence as that shown in the lower curve in Fig. 3 (111). On the basis of objective criteria which are described in detail in Appendix 11, a subgroup, CI, of 152 meteorites (those whose numbers carry prefixed asterisks in Table X I I ) was removed from the original group, CIII,and then smoothed distribution curves were prepared not only for Crbut also for the remainder subgroup, (711, of 165 meteorites, in exactly the same way as were the curves shown in Fig. 3 (111). Inspection of the resulting distribution curves in Fig. 3 (I) and (11), respectively, will disclose that the frequency of fall of the 152 meteorites in CIis very nearly independent of a, while that of the other subgroup, CIr, of 165 meteorites exhibits more perfect sinusoidal dependence on a than was shown by the original group, CIII,of 317 meteorites. I n fact, an even more exact dependence on the moon’s nearness in direction to that of Jupiter’s line of apsides is shown by the lower curve in Fig. 3 (11) than was shown by the analogous curve in Fig. 3 (111). In view of the fact (pointed out in Appendix 11) that, with very few exceptions, the meteorites entering into the subgroup Cr have orbital characteristics proclaiming them of interstellar origin, or present mineralogical or other features consonant with a nonsolar system origin; and that precisely the reverse may be said in regard t o almost all of the meteorites entering into the subgroup, CII, it would seem permissible to conclude that we have obtained rather convincing evidence of the existence of both solar system and interstellar meteorites. 6.6. B-Processes and the Hyperbolic Velocity Problem

The existence of two distinct meteoritic subgroups, CI and CrI,validates conclusions long ago reached by such pioneer investigators as G. von Niessl and W. F. Denning. Futhermore, examination of the positions on the celestial sphere of the cosmic quits of fireballs associated with certain of the most intensively investigated meteorite falls in CI, will only be found to redirect attention t o specific areas of hyperbolic radiation whose existence was signalized by von Niessl and Denning more than 75 years ago. At that early date, no notions of B-associations and B-processes were in existence. The writer wishes to conclude this section by pointing out the near coincidence in position between one of the most prolific ecliptic centers of hyperbolic radiation of von Niessl and Denning and the nearest of the aggregates, I1 Sco, contained in a tabulation recently published by Morgan et al. [142]. The significance of this relationship can best be appreciated in terms of the recent remarkable papers by Blaauw and Morgan [143, 1441 on the space motions of AE Aurigae and


307

Columbae. Blaauw and Morgan have shown that the observational data suggest that these two stars were formed in the same physical process 2.6 million years ago and that they now are moving away from their common point of origin in almost exactly opposite directions, each with the same high speed of 127 kmlsec. It is inconceivable that B-processes of such extreme violence should fail to eject, along with those giant masses rendered visible by their stellar character even in empty space, an enormous quantity of lesser, invisible debris, the smaller and far more numerous members of which comprise a new addition to the category of interstellar meteoritic material. It seems reasonable t o assume that equally potent B-processes must have occurred, possibly a t various times and at many centers in a n association a s rich and as widely extended as I1 Sco. If this be granted, then we have in the 11 Sco-aggregate a relatively nearby source of hyperbolic material situated, it must be emphasized, in proximity t o the ecliptic. I n view of the ultra-high-speed of masses ejected by B-processes, as revealed by Morgan and Blaauw’s study of AE Aurigae and p Columbae, one may infer that only rarely will meteoritic ejecta from the I1 Sco source survive, more than momentarily, head-on flight through the earth’s atmosphere. The great preponderance of direct orbits for the hyperbolic Scorpionids is thus immediately explained. The reader will recall that precisely the preponderance of direct over retrograde motions and the concentration t o the ecliptic shown by hyperbolic meteors have been cited in attempts t o discredit the very existence of interstellar meteorites. Two points remain that seem to merit emphasis. I n the first place, the results of this section give ample evidence that most meteor-orbit problems ought t o be regarded as three-body problems and not as the much simpler two-body problems alone attacked and solved by those classical procedures that have flooded the literature of both visual and photographic meteoric astronomy with extensive lists of assertedly highly accurate orbits. I n the second place, the ballistic potential of even the tiniest meteoritic particle, if i t be endowed with a strongly hyperbolic velocity, is formidable. p

6. CRATER-PRODUCING METEORITEFALLS

The refusal by scientists of the eighteenth century to give serious consideration t o the possibility that “stones might fall from the sky” is frequently cited as a prime example of learned blindness. One may feel sure, however, that scientific opinion of the future will concur in crediting meteoritics with the dubious honor of providing two still more inexplicable examples of scientific obtuseness: First, the failure to recognize until

308

LINCOLN LAPAZ

well into the twentieth century that certain remarkable topographic features had their origin in large-scale meteoritic impact and represent, indeed, an impressive and permanent testimonial t o the sensational nature of the effects produced by the most violent of all the interactions between the earth and infalling meteoritic material; and, second, the even more astonishing refusal of some geologists as late as 1953 to accept as a meteorite crater the first, terrestrial feature t o be recognized as such [145]. This refusal is the more singular because the crater in yuestion, that near Canyon Diablo, Arizona, is certainly a textbook example par oxcdlence of a meteorite crater, presenting as it does in clearly recognizable form almost all of the characteristics of this unique category of topographic. features. The current situation is rendered even more confused by the fact that, simultaneously with refusal to recognize the most obvious meteorite crater as such, recently both scientific and popular journals uncritically have publicized as meteorite craters a number of topographic features, none of which properly qualify for such as identihcation. The authentication of every meteorite crater recognized as such by meteoriticists rests on fulfillment of one or both of the following identification criteria: ( I ) Actual discovery in or near the crater of meteorites, either unaltered or, possibly, oxidized t o a lesser or greater degree (as a t IJssuri (Sikhote-Ah), Canyon Diablo, Wolf Creek), or of metamorphosed materials definitely known t o have resulted from meteoritic impact (e.g., the nickel-iron-bearing silica glass a t the Henbury and Wabar meteorite craters). (2) Actual observation of meteorite falls having earth-impact points in the watered areas (as in the case of the unparalleled Podkamennaya Tunguska fall of June 30, 1908, in Siberia).

In addition to the above-named principal criteria, auxiliary criteria which must be satisfied include the presence of bilateral symmetry (since strictly vertical infall is most improbable) and convincing evidence th a t the actual impact and explosion of a large meteorite is involved, such as a more or less regular decrease in the amount and the size of ejectamenta with distance outward from the crater; faulting, often roughly radial in character; the presence of radial percussion ridges or radially aligned jets of ejectamenta, including not only rock debris, but meteoritic fragments as well, or both; and the upturning and even the overthrow of strata, a t least where the impact has occurred in sedimentary beds. I n the case of such widely publicized craters as New Quebec (Chubb) in Canada, the Crater Elegante of Sonora, Mexico, and other craters recently asserted to be of meteoritic origin, neither the principal nor the


309

auxiliary criteria are satisfied. hi fact, in lieu of such evidence as is customarily presentfed to warrant assignment of a meteoritic origin to a crater, only less natural and less cogent reasons have been given for designating the Chub2, crater as a meteorite crater. For example, the mere existence of magnetic anomalies in rock admittedly rich in segregations of magnetite and allied minerals has been advanced by Meen (see [(ill, p. 24) as proof that Chubb is a meteorite crater. Again Millman, in summarizing the results of his exhaustive and valuable study of the profile of the Canadian crater [146],cites, as evidence considerably strengthening the meteoritic impact hypot’hesis of the origin of Chubb, what he regards as “ t he close agreement between the New Quebec crater and the normal series of explosion craters . . . ”-specifically, the series of terrestrial and lunar craters studied i n detail by Baldwin [147]. Both Baldwin’s ident’ification of the craters on the moon as explosion features resulting from the infall of meteorites and Millman’s extension of Baldwin’s argument to include the New Quebec crater rest, on a supposed concordance between various geomet>ricrat,ios exhibited by t he craters on t,he moon and the values of analogous geomet,ric ratios as measured in terrestrial craters known to have originated from explosions. The crucial point in establishing the supposed correlabion, as Baldwin himself recognized (see [147], p. 131), lay in bridging over the vast gap between relatively small military explosion cratjers, having diameters of at most a few hundred feet,, and lunar craters, having diamet’ers ranging from a few miles to as much as 146 mi. Baldwin asserted that the gap in question could be bridged over by use of the four terrestrial meteorite craters listed in his Table 5 (see [147], p. 125). But, if the correct measurements made at the most t’horoughly invest’igat>edof t’hese terrestrial meteorit,e crat.ers are employed, t,he concordance he sought to establish is found t.o break down completely. In view of t.he wide-ranging theore& ical considerations which Urey [148] and others have based on the coiicordance supposedly established by Baldwin, it seems well worthwhile t,o indicate precisely why it is invalid. Reference t o Table 5 and to the details concerniiig the main Odessa crat,er given in Chapter 4 of The Face qf the Moon, will disclose that by depth, Baldwin means t,he original interior relief of a crat>er;l* for example, in the case of the main crater at Odessa, he combines his estimak of -10 ft 11 Where several terrestrial meteorite craters of widely differing agcs and suhjec~t to a n annual aggraclat,ion variable within wide limits are under consideration, it is perfectly obvious t,hat use of any “depth ” other t.han t,he original interior crater relief cannot possibly he justified. So much the more so is this true as regards a cornparison involving not only several t,errestrial meteorite craters, but, also a series of crat,ers on a remote satellite where the very notion of aggradation is undefined and its annual rate is wholly unknown.

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for the original rim height above the surrounding plain (see [147], p. 73) with a figure of 90 ft for the distance at which the crater bottom originally lay below this plain, to obtain the depth of 130 ft entered in Table 5. It will be noted with surprise that this is not at all the procedure used to obtain the depth figure given in this table for the Barringer meteorite crater a t Canyon Diablo. Baldwin’s own estimate of the original rim height here is 300 ft. The original depth below the surrounding plain of the bottom of the Barringer meteorite crater is shown by the logs of the numerous drill holes put down inside the crater to amount to at least 1200 ft. Consequently, the original interior relief at the Barringer crater was at least 1500 f t [1491. In place of 1500 ft, however, Baldwin uses only 700 ft as the depth, thereby obtaining a (log-depth-log diameter) point whose coordinates (2.845, 3.618) place it exactly on the fundamental correlation curve he is seeking to establish (see [147], p. 132, Fig. 12). The correct (log depth-log diameter) point (3.176, 3.618), however, falls nowhere near the correlation curve in question-how wide is its departure from this curve will not be appreciated until one replots the data on a (depth-diameter) rather than on a (log depth-log diameter) diagram. Furthermore, the results obtained from the exhaustive program of drilling and excavation a t the Odessa meteorite crater, discussed in Section 6.1.2, quite discredit the almost perfect fit to the correlation curve of the main Odessa crater point, as plotted by Baldwin in his Fig. 12. Consequently, the concordance basic to all of Baldwin’s conclusions breaks down precisely in the region he recognized as critical. Even had Baldwin been successful in explaining Ebert’s Rule, and similar empirical relationships long ago discovered by the German selenographers, as consequences to be expected in case the lunar craters were of meteoritic impact origin, he would not have proved that all the craters on the moon are meteorite craters; for, as has been pointed out by J. Wasiutynski, relationships like those discovered by Ebert are also corollaries of another quite different theory of the origin of the lunar craters, namely, Wasiutynski’s convection-current theory of these features [150]. In the writer’s opinion, Wasiutynski’s new theory not only is physically the most plausible so far advanced (resting as it does on the fertile laboratory experimentation conducted by H. Benard and his associates) ; but also has the great additional advantage of not requiring (as the meteorite impact theory necessarily does) a random distribution of the centers of the lunar craters. That the distribution of the ordinary (i.e., the nonray) craters on the moon is not a random one was proved in 1942 by Scott [151]. Finally, as the writer has recently pointed out [152], Wasiutynski’s theory not only leads t o such empirical relations as Ebert’s


311

Rule, but also predicts both hexagonal crater shapes of the sort observed for many of the best preserved lunar craters and the highly significant rhomboidal network of dikes and rills-Puiseux’s network-discovered in 1907 by the French astronomer, P. Puiseux [153]. Unquestionably, a feature as remarkable as the Chubb crater merits the most exhaustive investigation, particularly on the part of those who seem determined to prove it is of meteoritic origin. It has been suggested that glaciation has obscured or destroyed all decisive evidence of the sort detailed in the principal criteria (1) and (2) given earlier in this section. However, as the writer recently pointed out [154], even as regards the exterior of Chubb crater, no support can be given to such a suggestion in view of discovery of typical meteorite-crater glass a t Mount Darwin in Tasmania in an area where the crater whence the glass in question was ejected is presumed “ t o have been destroyed by glacial erosion” (see Prior and Hey [22], p. 423). Certainly the suggestion in question is totally inapplicable t o the interior of the Chubb crater in view of the convincing explanation which has been given by Harrison [155] as to why the New Quebec crater was not filled with glacial debris; namely, that the continental ice sheet in part was deflected by the rim of the Chubb crater, and in part simply slid across the smooth, impenetrable ice floor provided by the frozen lake within this crater so that relatively little, if any, glacial debris was introduced into the crater by passage of the ice sheet. Furthermore, this ice sheet itself, together with the climatic conditions existing in far-northern Ungava, has spared the interior of the Canadian crater from such massive accumulations, chiefly of eolian origin, as mask much of the interior of the Barringer meteorite crater. Consequently, if the Chubb crater is a meteorite crater, it must contain within itself in relatively unadulterated state such evidences of meteoritic impact origin as have been found in profusion in the Barringer meteorite crater and in other recognized meteorite craters (see Sections 6.1 and 6.2). Of course, the deep water fill in Chubb constitutes a handicap to exploration of the interior of the Canadian crater, but modern devices and techniques for submarine sampling have overcome much more formidable obstacles. Consequently, until (and unless) systematic probing of the interior of the Chubb crater reveals undoubted evidence of a meteoritic origin, this remarkable Canadian feature should be excluded from the category of authenticated meteorite craters. 6.I . The Barringer Meteorite Crater

Approximately 20 mi west of Winslow, Arizona, near the Canyon Diablo, the large bowl-shaped depression of the Barringer meteorite

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crater is located. From a distance, the rim of this feature appears to the ground observer as a chain of low, hummocky hills, lighter in color than the surrounding reddish terrain. Viewed from the crest of the rim, the depression exhibits an approximately circular outline. This circularity is, however, largely illusory as air views of the crater have shown it to be

FI(:.4. Air view showing the plan-form of the Barringer meteorite crater.

squarish in shape (see Fig. 4), a fact recently stressed by Zimmerman 11561. This curious depression attracted the attention of the earliest settlers in the region and, as long ago as 1873, was named “Franklin’s Hole” by George M. Wheeler in honor of a well-known guide and Indian scout of this period. The topographic feature did not come under scientific scrutiny, however, until 1891, when the discovery of numerous masses of meteoritic iron scattered about the rim of the crater was announced by

EFFECTS O F METEOIZIT’ES Oh’ T H E EARTH

313

Footc [ 1571. Ititerest in the locality so011 \vas intensified by Moissaii’s subsequent ronfirmation of the discovery of small black diamonds in the sideritic material recovered a t the crater [158]. The Arizona depression has a major diameter of 3950 ft and a minor diameter of 3850 ft. The outer slopes of the crater rim rise a t a very gentle gradieut from the level of the dcscrt plain into which the crater is depressed; the itiiicr slopes, 011 the coiitrary, are exceedingly precipitous, so much so that only their lower portions have been covered by the widebpread deposit of talus bottoming the crater (see fiig. 5 ) . The height ot

FIG.5 The interior of thr Barringrr xnrteoritr cmtcr.

the rim above the surrounditig pl:iiii varies from 120 t o 160 it, while the greatest depth of the basin, measured from the crest of the rim t o the lowest point in the bed of the lake which occasionally occupies the basin, is approximately 570 ft. This depth is, however, only a fractioii of the original depth of the crater, as revealed by the mining operations carried 011 w i t h it; for thc ~ ~ u n i c r o u shafts s and bow holrs sunk in the interior of the carat er, after traversing crushed and metamorphosed masst’s of country rock aiid acwmiulations of rock flour, have in many cases penetrated into solid horizontally-bedded sedirnciitary strata. Undisturbed sandstone has thus been located a t depths of 800 ft or more below the central portion of the present crater floor (see [149], pp. 461-498). The country rock of the level plain on which the crater is located WHsists of the following horizontally-bedded geological formations: (1) The

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Kaibab (Permian) limestone, which in the Canyon Diablo area is approximately 250 ft thick, forms the surface of the plain. (Upon this surface, a few remnants of the reddish-purple Moenkopi (Triassic) formation are present as small, flattish hills, rising but slightly above the general level of the plain surrounding the crater.) (2) Below the Kaibab limestone lies approximately 1000 feet of soft, white Coconino (Permian) sandstone. (3) The Supai redbeds (Lower Permian), of variable thickness, underlie the Coconino sandstone. The outer slopes of the crater consist of loosely consolidated, generally angular, fragmental material, unquestionably derived from beds once filling the space now occupied by the crater. These outer slopes are covered with rock ejectamenta varying in size from great masses of limestone weighing approximately 4000 tons (such as Monument Rock and Whale Rock) to microscopic particles of crystalline quartz, known as “silica” or rock flour. The latter, as Barringer, Tilghman, and Merrill first clearly pointed out, had its origin in the comminution of the Coconino sandstone by the violence of the meteorite impact forming the crater. Some limestone fragments weighing from 50 to 100 pounds have been found a t distances of 1% to 2 mi from the crater, and both rock ejectamenta and meteorites have been found out to distances of at least 6 mi. The amount of rock material dislodged and wholly or partially thrown out of the crater has been estimated at between 300,000,000 tons [159] and 1,000,000,000 tons [160]. The extraordinary violence of the crater-forming process is shown not alone by the estimates just quoted from Barringer and t)pik; but also by such evidence as the following: (1) I n the south wall of the crater, a block of country rock, extending for half a mile as measured along the crater rim, and estimated by Barringer to weigh between 20 and 30 million tons, has been bodily uplifted through a vertical distance of approximately 100 ft. (2) As previously noted, the strata in the Canyon Diablo area, where undisturbed, are horizontally bedded. But, as exposed in the steep, inner walls of the crater, these same strata everywhere dip radially outward at angles ranging from 5” to 80” or even more. (3) At a number of locations, the entire succession of rock beds several hundred feet thick is observed to be faulted. With one exception, all of the many scientists who have closely examined the structure exposed a t the Barringer meteorite crater have reached the conclusion that these faults either are roughly radial, or are approximately at right angles to crater radii. The striking bilateral symmetry, which is evident not only in the dips of the uptilted stratified beds exposed along the inner walls of the crater, but also in the distribution of the major rock ejectamenta, was early noted by Barringer (see [159], p. 5). To some extent, this same bi-


315

lateral symmetry also is present in the distribution of the tens of thousands of pieces of meteoritic iron that have been found on the outer slopes of the rim and on the plain surrounding the crater; although nothing analogous to the very rich concentration of relatively small sideritic specimens present in the northeast quadrant has, as yet, been found elsewhere about the crater. The solid irons recovered about the Barringer meteorite crater range in mass from tiny gravel-sized specimens to individuals weighing as much as 1406 pounds. In addition to the almost unaltered nickel-iron, the meteoritic debris found at the Barringer meteorite crater includes an enormous number of fragments, large and small, of reddish-brown to chocolate-brown iron oxide, occasionally exhibiting a greenish stain as a result of the presence of nickel hydroxide; and, in far less profusion, rounded or pear-shaped masses of more or less incompletely oxidized meteoritic material-the so-called “ shale balls”-ranging in size from that of a marble to huge masses weighing hundreds of pounds. Specimens of all these types of meteoritic material, but particularly of the last variety, have been found buried not only in the heterogeneous aggregate of rock fragments and rock flour constituting the rim, but also in the moraine-like ridges of similar composition flanking the outer slopes of the crater. The attentive observer of the helter-skelter manner in which sizeable fragments of sandstone and limestone, “ rock flour” composed of particles of microscopic dimensions, and masses of meteoritic material of far greater density than that of the country rocks occur in the deposits just alluded to, will concur without hesitation in the following pronouncements of G. P. Merrill (see [149], pp. 466 and 495), one of the ablest and most critical students of the Barringer meteorite crater: The position they [ = the various components of the deposits under consideration] occupy is such as can be accounted for only on the supposition that all the material composing the deposit was in the air at the same moment of time and was deposited “pel1 mell,” wholly without order or reference to gravity . . . It is impossible to account for the position of these last [ = the shale ball irons] in any other way than to assume that they fell a t the same period of time as the material in which they lie embedded. The difference in specific gravity of the various materials is such that it is inconceivable that they should have traveled together for any great distance. Their association may be best explained on the assumption that all were poured out together over the crater rim . .

.

The above well-considered conclusions, voiced without qualification by a most conservative scientist who had had unparalleled opportunities to gain first-hand knowledge of subsurface conditions at the Barringer meteorite crater, are in striking contrast to the precipitate pronouncements of those few crater investigators who have advanced the unsup-

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ported conjecture that the wealth of meteoritic materials of diverse types found a t this crater stand in no causal relation to its origin; and, in fact, fell tens of thousands of years after its formation. I n addition to the mac,rometeoritic specimens considered above, a great deal of what might be termed micrometeoritic material, both metallic and oxidized, was first discovered half a century ago in and around the Barringer meteorite crater by its pioneer investigators, Barringer and Tilghman. The latter, in particular, as early as 1905 by use of a magnetic separator, recovered nickel-iron‘grariules not only from the fill inside the crater, butj also from samples collected on the north and south slopes of the crater rim. H e estimat’edthat the metallic nickel-iron granules thus isolated occurred ill the proportion of 1/80 t o 1/4 ounce per ton of material worked, results in line with modern determinations of the average concentration of such granules. The cJaims recently made by H. H. Nininger t-o have first identified such granules in 1948 after they had eluded all earlier investigators can best be judged against the background of the many earlier publications dealing with the collection, study, and identification of such granules collated and published by the present writer in 1953 [63]. Equally unjustified is t8hecredence accorded in some quarters to claims that work a t the Barringer meteorite crater in 1948 first suggested that, the impact of giant, crat>er-forming meteorites with our globe develops heat sufficient t o fuse and partially to vaporize immense masses of both projectile and target material with the consequence that much of the resulting melange is ejected from the impact crater and widely scattered about it. All of these ideas not only were carefully discussed in Merrill’s pioneer paper of 1908 on the Barringer meteorite crater [149], but were again elaborated in great, detail in the classical 1933 paper of Spencer [lSl] on the millions of tiny metallic nickel-iron spherules found in the closest association with masses of congealed silica glass a t the Wabar and Henbury meteorite craters; and on the curious, roughly globular, somewhat altered “crinkled peas” of similar origin and association found in the vicinity of the latter craters. If confirmation were needed of the validity of the interpretation which Barringer, Merrill, and others long ago placed on the then quite unique meteoritic granules, rock flour, and silica glasses discovered a t the Barringer meteorite crater, it is to be found in Spencer’s exhaustive discussions of similar meteoritic impact evidence later discovered a t the Henbury and Wabar craters. Only the few who now hypothecate a nonmeteoritic origin for the Barringer meteorite crater seem to have overlooked the peculiarly significant role played in discrediting all such hypotheses by the early discovery a t this crater of meteoritic granules associated not, only with rock flour (unquestionably derived from the

EFFECTS OF METEORITES O N THE EARTH

317

Coconino sandstone not by simple disintegration but, in the opinion of Merrill (see [149], p. 473), “through some dynamic agency acting like a sharp and tremendously powerful blow,” a view concurred in by Rogers [ 1621 and all other mineralogists, pet,rologists, and geologists who have carefully examined the rock and rock powder in question), hut also with heat-metamorphosed sandstone showing all gradations from the unaltered Cocoriino to various types of silica glass [l(i8]. As will be pointed out in the next sect,ion, certain significant resemblances bet,ween the Barringer meteorit,e crater and the second crater i n the world identified as of meteoritic origin, that near Odessa, Texas, can be notJiced at, once. Radiometric devices like the met>eoritedetectors, as well as visual observation, have cont~ributedto the list of similarities bet,ween t,he first two terrest,rial craters recognized as meteorite craters; for, as was noted during the instrumentd searches far buried meteorit.es made by the Ohio State University Meteorite Expeditions of I939 and 1941 t o Arizona and Texas, in addition t,o the quite localized signal maxima testifying to the presence of buried meteoritic material, the meteorit,e detectors occasionally revealed radiometric anonialies extending over areas of hundreds and even thousands of square feet. Several of the most marked anomalies found near the Odessa crater in September, 1939 later were rediscovered during a magnetometer survey of this region made by the Humble Oil and Refining Company as a contribution to the crater-exploration program initiated by the Bureau of Economic Geology of the University of Texas. Two of tfhemost prominent of these anomalies were later excavated and shown to be associated with subsidiary meteorite craterlets that had been so completely filled up by ejectamenta from the main crater and normal aggradation subsequent t o the infall of the Odessa meteorite that no trace of their existence was visible on the surface. In July, 1950, the Institute of Meteoritics of the University of New Mexico conducted a radiometric survey of one of the most pronounced of the anomalies earlier located near the Barringer meteorite crater. The isogramI2 derived from this survey (Fig. 6) confirmed that in extent, 12 The secondary (northern) high shown in this isogram was discovered before the primary but less symmetrical southern high was detected. The latter was, i n fact,, found during a systematic survey about the northern high. In the expanded survey of the entire anomalous area then undertaken, it proved expedient to choosr the intersection of the axis of the northern high with the axis of the southern high as t,he origin of the zy-coordinate system to be employed in the final survey. The directions given for these axes are only approximate since they are based on magnetic directions and the needle is not to be trusted over such terrain as that shown in Fig. 5. The scale of Fig. 5 can be det>ermined from the fact that t,he center of the circularly symmetrical northern high is 100 f t northeast of the origin of the xucoordinate system.

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shape, and radiometric relief, the Arizona anomalies were quite similar to those earlier discovered in Texas which were later proved to be the surface indications of small buried meteorite craters. To date, only shallow surface trenching has been conducted on the Pluto anomaly, as the feature shown in Fig. 6 is now called. Pending such

FIQ. crater.

a program of excavation as was instituted in and around the Odessa crater, the cause of the Pluto anomaly must remain undecided. It is, of course, possible that long-continued circulation, through subsurface fissures produced by the shock of the main crater-forming impact, of rain wash and of such scanty ground water as characterizes the Canyon Diablo region has brought about concentrations of oxidized and hydrated meteor-


319

itic materials sufficient to produce the anomalies detected a t the surface in the Canyon Diablo area. The close similarities between the radiometric indications observed about the Arizona and Texas craters, however, strongly suggest that both sets of anomalies have similar origins. On the basis of such evidence as that briefly summarized above, the Barringer meteorite crater universally is adjudged by meteoriticists as the best authenticated and the most representative of t,he recognized meteorite craters of the worId. 6.1. The Odessa Meteorite Crater

Located about 10 mi west-southwest of Odessa, Texas, is a much smaller and unquestionably a much older meteorite crater than the one near Canyon Diablo, Arizona. The Odessa crater now consists of a shallow basin, somewhat elliptical in outline, with an average diameter of about 550 ft. Scarcely noticeable from a distance, the crater rim rises only about 5 to 8 ft above the surrounding plain. Viewed from the rim, however, the interior of the crater is more impressive. As at the Barringer meteorite crater, the rim is an assemblage of loosely consolidated, generally angular fragmental material, unquestionably derived from beds once filling the space now occupied by the crater. Also as in the case of the Barringer crater, the inner walls are much steeper than the external slopes. Many solid iron meteorites, varying in size from the most minute fragments to specimens weighing hundreds of pounds, have been found on or outside the rim of the main crater or in smaller subsidiary craters located nearby. The siderites found at the Odessa crater seem to be identical in composition and structure with the Canyon Diablo octahedrites. Furthermore, as at Canyon Diablo, the octahedrites so far recovered at Odessa can be separated into two quite distinct categories: the first, comprised of individuals hurled out of the main crater in parabolic paths by the explosion of the principal crater-forming mass; and the second, consisting of what the writer [164] has termed ‘‘ outriders,” i.e., subordinate masses that trailed the principal mass in its swift flight through the atmosphere and reached the earth with some appreciable fraction of their original cosmic velocities. In most cases, the outriders were shattered to a greater or lesser degree by their impact with the earth target and thus became subject to excessive oxidation which resulted in the formation of shale balls identical in appearance and internal structure with those found in profusion near the Barringer meteorite crater. In one exceptional case, however, a metallic Odessa outrider, weighing approximately 200 pounds, was found buried at a depth of 43 in. in apparently solid rock which showed indisputable evidence of intensive metamorphic action where it was in contact with the meteorite. In fact, this outrider, as finally

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chiseled out of the surrounding rock, was found to be completely coated by a n armor apparently consisting of a genuine melt of country rock and nic kel-iron. The Bureau of Economic Geology of the University of Texas, to which great credit must go for taking the initiative in thoroughly investigating the Odessa crater, has reported that on the basis of the elaborate program of excavation and drilling which began as a W.P.A. project, the following materials have been recognized : (1) Impact ejecta thrown completely out of the crater; ( 2 ) folded, faulted, elevated, tilted, thrust, and otherwise dislocated rock strata in the rim of the crater; (3) post-meteorite impact fill, consisting chiefly of silt and fine sand, within the crater itself; (4) impact ejecta which fell back into the crater opening shortly after its formation; (5) BarringerMerrill type “rock flour” formed in place inside the crater; and (6) undisturbed country rock beneath the rock flour zone ( 5 ) . We shall discuss each of the above categories in sequence, calling attention to the chief similarities between the phenomena observed a t Odessa and those earlier detected a t the Barringer meteorite crater: (1) The accumulation of rock debris ejected by the impact of the Odessa meteorite completely surrounds the main crater in a deposit thickest in the rim and quite uniformly thinning out with distance from the crater. The surface debris consists chiefly of blocks of limestone often solidly cemented together by caliche. Pits and trenches extending outward from the rim disclose that large masses of shale are included among the limestone blocks, the maximum size of the blocks in each case being of the order of 3 to 4 ft. The existence of secondary accumulations of caliche cementing the ejected rock is evidence of the remoteness in time of the crater-forming impact. The peripheral deposit of ejecta contains rock specimens which have beeii identified as coming from Cretaceous formations t ha t must have been buried a t depths of nearly 100 ft prior to the meteorite impact. Except for the difference in scale incident to the quite different masses of the meteorites that fell a t Canyon Diablo and a t Odessa, there is the closest similarity between the out-throw deposits around the two craters. (2) The Cretaceous formations underlying t,he Odessa crater normally have the same horizontal positions as the strata in which the Barringer meteorite crater was formed. As a result of the impact of the Odessa meteorite, all strata in and immediately adjacent to the present Odessa crater, down t o a depth of approximately 90 ft, were moved from their original positions. As revealed in the inner walls of the Odessa crater, the rock strata were lifted, broken, folded, and faulted. In particular, one massive limestone, the top of which normally lies about 22 ft below the level


32 1

of the Odessa plain, was uplifted to a maximum extent of approximat,eIy 25 ft, evidence of the violence of the meteorite impact strictly comparable with that revealed in the arched-up south wall of the Barringer crater. Again, limestone strata once horizontally bedded show dips ranging up to go", analogous to those observed at the Barringer crater. Outside the crater, the dips in the rock strata produced by meteoritic impact) rapidly flatt>enout, as has been conclusively shown by the logs of test wells drilled on the north and south rims of the Odessa crater. In fact, in the north crater rim, the limestone stratum levels out t20almost, its normal position within a horizontal distance of no more than 110 ft. The extensive excavations carried out in the Odessa crater reveal rock beds so shattered, compressed, and disturbed that it is now quite impossible to determine accurately their original thickness or even to interpret their original st,ructural attitude. As evidence of the extreme structural dislocation present in rock formations a t the Odessa crater, a section of beds, including a resistant, fossiliferous limestone, has been folded into an asymmetrical a1it.icline, and evidence of thrust-faulting also is clearly shown. ( 3 ) The original depth, below outside ground level, of t.he impact. crater a t Odessa was about 106 ft. At the present time, the basin is filled to within 5 or 6 ft of the level of the surrounding plain, another indication of the great age of the basin. The lens-shaped fill within the cratclr (+onsists of fine sand mixed with red, incoherent silt,. Underlying t h r silt layer is a light-colored stratum consisting in part of silt,, hut, includitig material washed in from the rim of the crater, the whole bciiig partially cemented by caliche which has formed a t t'his level. The total thickness of these two post-met,eorite impact deposits is approximately 7lj ft. (4) Below the 75-ft t,hickness of upper fill iii t,he central part of t.he crat,er, there has been found a stratum of fragnieiitJalrock exteiidiiig down a distance of riot more t,hari 15 or 20 ftJto the original hottom of the crater. While this fragmental rock material is coarser in outlying areas aiid finer in the central portion of the crater, it is always easily dist.inguishable from the still finer material above and helow it,. (5) Exactly as a t the Barringer meteorite crater, rock flour has been formed from sand grains completely and suddenly shattered by the violent impact of t,he crater-forming meteorite. The rock flour exhibits the same almost incredible fineness earlier commetited on by Barringer, Tilghman, and Merrill in connection with their investigations a t the Arizona crater. The zone of rock flour at Odessa is thickest close to, butJ somewhat northeast of, the center of the original floor of the crater, thinning out in all directions so as to form a lens-shaped deposit lying well within the outer margins of the crater. Near the center of the crater the accumulation of rock flour underlies t,he zone of fragmental rock,

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indicating that in this region everything above the ceiling of the rock flour lens was thrown out by the impact of the meteorite. As the outer margins of this lens are approached, however, a few feet of unaltered sandstone and limestone are found to overlie the rock-flour layer. In fact, some of the exploratory bore holes within the crater penetrated Cretaceous rock underlain by rock flour, thus conclusively proving that the rock flour had been formed in situ. This fundamentally important discovery of rock flour in place in the Odessa crater, taken in conjunction with the essential identity of the rock flours found not only in the Barringer and Odessa craters but in other meteorite craters as well; and supplemented by the close similarities in the nature of the rock flour accumulations within those two meteorite craters most exhaustively explored, will enable the reader to appraise the likelihood of the suggestion recently made that the rock flour deposits at the Barringer meteorite crater are water-laid (see [145], pp. 846849). TABLE XIII. Thickness in feet Average ground level around crater, sea level datum 3050 Top of fill in crater at location of shaft 3044 First silt, top crater fill 0 . 0 to 2 5 . 5 Second stratum of crater fill 2 5 . 5 to 7 5 . 5 Ejecta which fell back into the crater 75.5 to 9 4 . 0 Rock flour in place 9 4 . 0 to 100 Undisturbed strata at depth 100 ft Bottom of shaft at depth 170 ft

25.5 50.0 18.5 6.0

(6) The depth to undisturbed rock strata, as well as the thickness and depth of burial of the other formations encountered during the construction of a shaft located near the center of the Odessa meteorite crater can be inferred from Table XIII. I n addition to the exhaustive excavations carried out at the main Odessa crater itself, two subordinate craters detected not only during the radiometric survey of the Odessa crater conducted by the Ohio State University Meteorite Expedition of 1939, but also by a magnetometer survey later made by the Humble Oil and Refining Company, were excavated under the supervision of the Bureau of Economic Geology of the University of Texas. The most interesting of these auxiliary craters was found to be approximately circular in outline and to have a diameter of about 70 ft and a depth of approximately 17 ft. This craterlet had been completely filled up, in part by ejecta, but also to a considerable extent by wash from its sides and from the rim of the main crater close to which


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it was located. Unlike the main crater, the craterlet had not penetrated far enough down to encounter solid rock, but had been formed entirely in clay-like, relatively incoherent materials. 6.3. Determinations of the Mass of Crater-Producing Meteorites and of the Age of the Craters

Stimulated by the identification of the Barringer and Odessa craters as meteorite craters, intensive search for other craters of meteoritic impact origin has been prosecuted all over the globe. Up t o the present, craters, generally recognized as meteoritic in origin by the date indicated, have been found a t the following 14 localities: 1905 1929 1932 1932 1933 1933 1933 1933 1937 1937 1938 1947 1948 1952

Barringer, Coconino County, Arizona Odessa, Ector County, Texas Henbury, McDonnell Ranges, Central Australia Wabar, Rub' a1 Khali, Arabia Campo del Cielo, Gran Chaco, Argentina Haviland, Kiowa County, Kansas Mount Darwin, Tasmania Podkamennaya Tunguska, Yeniseisk, Siberia Box Hole Station, Plenty River, Central Australia Kaalijarv, Oesel, Estonia Dalgaranga, Western Australia Sikhote-Ah, Eastern Siberia Wolf Creek, Wyndham, Kimberley, Western Australia Aouelloul, Adrar, Western Sahara

In addition to the numerous meteorite craters of prehistoric fall listed above, two meteorite crater-producing falls have been witnessed within the last half century within the U.S.S.R. The first of these, that of Podkamennaya Tunguska, occurred in Siberia on June 30, 1908. The writ8erand his collaborators in the Meteorite Bureau a t The Ohio State University and, later, in the Institute of Meteoritics of the University of New Mexico, have published translations of and critical commentaries on a considerable number of the papers in the very extensive literature that has appeared in the Russian language concerning the Podkamennaya Tunguska fall [165]. Among the many almost incredible phenomena produced by the Podkamennaya Tunguska meteorite, not the least intriguing was apparently complete annhilation of the impacting mass. So inexplicable on classical grounds was its disappearance without leaving even a trace of recognizable meteoritic debris in the area of fall, that its behavior led to the first suggestion of the possible existence of contraterrene meteorites [166]. The hypothesis of the existence of meteorites composed of

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reversed matter has received support from several different directions, the favorable evidence culminating quite recently in laboratory production of the final fundamental particle necessary for the fabrication of ‘‘ inside-out ” matter. On February 12, 1947, a second hardly less extraordinary meteoritecrater producing fall occurred in the Sikhote-Aliii mountain range not far from Vladivostok (see Fig. 7). The investigation of this so-called Ussuri, or Sikhote-Ah, fall, sponsored by the Meteorite Committee of the Academy of Sciences of the U.S.S.R., is by far the most elaborate and exhaustive of any meteoritical research project ever attempted, not excepting those at the Barringer and Odessa craters. Unfortunately, only Russians have participated in the Sikhote-Aliii investigations and, until 1956, none of the very numerous scientific papers written by those actually participating i n the investigations in the Sikhote-Alin area had appeared in any laiiguage other than Russian. A translation of and critical c.ommentary on a very important paper by E. L. Krinov was published by Boldyreff and the writer in 1950 [167]. Several other Russian papers relating tjo the Ussuri fall will soon appear in translation in forthcoming issues of the journal Mdeoritics. Had the Podkainenriaya Tunguska or Sikhote-Alin meteorites been of less exceptional iiature, they would have provided what might be described as very coiivenient if somewhat awebome field tests of the competency of meteorites of great mass to blast out full-scale meteorite craters. Such tests would have proved invaluable in deciding between rival theories concerning the impacting mass necessary to produce a meteorite crater of assigned magnitude [168, 1691. Fortunately, military development of lined shaped charges [128] and the concurrent perfection of satisfactory physical arid mathematical explanations of target penetration by 1.s.c. jets [170] have permitted attainment of a satisfactory solution of the very difficult problems of this sort posed by each recognized meteorite crater. For example, as regards the Barringer meteorite crater, a paper of fundamental importance by Rostoker [ 1711 discredits Rinehart’s assumption that crater volume is proportional to the kinetic energy of incident particles moving with velocities up to and including those of meteoritic order [172]. In striking contrast to the mass estimates of a few thousand tons which had been derived under the assumption that Rinehart’s postulate was valid, Rostoker has confirmed that the Canyon Diablo meteorite had a mass of several million tons, a figure of the same order of magnitude as that first found by Opik (see [160], p. 11) using a hydrodynamical approach. We come finally to the equally difficult problem of age determination

EFFECTS OF METEORITES ON TIIE EARTH

325

Fro. 7. The fall of the Sikhote-Alin (Ussuri) meteorite, as painted by the Russinn artist, Medvedev, an eyewitness of the phenomenon.

for meteorite craters. Rough estimates of the age of the Barringer meteorite crater have been based on tree ring counts for the oldest cedars found on its rim [173] or on the dating of artifacts found in pit houses built, in part, of rock debris ejected by the explosion that produced the Barringer crater [174]. On the basis of paleontological and geological evidence, ages

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for the Barringer meteorite crater ranging between 20,000 and 50,000 years have been suggested by Blackwelder [175] and others; and the discovery of a fossil horse (Equus conoersidens) in the crater fill at Odessa implies a much greater age for the Texas crater. These latter figures at once place the age determination problem for the Arizona and Texas craters well beyond the range of accuracy of the C-14 method. This is only one of the reasons for current interest in and intensive work upon a C1-36 analog of the C-14 method; for, if such a C1-method could be developed, accurate age determination for meteorite craters even as ancient as the one at Odessa would become possible. Lack of dependable radioactive age determination methods applicable to the meteorite crater problem would appear to justify the consideration given to an admittedly rough and provisional crater age determination method which will be dealt with in the next section. 6.4. The Diffusion of NiO in the Soil and the Age of the Rrenham

Meteorite Craters In a work on the oxidation and weathering of meteorites which has appeared as No. 3 in the Meteoritical Monograph Series of the University of New Mexico, J. D. Buddhue gives a tabulation of the percentage C of NiO found at various horizontal distances z from a large, buried Brenham, Kansas, meteorite. The graphical representation of C as a function of 2 given by Buddhue struck the writer as resembling a typical Fick diffusion curve. One purpose of this section is to show that, in spite of the complexity of the process detailed by Buddhue, whereby in time the nickel in buried iron meteorites leaches out into the encompassing soil, the concentrationdistance relation, at least for the compound NiO discussed by Buddhue, is well represented by the Fick diffusion curve C = C ( 2 ) shown in Fig. 8, in which the observed pairs of values (z, C) as given by Buddhue are at the centers of the small circles. Later in this section, the explicit form of the function C ( x ) defining the curve in Fig. 8, will be derived on the basis of an idealization of the Brenham diffusion problem described by Buddhue. The primary purpose of this section, however, is to point out that, once the equation C = C(z) is known for such a compound as NiO diffusing outward from a leaching meteorite, we are in position to make a determination of the approximate length of time the meteorite has been buried, provided only we have available a dependable numerical estimate of the average value, over the interval of burial, of the Fick diffusion coefficient D relevant to the particular compound and the particular subsoil in which diffusion of the meteoritic compound has occurred.


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It will be recalled that if it be assumed that the coefficient of diffusion D, a quantity representing the amount of substance in grams diffusing through a n area of one square centimeter in one second under a unit concentration gradient, is independent of the concentration C ; then, on the

RADIAL DISTANCE

-

CM

FIG.8. The Fick diffusion Curve for NiO in the soil around a buried 336,500-gram Brenham meteorite.

basis of Fick’s law of diffusion, the following partial differential equation can be derived :

ac -- D-a w _ at

ax2

Laboratory experiments on the validity of the assumption that D does not depend on C, clearly indicate that this assumption must be quite closely fulfilled in the case of the diffusion of compounds originating in a leaching meteorite, for the concentration in the subsoil of such compounds is very low indeed. Furthermore, since temperature changes in the subsoil are quite limited in extent, the coefficient D may be treated as independent of the temperature of the soil in which the leaching meteorite is buried. If we idealize the diffusion-problem posed by the buried Brenham

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meteorite described by Buddhue in the same manner as has been done by Van Orstrand and Dewey in a related case [176], then the relationship between Buddhue’s variables C and x can be obtained by solving the Fick partial differential equation (6.1). In this wise, we are led to the relation

C(x) =

co

[

X ~

1 - G: ~

1

cw2 dw

/:dG

where Co is the value of the function C(z) a t L = 0. It is expedient to express the solution (6.2) in tterms of Gauss’ error function, Erf ( 2 )in t)he form (6.3)

C ( x ) = Co[l - Erf ( Z ) ] ,

Z

=

9: ~

2f i t

Our problem is now so to determine the parameters COand Z that the resulting expression (6.3) will give a curve C = C(.c) fitting the observed pairs of (C, s) values as well as possible. Since the N O determination for J: = 1 ft was regarded as the most accurate of the observed C’s, 2 was evaluated for various choices of Co with .c always assigned the value 30.5 em. I n this manner, the associated pair of values (Co = 0.19%, 2 = 0.34) was found to give a reasonably satisfactory fit to all of the observed data, as may be seen from Fig. 8 which is based on the choice indicated. If we suppose that the numerical value of the diffusion constant D is known, then the total time t required for diffusion t o build up the concentration observed at a distance x = 30.5 cm can be calculated a t once from the relation (6.4)

I n the sequel, reasons for taking D = cm2/sec in the Brenham case will be giveii. With this choice of D,one finds t = 637,000 years, a value consonant with the almost complete effacement of the Brenharn craters by the very slow action of aggradation and rim weathering. There remains the problem of justifying the choice just made for the value of D a t Brenham. Evidently, if a Fick diffusion curve of the sort pictured in Fig. 8 is available for a crater of k n o w n age, then by simply reversing the argument. of the last paragraph, we can determine the numerical value of the diffusion constant D. The Odessa crater has tentatiuely been assigned an age of t = 200,000 yr on the basis of the discovery of bones of Equus conversidens buried in the crater fill.

E F F E C T S O F M E T E O R I T E S ON T H E E A R T H

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I n the case of the Odessa crater, the relation analogous t o (6.4) is 03.5)

=

Hence, adoption of the value t

=

D

=

16.55 200,000 years leads to

0.837 X 10-'" cni2/sec

Values of D determined in diffusioii experiments ill the laboratory have a wide range but the few values ('orresponding to such low temperatures as those i n the earth a t Brenham, duster around

D = 10-'" cni2/ser the value adopted above. The first serious objection that can be raised against the age-deterniinatioii method developed above is that we have replaced a threedimensional physical situation by a one-dimensional idealization. ,Justification for limiting attention t o the simplified problem is given by such results as those of Van Orstrand and Dewey (see [176], p. 90) who found that predictions based on the one-dimensioiial idealization ' I represented the observed facts t o a high degree of precision," a t least for solid gold diffusing into solid lead a t moderate temperatures. A much more serious objection, can be based on the fact that ube was made of a value of D that may very poorly represent the average diff usiori conditions existing i n the Brenham subsoil over the interval cxt ending from the time the meteorite fell there t o the present. No defeiibe against this objection will be attempted. The writer prefers t o cite the objection in question as evidence of the urgent need for laboratory and field programs directed toward accurate determination of 11 for various compounds of meteoritic origin, diffusing under R wid(. iraric+y of wbsurfaw conditions. APPENDIX

I.

n/lETEORITICAL P I C T O G R A P H S 4 N D T H E \'ENERATIQY

EXPLOITATION O F METEORITES A first purpose of this section is t o describe meteoritical pictographs and petroglyphs of undoubted authenticity and considerable age, and to suggest that a meteoritical significance can with propriety be attached to certain pictographs of very much greater age. A4second purpose is to present briefly evidence of meteorite worship, and t o show that, meteorite cults have persisted up t o the present time. The section concludes by pointing out that while the more imaginative among primitive and medieval men venerated the objects that fell from heaven, the more practical had no hesitation in utilizing the heaven-sent material i n the fabrication of ornaments, weapons, and tools. AND

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‘ Those pictographs and petroglyphs left by Paleolithic man do not suggest that the moon, the planets, the fixed stars, or the sun in any notable way forced themselves upon the developing consciousness of Man. Soundless and endowed with languid motion, these bodies remained relatively unnoticed by him in the remoteness of the sky. The sudden, startling appearances of meteor showers and meteorite falls, however, must have had an entirely different effect upon him. Darting swiftly out of the heavens, shooting stars fall like a fiery snowstorm in the very faces of the beholders, and earth-shaking tumult accompanies the passage of a meteorite through the atmosphere. T o these unfamiliar and violently obtrusive visitations, early man no doubt reacted with frightened curiosity, as do the peoples of today. Because meteor showers and meteorite falls alarmed him, however, it does not follow that he would refrain from permanently recording them. Poisonous snakes and ferocious animals must have similarly terrified him, yet drawings, paintings, and carvings of snakes and wild beasts were made by him in profusion. It is to be expected that the records of primitive man’s attempts a t the earliest depiction of meteoritical phenomena will be much rarer than those devoted t o the animals and reptiles with which he fought for food and place. Even in Paleolithic times, meteor showers and meteorite falls were no doubt infrequent, and still less frequently would such phenomena be witnessed by men having the ability, the facilities, and the opportunity to make faithful and enduring likenesses of them. Such rarity may explain the failure, so far, to discover meteoritical pictographs among the artistic records preserved from the earliest times, but it seems likely to the writer that this failure may be due in part to misinterpretation of some of the evidence actually a t hand. Certain undoubtedly genuine meteoritical pictographs of moderate age are known. It seems probable that a meteoritical significance properly can, and should, be associated with other known pictographs of much greater age. Active collaboration between specialists in archeology, anthropology, ethnology, and meteoritics in exploring this promising and apparently neglected field would prove of much value. Today, when studies of radioactive carbon in the bones of a long-dead artist may serve indirectly to date one of his representations of a meteor shower, or when measures of the effects produced in a meteorite by exposure to cosmic radiation may, reciprocally, serve to determine the age in which a paleolithic artist painted the meteorite’s fall, i t is clear that close cooperation between those who study ancient man and those who investigate early meteorite falls and meteor showers is likely to be of mutual profit. As a modern instance of the pictorial recording of a n event of meteoritical importance, it may be noted that the Navajos of the Four Corners


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area of New Mexico, Arizona, Colorado, and Utah turned to pictographic means in describing the phenomena accompanying the fall of the meteorite of October 30,1947, in that region. Lacking words, other than “moving star,” with which to characterize this remarkable fall, the Navajos proved surprisingly adept a t tracing in the dust, or on wooden, rock, or paper surfaces, pictures that told the story more effectively than their speech could have done. Stimulated by the tendencies observed among the Navajos, the writer undertook a study of the relevant literature. This investigation disclosed that (1) a great meteorite fall occurring in the winter of 1821-1822 and (2) the spectacular Leonid meteor shower of November 13, 1833, were permanently recorded by historians of the Dakota tribe in the Winter Counts, by means of which the Indians established a chronological system. These Winter Counts were pictographic records, i.e., paintings, arranged on a spiral or similar curve on the tanned inner skin of a buffalo robe, of the year-characters whereby the Dakotas established a continuous designation of the years without having to resort to a scheme involving assignment of consecutive numerals to them [177]. Scientific records of the actual fall of a meteorite in the winter of 1821-1822 seem t o be lacking. As Mallery remarks, “There were not many correspondents of scientific institutions in the Upper Missouri region a t the date mentioned.” Furthermore, because of the enormous territory ranged over by the various tribes or bands of the great Dakota Nation, it is difficult to decide which one of several almost unweathered meteorites found in Dakota-land in the later 1800’s is most likely to have been associated with the fall the Indians observed in 1821-1822. The Fort Pierre, South Dakota (CN = 1003,344), and the Iron Creek, Alberta, Canada (CN = 1115,530:), octahedrites would seem to deserve careful consideration in this connection. As regards the Leonid shower of 1833, however, a wealth of hiRtorica1 and descriptive material is preserved in scientific journals and elsewhere. Intercomparison of the white- and red-men’s records of this meteor shower shows that both received the same impression of a veritable snowstorm of stars, descending into the very faces of the observers. Furthermore, the printed testimony justifies the Dakotas in representing the individual Leonid paths as luminous hairlines ending in bright bursts or flares similar to those observed for about 15% of the brighter Nu-Draconid meteors seen during the recent meteor shower of October 9, 1946. The crescent shown in the Dakota record of the 1833 Leonid shower may have been intended to represent the young moon, which set about an hour after the sun in the early evening of November 12, 1833. If this interpretation is indeed correct, then the association of this crescent with

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the Leonids indicates that in Dakota-land, as a t certain other places in the United States, the forerunners of the Leonid shower were observed in sufficient numbers t o attract attention even before moonset. It is more likely, however, that this crescent was intended to represent an extremely bright, long-enduring Leonid train. Such trains attracted universal attention among white observers and led to such printed descriptions as the following: " . . . it was then very brilliant, in the form of a pruning hook . . . I first saw it at 5 o'clock, when it resembled a new Moon, 2 or 3 hours high, shining thru a cloud . . . ) l I n addition to the portraitures of meteorite falls and meteor showers painted by the Amerinds-crude examples of a process of direct representation which culminated in such magnificent fireball paintings as those of Raphael [178] and Medvedev [179]-meteoritical pictographs of a conventionalized form have been found in the records left, for example, by the ancient Mexicans. These people used symbols to represent meteors which are reminiscent of those employed by the Europeans of the Middle Ages to portray comets. I n plates 29 and 30 of the first volume of Lord Kingsborough's compendious work on Mexican antiquities [180], one device-figure is reproduced which contains a circle enclosing an %pointed star with black spiraling plumes trailing from it, and another such figure shows a serpent darting out of a bowl of stars. I n the codex Telleriano Remensis (see [180], Vol. 6, p. 138), the first of these symbols is referred to as a smoking star, and the second as a serpent descending from the sky. Mallery believes that the first symbol represents a meteor that fell in 1534 (the year that Don Antonio de Mendoqa became Viceroy of New Spain, according to the codex quoted), and that the second symbol represents a meteor that fell in 1529 (the year that Nuiio de Guzman set out to conquer the Province of Yalisco, according to the same codex). Doubtless a careful search of the voluminous records left by the ancient Mexicans would disclose many other meteoritical pictographs. The examples just given show that meteoritical pictographs may be either portraitures or conventionalized forms. If, with this fact in mind, an examination is made of such pictographic collections as those published by Kingsborough and Mallery, a considerable number of drawings and paintings will be noted that appear t o have meteoritical significance. Furthermore, certain petroglyphs, or rock carvings, believed t o be of greater antiquity than the Mexican and Amerind pictographs, also seem to have such significance. Thus, a petroglyph found near San Marcos Pass, California, pictures two star-like figures joined by a trail, suggesting the path of a meteor from one position in the sky to another; a rock carving in Kansas shows a five-pointed star surrounded by a cloud; and


333

a petroglyph a t Ojo de Benado, New Mexico, depicts what appears to be a crescent moon crossed by a hairline path with a terminal burst, closely resembling Dakota pictographs of typical bright Leonid meteors. Finally, although published collections of late Paleolithic pictographs (devoted as they primarily are to the more remarkable of innumerable animal portraitures) can scarcely be regarded as giving a typical sample of the work of the artists of that period, nevertheless they contain many forms suggestive of meteors and meteorites. One type in particular seems worthy of consideration in this connection, namely, the category of socalled radiate structures (figures rayonnantes) found in the cavern at Altamira, Spain, and studied with much care by Cartailhac and Breuil [181]. These investigators point out that, in spite of their best efforts, the meaning of these drawings is not understood. Various interpretations of these figures have been given, ranging from “ a bit of reed-grass” (see [181], p. 637) to a “magic weapon” [182]. It would seem quite possible that some of the “sheaves of rays” at Altamira-namely, those associated with certain reddish markings-are portraitures of auroral wreathes (i.e., incomplete auroral crowns) ; but others may be portrayals of showers of shooting stars during which a few curved and otherwise abnormal paths and several nonconforming meteors also were seen. The fact that the represented meteors are not shown shooting outward in all directions from the radiant is not fatal to this interpretation, for a similar-although somewhat less restrictive-limitation in directions of departure from the radiant is shown, e.g., in the Greenwich drawing of the Leonid shower of November 13, 1866, reproduced in Chambers’ “Astronomy” [183]. Furthermore, the artist may have observed only that limited portion of the shower visible through the entrance arch of the grotto of Altamira. If we seek to establish the identity of the shower which may have beeii depicted in the Altamira pictograph, it is interesting to note that twice within the period during which it is believed the artists of the Reindeer Age were active, the radiant of the annual Lyrid shower was near the north celestial pole for long intervals of time. Hence, as seen from the latitude of the Pyrenees Mountains, it would have occupied a nearly stationary position high in the northern sky, quite favorably placed for observation from the old entrance of the cavern at Altamira, since, according to Cartailhac and Breuil, this cavern opened toward the north a t the top of a hill (see [181], p. 626). As there is convincing evidence that earlier, richer showers of the Lyrids can be traced back into the past for more than 2500 yr 11841, it is not inconceivable that spectacular Paleolithic returns of this perennial shower inspired such portraitures as the Altamira pictograph. Much later in the evolution of mankind than the Altamira paintings,

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the development of religious instincts led the more cultured people of antiquity to regard the stars as the domiciles of deity. Consequently, the fall of a meteorite was not unreasonably interpreted as the arrival on earth of divinity or an image thereof. Meteorites of observed fall were received with all due honor, and often were embalmed, draped, housed, and worshipped in temples built especially for them. As late as December 2, 1880, a new worship of this sort was inaugurated as a result of the fall of the Andhara, India, aerolite. Immediately after the fall and recovery of this meteorite, two Brahmans established themselves as its ministering priests and many thousands of the natives flocked to see it and to contribute funds for the construction of a temple in which to house it [185]. The meteorite temples of India and Japan, traceable to local meteorite falls like the one at Andhara, have such African analogs as the himmas or fetish temples of the Ashanti Negroes, the sacred stones of which are also meteorites or reasonable facsimilies thereof (see [2], pp. 50-51). Here, however, the origin of the cult apparently is lost in remotest antiquity. Since the giant water-filled crater, Lake Bosumtwi, has been seriously considered to have originated in large-scale meteoritic impact 11861, it is conceivable that the widespread and tenacious character of Ashantian meteorite worship is connected with racial memory of a past event of colossal proportions, lethal to many of the country’s early inhabitants. Among the most highly cultured Greek and other Mediterranean races, practices analogous to the worship of the Black Stone of Kaaba in Mecca (which persists even to the present day [187]) flourished and are recorded not only in the historical chronicles, but in more permanent, if less profuse, form in the various medals or Betyl coins struck to commemorate the arrival of meteorites on the earth. In his treatment of “numismatic meteoritics,” Brezina regards the term “ Betyl” as derived, by the way of the Greek PETVXOS,from the early Hebraic ‘‘Bethel,” or “home of God” [188]. As in the case of the pictographs earlier discussed, these Betyls, or meteoritic medals, show a transition from a primitive form-in which, for example, a black, conical aerolite found a naturalistic expression on the coin struck to commemorate its falllg-to a more sophisticated phase in which the meteoritical representations were increasingly conventionalized. But if man venerated meteorites, as the above evidence shows, he also exploited their properties of malleability and durability. In an inter1s Brezina gives the following description by Herodotus in regard to the Betyl of Emisa: “A large stone, which on the lower side is round, and above runs gradually to a point. It has nearly the form of a cone, and is of a black color. People say of it i n earnest that it fell f ~ o mHeaven.”


335

esting paper, Rickard stresses that primitive man everywhere used meteoritic iron in the earliest stage of his metal culture [189]. As evidence proving that meteoritic iron was used in antiquity, Rickard calls attention to a number of the ancient names for iron:11 The Sumerian name for iron was an-bar, meaning “fire from heaven.” The Hittite ku-an has the same meaning. The Egyptian name, ‘bia-en-pet, has been variously translated; probably the first meaning of bia was “thunderbolt” . . and pet stands for “heaven,” so here also we have the plain intimation that the earliest iron was of celestial origin. A Hittite text says that whereas gold came from Birununda and copper from Taggasta, iron came from heaven Likewise the Hebrew word for iron, parZil, and the equivalent in Assyrian, barzillu, are derived from barZu-ili, meaning “metal of god” or “of heaven.” Even today the Georgian name for a meteorite is his-natckhi, meaning “fragment of heaven” (see “91, p. 55).

.

...

Various interesting uses of meteoritic iron may be cited: Fragments considered to be parts of a dagger found by Woolley at Ur of the Chaldees in 1927 have been ascribed to the first dynasty of Ur before 3000 B.C. According to Desch, these fragments upon analysis, were found to contain 10.9% nickel [191]. Furthermore, the predynastic iron beads found by Wainwright in 1911 at Gerzah, Egypt contained 7.5% nickel [192]. Similarly, microanalysis of a minute fragment of a thin blade of iron inserted into a small silver amulet in the form of a Sphinx head (XI dynasty) showed that iron and nickel were present in about the ratio 10 to 1 [193]. Atilla and other early conquerors of Europe possessed swords from heaven”; and the weapons of the Caliphs were of the same meteoritic material as the Kaaba stone. Emperor Jehangir (1605-1627) has put on record in considerable detail the circumstances of the fall on April 10, 1621, of the Jalandhar meteorite, from which by his orders two sword blades, one knife, and one dagger were smelted [194]; and as late as the beginning of the 19th century, J. Sowerby manufactured a sword from the Cape of Good Hope meteorite for presentation to Alexander, Emperor of Russia [195], while the Javanese armourers worked up the Prambanan siderite into weapons for royalty [196]. In various primitive areasSenegal, Mexico, and Chile-the natives have long utilized meteoritic iron €or agricultural implements, spurs, stirrups, spearheads, and knives. 14 In connection with the transition from picture writing to cuneiform script, it is interesting to note that one of the symbols in the earliest Egyptian hieroglyphic term for iron (min) is a good depiction of the teardrop shape of a falling fireball [190]. Some question has been raised as to whether or not the primitive Egyptians knew of the origin of meteoritic iron at the time this term was in use. This issue is still unresolved by Egyptologists. The resemblance of the hieroglyphic form to the actual phenomenon depicted would seem to indicate a “holdover” from the earlier picture writing of a people possessing knowledge a8 to the origin of meteoritic iron.

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The Eskimos of Greenland, using the famous Cape York meteorites as their source of supply, made metal knives, harpoon points, and other implements. Many artifacts composed of meteoritic iron have been found in the mounds erected by the prehistoric inhabitants of the central United States. Rickard (see [189], p. 59) has noted that, “The most remarkable objects found in these mounds were copper ear-ornaments, of spool shape, that had been covered with a thin plate of meteoric iron.”l6 The analyses of this overlaid material made by Kinnicutt [197] proved the presence of nickel in the proportion characteristic of meteoritic iron. More recently, members of the Illinois State Museum, excavating a group of Indian burial mounds in the Havana, Illinois area, discovered “22 rounded bead-like objects, composed of strongly oxidized iron, together with slightly more than 1000 ground shell and pearl or pearl slug beads” [198]. Several of the metal beads were analyzed, ground, and etched. They showed not only the characteristic nickel content of meteoritic iron, but also an interesting relict Widmanstatten structure. The examples cited here are but a few of the many well-substantiated instances of man’s utilization of meteoritic iron for practical and ornamental purposes. They clearly show, however, the important role played by cosmic materials in primitive, ancient, and medieval cultures. APPENDIX11. BASICMETEORITIC DATAAND CLASSIFICATIONAL CRITERIAEMPLOYED IN SECTION5 The reasons for selection of the half-century between 1896 and 1945 to provide the main body of data tested in Section 5 have already been given, together with the circumstances under which 17 meteorite falls of later data (1946-1950) came to be included during progress of the investigation. Use of Astapowitsch’s valuable summary of critically evaluated orbital elements in connection with work on the 1896-1950 data led t o the decision to take advantage of other orbits included in Astapowitsch’s paper, specifically those of the famous falls of Barbotan, L’Aigle, Pultusk, Hessle, MOCS,and Khairpur, occurring in the years 1790, 1803, 1868, 1869, 1882, and 1873, respectively. Nautical almanacs for these early years (none of which were in the files of the University of New Mexico Library) were very kindly loaned to the Institute of Meteoritics by the Yale University Observatory. Availability of these almanacs permitted treatment not only of the falls for which orbits had been computed, but 16 Through the courtesy of the Ohio State Museum, typical artifacts of this sort were displayed by the Society for Research on Meteorites at the American Association for the Advancement of Science convention in Columbus, Ohio, in December, 1939.

’


337

also for all meteorite falls witnessed in the particular year covered by each almanac. Furthermore, since some of the bound volumes loaned by Yale contained almanacs for several years in addition t o the one requested, advantage also was taken of the opportunity thus afforded to treat several additional early witnessed falls. These facts make clear the quite fortuitous manner in which the meteorites with dates of fall prior t o those of the main body of data came to be included in our list. Furthermore, one can easily verify that exclusion from Table XI1 of the 36 meteorites with dates of fall prior to January 1, 1896, would produce no significant change in the results stated in Section 5. Much more important than the fortuitous circumstances determining the composition of the group C,,,, are the criteria deliberately chosen to effect a subdivision of CrIIinto the subgroups, CI and CII, of Section 5. First, it must be emphasized that objective considerations, which will be discussed below, led to adoption of these criteria; and, second, that the criteria were adopted before the subdivision of CIII was attempted and, once chosen, were transgressed only 7 times in the treatment of the totality of 264 classified meteorites. The reasons for the 7 transgressions will be given below. I n the writer’s considered opinion, the reasons given are valid ones; but, whether they are or not, it cannot be too strongly emphasized that the exceptions permitted, amounting t o only 2.6% of the total number of classified meteorites, are too few in number to affect materially the validity of the conclusions based on consideration of the subgroups CI and CII, as finally constituted. For 53 of the 317 meteorites treated, only such indefinite classifications as stone (AE), chondrite (C), etc., were known. For the remaining 264 meteorites, definite meteoritic classifications, often involving numerous qualifiers-to employ Leonard’s useful term [ 1991-were available. Except for 7 of the 264 classified meteorites, certain of these qualifiers (those italicized in the summaries given below) were taken as providing criteria of decisive importance in allocating a given meteorite either t o C, or to CIr.I n this manner, the following assignments were made of all meteorites for which a classification was known:

To the Subgroup, CI All 26 irons and iron-stones; all 10 carbonaceous chondrites; all 4 howarditic chrondrites of Brezina; all 38 white chondrites; and 44 of the 46 gray chondrites. I n addition, 3 mineralogically quite similar bronzite chondrites (Pultusk, Nassirah, and Queen’s Mercy) and 2 mineralogically quite similar enstatite chondrites (Khairpur and Leonovka) , which normally would have been allocated t,o Crr were placed in Cr because, in the writer’s

338

LINCOLN LAPAZ

judgment, the evidence that the extra-atmospheric velocities of these bodies were strongly hyperbolic outweighs all other considerations.

To the Subgroup, CII All 31 achondrites; all 17 spherulitic chondrites; all 37 intermediate chondrites; all 5 crystalline chondrites; all 7 black or veined chondrites; all 14 classified hypersthene chondrites not previously allocated; 17 of the 20 bronzite chondrites; and 7 of the 9 enstatite chondrites. In addition, 2 mineralogically quite similar gray chondrites (Ekeby and Kendleton), which normally would have been allocated to CI were placed in CII because, in the writer’s opinion, there is compelling evidence that the extra-atmospheric velocities of these meteorites were definitely elliptic. Choice of the criteria employed above in subdividing the 264 classified meteorites did not rest solely on such evidence as that furnished by compilations of meteorite orbits (like that of Astapowitsch) and the writer’s determinations of extra-atmospheric velocities for various meteorites. In addition, careful consideration was given to such convincing non-orbital evidence as the following: Watson’s [a001 laboratory demonstration that exposure of lightish-colored chondrites to a temperature of 800°C will turn them into black chondrites, and his sound probabilistic argument that the agency responsible for the heat darkening of the chondrites picked up by the earth is the sun and not the stars; and arguments, some dating back to Sorby [201] and Cohen [202]-and others recently published by Urey [148]-which indicate that the spherulitic chondrites had their origin in the crowded confines of the solar system. There remains to be considered the manner in which the 53 unclassified meteorites were allocated to either CI or CII.Since in these 53 cases, no clue whatever could be found permitting use of the precepts employed on the other 264 meteorites, the only alternative was t o leave the matter to chance. The 53 unclassified meteorites appeared in alphabetical order in the meteorite catalog employed in drawing up a polar diagram which served as a basis for preparation of Table XII. As each of the stones in question was encountered in the catalog, the toss of a coin determined whether the meteorite was to be allocated to CI or to CII.As a result, 25 of the unclassified meteorites went into the first of these subgroups and 28 into the second. Interestingly enough, the ratio 25/28 is very close to to that in which the classified chondrites were distributed between CI and CII.Since reliance on chance to distribute the unclassified meteorites conforms to the natural random fall of interstellar meteorites with respect to a, there is no reason to believe that the 25 stones assigned to CI by this process could significantly alter the distribution with respect to a of


339

the classijied component of this subgroup. Similarly, since, as the reader can verify, chance provided a fairly uniform distribution with respect to cy of the 28 stones assigned to CII, the unclassified component of CIr cannot be held responsible for the sinusoidal excellence of the lower curve in Fig. 2 (11) of Section 5. LIST OF SYMBOLS (According to the Section in which they appear)

Section Symbol 1.2.1 HTA N

P

P e n

P* 1.2.2 BTA

N P P e

1.2.3 VTA d V

V II

3.2

R

Definition human target area number of meteorites reaching the surface of the earth per century probability t h at at least one of these N meteorites will strike the specified HTA area of HTA (in square miles) divided by 2 x 108 Naperian base of logarithms number of centuries after 2100 A.D. probability t h at in a t least one of the n centuries after 2100 A.D. at least one meteorite will hit in the HTA of the world built-over target area number of meteorites reaching the surface of the earth per century probability that at least one meteorite will hit in the New York City area in the 20th century area of BTA (in square miles) divided b y 2 x 108 Naperian base of logarithms vehicular target area diameter of rocket cruising speed of rocket velocity of visual meteors space density of visual meteors region within which the fallen meteorites constitute a n essentially homogeneous group total number of meteorites in R independent, equally intensive searches of R number of meteorites found during first search S1 number of meteorites found during second search Sn number of discoveries common to both SI and 5 2

nl/N ndN PZ X n: = (n2. n d / N pl X nl = (nl n z ) / N a real number such that 0

<e 6 1 integer nearest to the fraction, f recovery index actual number of individuals recovered in a strewn field of which the (probable) population is N

LINCOLN LAPAZ

Section

Definition

Symbol

3.4.2 P'

tangent plane to earth a t a point, 0, chosen as origin of an xyz-coordinate system of which the y and z axes lie in the plane P' inradius of spherical cap with center 0 on earth's surface area of circle with center 0 in plane P' radius of spherical iron meteorite the distance beneath the plane P' of the center of a spherical iron meteorite of radius r radius of the smallest meteorite which just attains a penetration of depth x into the earth radius of the largest meteorite which does not penetrate too far into the earth to touch the plane, x = x (ra - [ d ( r ) - zI2)% density of meteoritic nickel-iron number of years of sideritic infall number of iron meteorites with radius between r and T dr striking the entire earth annually depths in the earth beneath the plane I" ratio in right member of equation (8) of this section total observed increase in the length of the day a real number such that 0 < e 5 1 moment of inertia of earth and corresponding duration of diurnal rotation a t any time t value of Z at beginning of year for which we seek to calculate the meteoritic accretion, and corresponding duration of diurnal rotation increment in I due to the infall of meteoritic material radius of earth mean density of the earth thickness of shell of meteoritic dust density of meteoritic dust density of the fictitious auxiliary deposit D defined in Section 3.5 number of meteoritic particles falling on the earth annually average mass of these meteoritic particles mass of the earth when Z = ZO total observed annuaE increase in the length of the day radius in centimeters of interplanetary particle number of interplanetary particles per cubic centimeter in the vicinity of the earth's orbit with radii between r and r +dr electron density Harvard Meteor Program angular velocity time of apparition observed velocity eccentricity point of initial visibility midpoint of the photographed path of H.M.P. meteor No. 670

+

lo, 7 0 AI

r PO

A? P P'

N -

m

M AT

3.6.3 r n(r ) dr

4.2

n. H.M.P. w

T VO e

4.5.1

B B'


Section

Symbol

H' A 6

4.5.2

4.5.3

5.1

6.4

c

= C(z)

co = C(0) D t

z=

341

Definition end point of meteorite path deceleration experienced by the meteorite deceleration determined from the H.M.P. photograph of meteor No. 670 constants time value of t when meteorite reaches end point H i velocity as a function of t position as a function of t velocity of meteorite a t the point H' Naperian base of logarithms coordinates of meteorite a t time t in the wv-coordinate system speed with which meteorite penetrates air effective speed of sidewise escape of compressed gaseous cap average values of 6 ( t ) and i ( t ) taken over the short time interval, AT constants smoke formation pressure atmospheric density in grams per cubic centimeter constants velocity of meteorite right ascension of the moon quadrants centering on CY = O", QO",180", 270°, respectively probability of success counter-probability = 1 - p number of trials discrepancy probability that in n trials the discrepancy will not exceed numerically the number d percentage of NiO found at horizontal distance z from buried meteorite Fick diffusion coefficient time X

2

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67. Such use is described in a personal letter received from Shirl Herr in 1935. 68. For a description of the Hotchkiss Superdip, see: Stearn, N. H. (1929). The Hotchkiss Superdip: A new magnetometer. Bull, Am. Assoc. Petrol. Geol. 13, 659; (1932). Practical geomagnetic exploration with the Hotchkiss Superdip. Trans. Am. Znst. Min. Met. Engrs. 97, 169-199; Swanson, C. 0. (1934). Use of magnetic data in Michigan iron ranges. Trans. Am. Inst. Min. Met. Engrs. 110, 29@312; DeBeck, H. 0. (1934). An accurate simplified magnetometer field method (employing the Superdip). Trans. Am. Inst. Min. Met. Engrs. 110, 326-333. 69. Theodorsen, T. (1930). Instrument for detecting metallic bodies buried in the earth. J. Franklin Inst. 210, 311; (1930). A sensitive induction balance for the purpose of detecting unexploded bombs. Proc. Natl. Acad. Sci. U.S. 16, 685. 70. Wisman, F. 0. (1942). Synthetic symmetry in mutual-induction balances: A practical problem with meteorite detectors. Contrib. SOC.Res. Meteorites 3(1), 23-30. 71. Watson, F. G. (1956). “Between the Planets,” rev. ed., Plate 34. Harvard Univ. Press, Cambridge, Mass. 72. Compare LaPaz, L. (1944). Meteoritical position problems. Contrib. Soc. Res. Meteorites 3(3), 148-153. 73. Buddhue, J. D. (1950). “Meteoritic Dust,” pp. 12-27. Univ. New Mexico Press, Albuquerque, New Mexico. 74. Hoffleit, D. (1952). Bibliography on meteoritic dust with brief abstracts. Harvard Coll. Obs. Tech. Rept. No. 9, 5-45. 75. Nordenski:ild, N. A. E. (1873). Observations sur les poussieres charboneuses, avec fer m6tallique, observ6es dans la neige. Compt. rend. 77, 463; (1874). On the cosmic dust which falls on the surface of the earth with the atmospheric precipitation. Phil. Mag. [4] 48, 546; (1874). Ueber kosmischen Staub, der mit atmosphiirischen Niederschlagen auf die Erdoberflache herabfiillt. Ann. d. Phys. 161, 154-165. 76. Lasaulx, A. von (1881). ‘her sogenannten kosmischen Staub. Tschermak’s mineralog u. petrog. Mitt. S , 517. 77. Krinov, E. L. (1955). “Bases of Meteoritics,” pp. 117-130. Press for TechnicalTheoretical Literature, Moscow. 78. opik, E. J. (1951). Astronomy and the bottom of the sea. Irish. Ast. J. 1, 145158. 79. Whipple, F. L. (1955). Meteors. Publ. Ast. SOC.Pacif. 67, 367-386. 80. Watson, F. G. (1937). Distribution of meteoric masses in interstellar space. Harvard Ann. 106, 628. 81. Watson, F. G. (1939). The mean chemical composition of meteorite accretion. J. Geol. 47, 426-430. 82. Millman, P. M. (1952). A size classification of meteoritic material encountered by the earth. J. R o y . Ast. SOC.Can. 46, 79-82 (ref. on p. 81). 83. Woodward, R. S. (1901). The effects of secular cooling and meteoric dust on the length of the terrestrial day. Ast. J . 21, 169-175. 84. Stoney, G. J. (1902). The effect of meteoric deposits on the length of the terrestrial day. -4st. J . 22, 85-87. 85. Brouwer, D. (1952). A study of the changes in the rate of rotation of the earth. Ast. J . 67, 125-146. 86. Hoffmeister, C. (1937). “Die Meteore.” Akademische Verlagsges., Leipzig. 87. Goubau, F., and Zenneck, J. (1931). Eine Methode zur selbsttiitigen Aufzeichnung der Echos aus der Ionosphare. Hochfrequenztech. u. Elektroakusl. 41, 77.

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88. Vestine, E. H. (1934). Noctilucent clouds. J . Roy. Ast. SOC. Can. 28, 249-272. 89. Bowen, E. G. (1953). The influence of meteoritic dust on rainfall. Australian J . Sci. Research AB, 490-497. 90. Landsberg, H. E. (1938). Atmospheric condensation nuclei. Erg. kosm. Phye. 3, 155. 91. Bowen, E. G. (1956). The relation between rainfall and meteor showers. J . Meteorol. 13, 142-151 (ref. on p. 142). 92. Crozier, W. D. (1956). Rate of deposit in New Mexico of magnetic spherules from the atmosphere. Bull. Am. Meteorol. Soc. 37, 308. 93. Bowen, E. G. (1956). The relation between rainfall and meteor showers. J . Meteorol. 13, 146-147. 94. Mason, B. J. (1956). The nucleation of supercooled water clouds. Sci. Progr. 44, 479-499 (see particularly pp. 491-492); Whipple, F. L., and Hawkins, G . S. (1956). On meteors and rainfall. J . Meteorol. 13, 236-240. 94a. Upik, E. J. (1956). As reported in Weather XI, 196-197. 95. Kaiser, T. R. (1953). Phil. Mag. Suppl. 2, 495. 96. Kaiser, T. R., and Seaton, M. J. (1954). Interplanetary dust and physical processes in the earth’s upper atmosphere. M h . SOC. roy. sci. Likge [4] 16, 48-54. 97. Whipple, F. L. (1954). Photographic meteor orbits and their distribution in space. Ast. J . 69, 202-207. 98. upik, E. J. (1950). Interstellar meteors and related problems. Zrish Ast. J . 1, 80-96 (ref. on p. 92). 99. Whipple, F. L. (1938). Photographic meteor studies. I. Proc. Am. Phil. SOC. 79, 499-548. 100. Compare Harvard Meteor Program Tech. Rept. No. 7, 20 (1951). 101. Niessl, G. von (1881). Theoretische Untersuchung uber die Verschiebungen der Radiationspunkte aufgeloster Meteorstrome. Sitzber. Akad. Wiss. W i e n , Math.-naturw. KZ. Abt. ZI, 83, 96-143; (1912). tfber die Bahn des grossen detonierenden Meteors vom 23 September, 1910, 6 h 30.9 min. mitteleuropaischer Zeit. Sitzber. Akad. Wiss. Wien, Math.-naturw. K1. Abt. I I a , 121, 1883-1936. 102. Compare Harvard Meteor Program Tech. Rept. No. 4, 10 (1949). 103. Almond, M., Davies, J. G., and Lovell, A. C. B. (1950). Observatory 70, 112-113. 104. Lovell, A. C. B. (1954). “Meteor Astronomy,” pp. 236-237. Oxford Univ. Press, London and New York. 105. Z)pik, E. J. (1955). Book Review of Lovell’s Meteor Astronomy. Zrish Ast. J . 3, 144-152 (ref. on p. 150). 106. t)pik, E. J. (1955). Meteors and the upper atmosphere. Irish Ast. J . 3, 180. 107. McKinley, D. W. R., and Millman, P. M. (1949). A phenomenological theory of radar echoes from meteors. Publ. Dom. Obs. Ottawa XI, 329-340 (ref. on p. 336). 108. Millman, P. M., and McKinley, D. W. R. (1949). Three-station radar and visual triangulation of meteors. S k y and Telescope, 8, no pp. given. 109. up&, E. J. (1934). On the distribution of heliocentric velocities of meteors. Haward Cott. 06s. Circ. 391, 1-9 (ref. on p. 8). 110. t)pik, E. J. (1937). Researches on the physical theory of meteor phenomena, 111; Basis of the physical theory of meteor phenomena. Publ. obs. ast. univ. Tartu 29, 1-59 (ref. to Section 3, g). 111. Whipple, F. L. (1950). The theory of micro-meteorites. I. I n an isothermal atmosphere. Proc. Natl. Acad. Sci. U.S. 36, 687-695; (1951). Part 11. I n heterothermal atmospheres. Zbid. 37, 19-30. 112. Clegg, J. A. (1948). The determination of meteor radiants. Phil, Mag. [7] 39, 580; Clegg, J. A., and Davidson, I. A. (1950). A radio echo method. Phil. Mag. [7] 41, 84.


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113. Opik, E. J. (1934). Results of the Arizona expedition for the study of meteors. 11. Statistical analysis of group radiants. Harvard COX Obs. Circ. 388, 1-38; Part 111. Velocities of meteors observed visually. Zbicl. 389, 1-9; Part V. On the distribution of heliocentric velocities of meteors. Zbid. 391, 8. 114. Boothroyd, S. L. (1934). Results of the Arizona expedition for the study of meteors. IV. Telescopic observations of meteor velocities. Haruard Coll. Obs. Circ. 390, 1-12. 115. Opik, E. J. (1940). Meteors. Monthly Not. Roy. A s t . SOC.100, 317. 116. McKinley, D. W. R. (1951). Meteor velocities determined by radio observations. Astrophys. J . 113, 225-267 (ref. on p. 247). 117. Stromgren, E. (1914). Ueber den Ursprung der Kometen. Publ. mindre Meddelelser Klbenhavns Obs. No, 19, 193-250. 118. Armellini, G. (1922). The secular comets and the movement of the sun through space. Popular Ast. 30, 280-286. 119. LaPaz, L. (1955). Book review of J. G. Porter’s Comets and Meteor Streams. J . Opt. SOC.Am. 46, 231-233. 120. Whipple, F. L. (1952). Meteoritic phenomena and meteorites. I n “Physics and Medicine of the Upper Atmosphere” (C. S.White and 0. 0. Benson, Jr., eds.), Chapter X, p. 141. Univ. New Mexico Press, Albuquerque, New Mexico. 121. Nielsen, A. V. (1943). The velocity of the Pultusk meteor. Meddelelser Ole Roemer-Obs. No. 17, 224. 122. Wylie, C. C. (1940). The orbit of the Pultusk meteor. Popular Ast. 48, 306311 (in particular, see p. 307). 123. W-ylie, C. C. (1938). Real heights of bright meteors according to magnitude. Contrib. Uniz.. Iowa Obs. 8, 261-264. 124. Whipple, F. L. (1943). Meteors and the earth’s upper atmosphere. Revs. Mod. Phys. 16, 248. 125. Carmichael, R. D., Weaver, J. H., and LaPaz, L. (1937). “The Calculus,” pp. 275-278. Ginn, Boston, Mass. 126. Wylie, C. C. (1038). On von Niessl’s velocities for meteors. Contrib. Univ. Iowa Obs. 8, 253-260. 126a. Rinehart, J. S., and O’Neil, R. R. (1957). Observations on ablations andmetallurgical effects produced by surface heating of the Algoma meteorite. Tech. Rept. N o . 1, Smithsonian fnst. Astrophys. ODs., AFSOR-TN-57-541, ASTIA Doc. No. AD 136 529, 24 Sept. 127. Dodwell, G. F., and Fenner, C. (1942-1943). The Kybunga daylight meteor. Proc. Roy. Geograph. SOC.Australasia, S. Aurtralian Br. 44, 6-19. 128. von Heine-Geldern, R., and Pugh, E. M. (1953). The photography of highspeed metallic jets. Meteoritics 1, 5-10. 129. Epstein, P. S. (1931). On the resistance of projectiles. Proc. Natl. Acad. Sci. U.S. 17, 532-547. 130. Kent, R. €1. (1936). The smokiness of “smokeless” powder. U.S. Ballistic Lab. Rtpt. No. 33, Aberdeen Proving Ground. 131. Olivier, C. P. (1954). The Kentucky meteorite of 1950 September 19/20: A.M.S. No. 2326. Meteoritics 1, 247-250. 132. Horan, J. R. (1953). The Murray, CallowayICounty, Kentucky, aerolite (CN = $0881,366). Meteoritics 1, 114-121. 133. Opik, E. J., as quoted in the summary on the Jodrell Bank conference on meteor astronomy (1948). Observatory 68, 229. 134. Lowell, P. (1908). On the velocity with which meteors enter the earth’s atmosphere. Ast. J. 26, 1-3. 135. opik, E. J. (1951). Collision probabilities with the planets and the distribution

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of interplanetary matter. Proc. Roy. Irish Acad. 64, 165-199 (ref. on pp. 186189). Chant, C. A. (1913). An extraordinary meteoric display. J . Roy. Ast. Soc. Can. 7, 145-215. Wylie, C. C. (1953). Those flying saucers. Science 118, 125. See also his letter in “Comments and communications’’ section, Zbid. p. 726. Pickering, W. H. (1923). The meteoric procession of February 9, 1913. Part IV. Popular Ast. 31, 501-505 (ref. on p. 503). Astapowitsch, I. S. (1939). Concerning results of the study of orbits of 66 meteorites. Ast. J. Sou. Un. 16, 15-45. Leonard, F. C., and DeViolini, R. (1956). A classificational catalog of the meteoritic falls of the world. C d i j . Univ. Publ. Ast. 2(l), 1-79. Fisher, W. J. (1933). On the finding of newly fallen meteorites. Hurvard Reprints 89, 1-8. Morgan, W. W., Whitford, A. E., and Code, A. D. (1953). Studies in galactic

structure. I. A preliminary determination of the space distribution of the blue giants. Astrophys. J. 118, 318-322. 143. Blaauw, A., and Morgan, W. W. (1953). Note on the motion and possible origin of the 0-type star H D 34078 EE AE Aurigae and the emission nebula I C 405. Bull. Ast. Znst. Netherl. 12,76-79. 144. Blaauw, A,, and Morgan, W. W. (1954). The space motions of AE Aurigae and p Columbae with respect to the Orion nebulae. Astrophys. J . 119, 625-630. 145. Hager, D. (1953). Crater Mound (Meteor Crater), Arizona, a geological feature. Bull. Am. Assoc. Petrol. Ceol. 37, 821-857. 146. Millman, P. M. (1956). A profile study of the New Quebec crater. Publ. Dom. Obs. Ottawa 18,61-82. 147. Baldwin, R. B. (1949). “The Face of the Moon.” Univ. Chicago Press, Chicago, Illinois. 148. Urey, H. C. (1952). “The Planets: Their Origin and Development,” p. xi and pp. 26-29. Yale Univ. Press, New Haven. 149. Merrill, George P. (1908). The Meteor Crater of Canyon Diablo, Arizona: its history, origin and associated meteoric irons. Smithsonian Inst. Misc. Coll. 60, 471. 150. Wasiutynski, J. (1946). Studies in hydrodynamics and structure of stars and planets, Astrophys. Norveg. 4, 182-205. 151. Scott, William (1942). The distribution and probable origin of the lunar craters. Contrib. SOC.Res. Meteorites 3, 31-36. 152. LaPaz, L. (1949). The craters on the moon. Sci. American 181, 2-3. 153. Puiseux, P. (1906). Les formes polygonales sur la Lune. Bull. SOC.ast. France 20, 465-480. 154. LaPaz, L. (1954). Evidence on the nature of the Ungava crater unobscured b y glaciation. Meteoritics 1, 228. 155. Harrison, J. M. (1954). Ungava (Chubb) Crater and glaciation. J . Roy. Ast. SOC.Can. 48, 16-20. 156. Zimmerman, W. (1948). The non-circularity of the Canyon Diablo, Arizona, meteorite crater. Contrib. Meteorit. SOC.4, 148-150. 157. Foote, A. E. (1891). A new locality for meteoric iron with a preliminary notice of the discovery of diamonds in the iron. Proc. Am. Assoc. Adu. Sci. 40,279-283. 158. Moissan, H. (1893). Etude de la mbthrite de Canon Diablo. Compt. rend. 116, 288-290. 159. Barringer, D. M. (1909). “Meteor Crater (Formerly Called Coon Mountain or


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Coon Butte) in Northern Central Arizona,” p. 4. Privately printed. Read before the National Academy of Sciences meeting, Princeton Univ. November 16, 1909. 160. opik, E. J. (1936). Researches on the physical theory of meteor phenomena. I. Theory of formation of meteor craters. Acta et Comment. Univ. Tartu. A XXX1, 3-12. 161. Spencer, L. J. (1933). Meteoric iron and silica glass from the meteorite craters of Henbury (central Australia) and Wabar (Arabia). Mining Mag. (London) 23, 387-404. 162. Rogers, A. (1930). A unique occurrence of lechatelierite or silica glass. A m . J . Sci. 19, 195-202. 163. Blackwelder, Eliot (1953). Crater mound-meteor crater. Bull. A m . Assoc. Petrol. Geol. 37, 2577. 164. LaPaz, L. (1944). Meteoritical position problems. Contrib. SOC.Res. Meteorites 3, 152, footnote 2. 165. LaPaz, L., and Wiens, G. (1935). “On the History of the Bolide of 1908, June 30,” by L. Kulik. Contrib. SOC.Res. Meteorites 1(1), 29-34; (1935) “On the Fall of the Podkamennaya Tunguska Meteorite in 1908,” by L. A. Kulik. Ibid. 1(1), 3539; (1936). “Preliminary Results of the Meteorite Expeditions Made in the Decade 1921-31,” by L. A. Kulik. Ibid. 1(2), 15-20; (1937). “Instructions for the observations of [High Temperature Effects due to] Lightning [and other agencies],” by L. A. Kulik. Zbid. 1(3), 29-33; (1940). “New Data Concerning the Fall of the Great [Tungus] Meteorite on June 30, 1908 in Central Siberia,” by I. S. Astapowitsch. Ibid. 2(3), 203-226. 166. LaPaz, L. (1941). Meteorite craters and the hypothesis of the existence of contraterrene meteorites. Contrib. SOC.Res. Meteorites 2(4), 246. 167. Boldyreff, A. W., and LaPaz, L. (1950). Some characteristic features of the Sikhota-Alin (Ussuri) iron-meteorite shower [of the U.S.S.R.: ECN = T 1347,4621. Contrib. Meteorit. Soc. 4(4), 264-269. 168. Wylie, C. C. (1943). Calculations on the probable mass of the object which formed Meteor Crater. Popular Astron. 61, 97-99; (1943). Applying mine-crater formulas to Meteor Crater in Arizona. Ibid. 61, 220-222. 169. LaPaz, L. (1943). Probable mass of Canyon Diablo meteorite. Contrib. SOC.Res. Meteorites 3, 95. 170. Birkhoff, G., McDougal, D. P., Pugh, E. M., and Taylor, G. (1948). Explosives with lined cavities. J . Appl. Phys. 19, 563-582. 171. Rostoker, N. (1953). The formation of craters by high-speed particles. Meteoritics 1, 11-27. 172. Rinehart, J. S. (1950). Some observations on high-speed impact. Contrib. Meteorit. SOC.4, 299-305. 173. Barringer, D. M. (1905). Coon Mountain and its crater. Pror. Acad. Sci. Phila. 67, 868. 174. LaPae, L. (1950). A preliminary report on Indian ruins discovered near the crest of the Barringer Meteorite Crater, Arizona. Contrib. Meteorit. SOC.4(4), 285-286. 175. Blackwelder, E. (1932). The age of Meteor Crater. Science 76, 557-560. 176. Van Orstrand, C. E., and Dewey, F. P. (1916). Preliminary report on diffusion of solids. U.S. Geol. Surv. Profess. Pap. 96, 83-96. 177. Mallery, G. (1893). Picture writing of the American Indian. Ann. Rept. Bur. Am. Ethnol. 10, 25-777. 178. DaubrBe, G. A. (1891). Bolide peint6 par Raphael. Astronomie. Revue mens. populaire 10, 201-206; Newton, H. A. (1891). Fireball in the Madonna di Foiigno. Am. J . Sci. [3] 41, 235-239.

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179. A black and white reproduction of Medvedev’s painting appears as a frontispiece in E. L. Krinov’s “Bases of Meteoritics,” Moscow, 1955. 180. Kingsborough, E. K. (1831-1848). “Antiquities of Mexico,,’ 9 vols. Have11 and Colnaghi, London. 181. Cartailhac, E., and Breuil, H. (1904). Les peintures et gravures murales des Cavernes Pyreneennes. Anthropologie Paris 16, 625-644. 182. Raphael, M. (1945). ‘[Prehistoric Cave Paintings,” p. 78. Pantheon, New York. 183. Chambers, G. F. (1877). “Handbook of Descriptive Astronomy,” 3rd ed., p. 805. Oxford Univ. Press, London and New York. 184. Biot, E. (1841). Catalogue general des Qtoiles filantes et des autres met6ores observhs en Chine pendant 24 siitcles. Mbm. Savants Etrangers 1, 74. Academie des sciences morales et publiques, Paris. 185. Cunningham, A. (1883). The Andhara meteorite. Repls. Archaeol. Surv. Zndia (new Imperial series) 16, 32-34. 186. Maclaren, M. (1931). Lake Bosumtwi, Ashanti. Geograph J . 78,270-276; Junner, N. R. (1933). Lake Bosumtwi. Rept. Geol. Survey. Gold Coast 4-7; (1937). The geology of the Bosumtwi caldera and surrounding country. Bull. Gold Coast Geol. Surv. 8,5-38. 187. Sheikh, A. G. (1953). From America to Mecca on airborne pilgrimage. Natl. Geograph. Mag. 104(1), 1-60 (especially 26-27). 188. Brezina, A. (1889). Darstellung von Meteoriten auf antiken Munzen. Monatsbl. Numismat. Ges. Wien 70, 212-214. 189. Zimmer, G. F. (1916). The use of meteoritic iron by primitive man. J. Iron Steel Inst. (London) 94(2), 306-356; Rickard, T. A. (1941). The use of meteoric iron. J . Roy. Anthropol. Inst. Gt. Brit. and Ireland 71, 55-66. 190. Mellor, J. W. (1932). “ A Comprehensivc Treatise on Inorganic and Theoretical Chemistry,” Vol. XII, p. 484. Longmans, New York. 191. Desch, C. H. (1928). Report on the Metallurgical examination of specimens for the Sumerian Committee of the British Association. Rept. Brit. Assoc. Adu. Sci. 437-441. 192. Wainwright, G. A. (1932). Iron in Egypt. J . Egypt. Archaeol. 18, 3-15. 193. Coghlan, H. H. (1941). Prehistoric iron prior to the dispersion of the Hittite Empire. M a n 41, 74. 194. Khan, M. A. R. (1934). “Meteors and Meteoric Iron in India.” Privately printed by Moses and Co., Secunderabad. 195. Sowerby, J. (1820). Particulars of the sword of meteoric iron presented b y Mr. Sowerby to Emperor Alexander of Russia. Phil. Mag. 66, 49-52. 196. Berwerth, F. (1907). Javanische Waffen mit “ Meteoreisenpamor.” Tschermak’s Mineralog. u. petrog. Mitt. 26, 506-507. 197. Kinnicutt, P. (1884). Report on the meteoric iron from the Altar Mounts in the Little Miami Valley. Ann. Rept. Peabody M u s . 3, 381. 198. Grogan, Robert M. (1948). Beads of meteoric iron from an Indian mound near Havana, Illinois. Amer. Antiquity 13, No. 4, 302. 199. Leonard, F. C. (1948). A simplified classification of meteorites and its symbolism. Contrib. Meteorit. SOC.4, 142, 200. Watson, F. G. (1956). “Between the Planets,” rev. ed., p. 161. Harvard Univ. Press, Cambridge, Mass. 201. Sorby, H. C. (1877). On the structure and origin of meteorites. Nature 16, 496498. 202. Cohen, E. (1894). “Meteoritenkunde,” Vol. 11. Koch, Stuttgart.

SMOOTHING AND FILTERING OF TIME SERIES AND SPACE FIELDS J. Leith Holloway, J r . U. S.

Weather Bureau, Washington

D.C.

Page Introduction.. . . . . . . . . . . . ............................. 351 Time Smoothing and Filtering., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Equalization, Pre-emphasis, and Inverse Smoothing. . . . . . . . . . . . . . . . . . 353 Smoothing and Filtering Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Frequency Response of Smoothing Functions and Other Filters.. . . . . . . . . . . 355 Design of Smoothing Functions and Filters with Specified Frequency .... 363 Response. . . . . . . . . . . . . . . . . . . . . . . . . . . 7. High-Pass and Band-Pass Filtering Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 365 8. Elementary Smoothing and Filtering Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 369 9. Design of Inverse Smoothing Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 10. Design of Pre-emphasis Filters.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 11. Filtering by Means of Derivatives of Time Series.. . . . . . . . . . . . . . . . . . . . . . . 378 12. Space Smoothing and Filtering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Acknowledgments.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 List of Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

1. 2. 3. 4. 5. 6.

1. INTRODUCTION

Time and space smoothing are widely used in the study of geophysical problems. For example, the drawing of smooth isolines through data plotted on a map of some geophysical variable is a type of space smoothing. The computation of consecutive monthly mean values of a series of measurements and the use of long-lag instruments for suppressing rapid fluctuations in readings are examples of time smoothing. The purpose of this paper is t o give the reader a better understanding of just what these smoothing methods really accomplish so as to provide a rational basis for the selection of any particular smoothing method. Smoothing will be shown t o be a special type of filtering, and the analysis will therefore be extended t o cover numerical filters of all types. Much of the information in this paper is not new, but is scattered rather widely in the mathematical, statistical, and scientific literature. An attempt is made here to combine all pertinent information on smoothing and filtering in a concise, simple, underst,andable form for the benefit of geophysicists concerned with these problems. Since this paper is pri351

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J. LEITH HOLLOWAY, JR.

marily a survey of the field, rigorous mathematical derivations of the equations are not given. 2. TIMESMOOTHING AND FILTERING In statistical work a series of data arranged chronologically is commonly called a time series. For example, a series of daily mean temperatures in order of date is a time series. Generally the data in time series are equally spaced in time, and therefore all time series discussed in this paper will have equally spaced data. The variations in the data with time may be relatively smooth and orderly or may be rather complex and without apparent pattern. It is convenient to consider that the variations in time are produced by superimposed sinusoidal waves of various amplitudes, frequencies, and phases. The Fourier theorem states that no matter how complicated the fluctuations in the data may be they can be accounted for by the superposition of a number of simple component sinusoidal waves [l]. The amplitudes, frequencies, and phases of these waves are generally changing constantly with time. The exception to this is where fixed cyclic processes such as diurnal and annual influences tend to induce waves of constant frequency (one cycle per day or per year) into the data. The purpose of time smoothing is to attenuate the amplitudes of high-frequency waves in the data without significantly affecting the lowfrequency components. The attenuation is roughly proportional to frequency. Above some high frequency, depending upon the properties of the smoothing method used, the attenuation is complete for all practical purposes. The assumption upon which the use of smoothing is justified is that high-frequency oscillations in the data are either random error (“noise”) or are of no significance to the particular type of evaluation of the data to be carried out after the smoothing. Smoothing thus enables one to concentrate on the low frequencies without the distraction caused by high-frequency noise and other irrelevant fluctuations. A smoothed value of an observation in a time series is merely an estimate of what the value would be if noise and other undesired high frequencies were not present in the series. Smoothing is a special case of the broader general process of filtering, a concept brought into the field of time series analysis from electrical engineering. An electrical filter, such as a simple resistance-capacitance network, separates various sinusoidal components of an “ electrical time series” according to frequency. An electrical analog of a numerical time series is a continuously varying voltage or current-usually referred to as a “signal” in electrical engineering. Electrical filters can be designed to pass only low frequencies of the signal while attenuating or eliminating

SMOOTHINQ AND FILTERING

353

high frequencies. This type of filter is commonly called a “low-pass filter.” Thus smoothing of a numerical time series is analogous to lowpass filtering of an electrical signal. Filters can also be designed to pass only high frequencies and attenuate low frequencies (“ high-pass filter ”). Finally other filters pass a band of intermediate frequencies ( ‘ I band-pass filter”) and attenuate both very low and very high frequencies in the input signal. Numerical band-pass and high-pass filters will be described in Section 7. 3.

EQUALIZATION, PRE-EMPHASIS, AND

INVERSE SMOOTHING

Two other terms which may be borrowed from electrical engineering and applied to time series analysis are “equalization” and “pre-emphasis.” Equalization is the process of restoring the original balance of amplitudes of sinusoidal waves in a signal which has previously been altered by filtering or pre-emphasis. Pre-emphasis is the amplification before transmission of certain bands of frequencies in a signal above a standard amplification. For example, the high audio-frequencies are pre-emphasized before transmission by a frequency-modulation transmitter to override high audio-frequency noise introduced by the transmitter, the atmosphere, and the receiver. This can be done since high audio-frequencies are generally of low amplitude in the original signal. At the receiver the correct balance of frequencies in the audio signal is restored by an equalization filter (“de-emphasis” filter) which in this case is merely a low-pass filter having an attenuation which is the inverse of the frequency characteristics of the pre-emphasis at the transmitter. A second example of electrical equalization is the compensation for the attenuation of high frequencies in a long transmission line. I n this case the line itself acts as a low-pass filter. The original balance of frequencies in the signal is approximately restored at the end of the line by an equalization amplifier which has greater amplification a t high than at low frequencies. As long as some fraction of the signal a t a given frequency is received above the noise in the line, the original relative amplitude at this frequency can be approximately restored. Of course, in order to equalize properly it is necessary to know how much the line attenuates the signal a t each frequency. It follows that it should be possible to equalize a time series previously smoothed provided that the characteristics of the smoothing were known. Later in this paper, a method for performing this equalization numerically will be given. The term “inverse smoothing” will be used for the numerical equalization used for accentuating high frequencies in respect to low frequencies in order to restore the original balance of frequencies to a smoothed time series. A good deal of work on inverse smoothing already

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J. LEITH HOLLOWAY, J R .

has been done in several scientific fields, notably in radio-astronomy 12-10].

4. SMOOTHING AND FILTERING FUNCTIONS Smoothing of a time series is performed by a type of numerical or mathematical operator which will be termed a ‘‘ smoothing function” in this paper. Generally the smoothing function consists of a series of fractional values, called weights. I n this paper, the term “filtering function” will be applied t o operators which perform filtering of time series other than smoothing. Filtering functions also generally consist of various weights similar t o those of smoothing functions. The weights determine in what proportion each observation in the time series contributes to the estimate of the smoothed or filtered value. In the process of smoothing or filtering, successive observations in a time series are cumulatively multiplied by these weights. The smoothed or filtered value corresponding t o the observation x t in the time series is computed from observations xt-,, through xt+, by the following linear equation: (4.1) & =

5

k=-n

Wkxi+k

= w-nX1-n

+

’ ’

+

+

w-1%-1

+

W O X ~ WlXt+l

+

*

*

+

+

WmXt+m

where w k is a particular weight in the smoothing or filtering function. The weight W O , which is multiplied by the observation xt, will be termed the principal weight in this paper. The principal weight is the central one in the case of the equally-weighted running mean where all weights are identical and equal to 1,” where N is the number of observations used in computing the mean. The use of smoothing and filtering functions is illustrated in Fig. 1. The weights of the smoothing function in the block shown in this figure are cumulatively cross-multiplied by the adjacent values in the time series and the resulting product is entered opposite the time series value multiplied by the principal weight. Then the smoothing function is moved down one time increment (data interval) along the time series and t h e cross-multiplication is repeated t o obtain a second smoothed value. This process is repeated unt,il the lowest weight in the smoothing function reaches the end of the series. The sum of the weights of a smoothing or filtering function determines the ratio of the mean of the smoothed or filtered series t o the mean of the original series assuming that these means are computed over periods long enough to insure stable results. I n smoothing it is generally desired

355

SMOOTHING AND FILTERING

t o leave the mean. of the series unchanged, and consequently the sum of the weights of most smoothing functions is made equal t o unity. With some filters to be discussed later it is not necessary to preserve the mean of the series, and in these cases the sum of the weights may be different from unity. For theoretical evaluation of smoothing and filtering functions it is often useful t o consider th at these functions are continuous rather than composed of discrete weights. The continuous analytic form of the smoothing or filtering function will be some function w(t) describing the envelope of the discrete weights. The time origin here is the time of the principal weight; namely, the time of the observation for which a smoothed value is being computed. The area under the continuous Time Series Smoothing Function\ Principal Weight

--f

,016 +.094 +.234 +.312 +.234 +.094 +.016

X X X X X X X

Time Series

28 23 21 24+ 22.3 22 20 17 18 29 35

.016 + . 094 + . 234 + . 312 + . 234 + . 094 +.OlS

X X X X X X X

28 23 Smoothed 21 24 2 2 . 3 Values 22 4 2 1 . 4 20 17 18 29 35

FIG. 1. Illustration of smoothing a time series by means of a typical smoothing function.

smoothing or filtering function corresponds t o the sum of the weights of a discrete-valued function. Thus the ratio R of the mean of the filtered series t o that of the original series is

R

=

-w

w(t) at

in the case of continuous functions.

5. FREQUENCY RESPONSE OF SMOOTHING FUNCTIONS AND OTHERFILTERS The ratio of the amplitude of a wave of a given frequency in the time series after filtering t o the original amplitude before filtering is the frequency response of the filter a t this frequency. The frequency response is a function of frequency. For example, the value of the frequency response of a smoothing function is near unity at low frequencies and near zero a t high frequencies.

356


The frequency response of a mathematical filter such as a smoothing function can be derived by determining the effect of this filter on a unitamplitude sinusoidal wave of frequency f. This wave may be represented by a unit vector rotating about the origin with angular velocity of 27rf in the complex plane. The projections of this unit vector on the real and imaginary axes define the phase of the unit amplitude wave a t any time. The projection of this vector on the real axis is cos (27rft), and on the imaginary axis, sin (27rft), where t is time.' After smoothing or some other form of filtering, the modified wave can be represented b y another vector rotating about the origin in the complex plane with the same angular velocity but having a magnitude and phase angle generally different from that of the original unit vector. The amplitude of the smoothed or filtered wave and its phase angle a t any time is determined by the instantaneous projections of this modified vector on the real and imaginary axes. These projections can be shown to be merely the weighted mean of the projections of the original unit vector on these axes averaged over the interval during which the smoothing or filtering function is operating, hereafter t o be referred t o as the filtering intervaL2 The weighted mean projection of the unit vector on the real axis averaged over the filtering interval is obtained by summing the products of each weight and the corresponding value of cos (27rf.t) a t time t. The weighted mean projection on the imaginary axis is obtained in a similar manner but with the sine substituted for the cosine. I n the case of continuous analytic forms of the smoothing and filtering functions these mean projections must be computed by integration. The ratio of the magnitude of the modified vector t o the magnitude of the unit vector (unity) is the frequency response of the filter. Thus, the magnitude of the modified vector is the frequency response. This magnitude is the absolute value of the complex quantity R ( f ) given by

The cosine term of (5.1) is a weighted time mean projection of the unit vector on the real axis, and the sine term is the simultaneous weighted time mean projection of this vector on the imaginary axis. This equation will be recognized as the inverse Fourier transform of w(t). Therefore, the inverse Fourier transform of the smoothing or filtering function is the filter's frequency response function. The absolute value of R ( f ) is the The arguments in this paragraph are valid for any time origin in addition to tha t used in the last section. * For example, the filtering interval in Fig. 1 is seven data intervals.

357


square root of the sum of the squares of the real and imaginary parts of R ( f ); namely

IR(nl = uw-w" + [ I m { R ( f ) W

(5.2)

The angle between the original and the modified vector is the phase shift which the filtering function w(t) produces at frequency f. This angle 4 is given by

4 = tan-' [Im(R(f))/Re{R(f)Jl

(5.3)

+ +

For smoothing and filtering functions having (n m 1) discrete weights the frequency response is computed by the following form of equation (5.1)

where units of frequency f are cycles per data interval. The absolute value of R(f) obtained from the above equation is again computed by equation (5.2). It is desirable that smoothing and filtering functions not shift the phase of waves of any frequency. The phase shift angle can be made equal to zero by requiring that the numerator of the argument in equation (5.3)(namely, the specified weighted time mean projection of the unit vector on the imaginary axis) be zero. This in turn can be accomplished by requiring that the function w(t) be even (namely, that w(-t) = ~ ( t ) ) , ~ for if w(t) is even, the terms containing the sines in equations (5.1) and (5.4) are zero, and R ( j ) is a pure real quantity computed by

R(f)

(5.5)

=

w(t) cos (27rft) dt = 2

/om

w(t) cos (27rft) dt

for continuous w(t) functions or

+

for smoothing and filtering functions having ( 2 n 1) discrete weights. The frequency response of an equally-weighted running mean of (2n 1) consecutive terms may be computed from equation (5.6) be-

+

* I n the case of discrete-weighted is that

W-E

=

smoothing and filtering functions this condition wk.The smoothing function in Fig. 1 is even.

358


cause w-k = wk; in fact, in this case every weight Equation (5.6) gives R ( f ) = wo 2 ~ cos 1 (2~jA.t) 2 ~ cos 2 (&!At) (5.7)

+ = (2n +

wk

+

l)-I[l

= 1/(2n

+ 1).

+. + 2w, cos (2n7rfa.t) + 2 cos (27rfAt) ++2 cos (&!At) . . . + 2 cos (2n?rfAt)] *

where At is the data interval. The use of At here avoids the requirement that the units of frequency be cycles per data interval; the units of frequency in (5.7) are thus cycles per particular unit of time used. A convenient approximation of the frequency response of the equallyweighted running mean can be computed from equation (5.5) by using the analytic form of the envelope of the weights, w ( t ) ; namely, (5.8)

where T is the filtering interval. Equation (5.5) gives (5.9)

R(O

=

2

,l""z T-'

cos (2n;fl) c~t=

(Z!T)-~

sin

(.trf~)

Equation (5.9) gives a very accurate approximation of the frequency response of equally-weighted running means.4 The exact frequency response of an equally-weighted running mean having five weights of one-fifth each is shown in Fig. 2 as a dotted line. This response is computed from equation (5.7) assuming a data interval of T/5 so as to make the filtering interval equal to T. For comparison, the approximate frequency response for this type of running mean computed from equation (5.9) is shown in this figure as a solid line. Notice that the agreement between these two curves is quite good; the agreement would be even better with running means of more than five terms. The physical meaning of the negative response a t some frequencies in Fig. 2, is that the input waves of these frequencies are reversed in polarity in addition to being changed in amplitude. A reversal of polarity of a wave means that its maxima are changed into minima and vice versa.6 Positive and negative values of the frequency response above the frequency of the first zero response point are undesirable because they will introduce many unwanted and misleading high-frequency ripples into the smoothed output. A method for suppressing or eliminating these undesirable responses By L'Hopital's rule this function has the value unity at f = 0. Reversal of the polarity of a wave corresponds to a reversal of the direction of rotation of the vector representing this wave. This is equivalent to a 180-degree phase shift of the wave. 4 6

359


1.0

w u)

0.5

z P 0 W u) K

0

0.0

I .o

2.0

FREQUENCY IN CYCLES PER FILTERING INTERVAL

T

FIQ.2. Frequency responses of equally-weighted running means and of a normal probability curve smoothing function. Solid line is response of the running mean computed from equation (5.9). Dotted line is response of the five-term running mean computed from equation (5.7). Dashed line is response of normal curve smoothing function having u = T / 6 computed from equation (5.11).

is t o provide a smoothing function having weights decreasing in magnitude outward from the principal weight. For example, the smoothing function weights may be made proportional t o the ordinates of the normal probability curve. A continuous analytic form of this smoothing function is (5.10)

w ( t ) = (27rg2)-%exp ( - t 2 / 2 a 2 )

The area under the above function is unity so that the mean of the smoothed series will be conserved. The frequency response of this smoothing function is obtained from equation (5.5); namely, (5.11)

R ( j ) = exp (-2?r2a2f2)

360


The frequency response given by equation (5.11)for the normal curve smoothing function decreases smoothly with increasing frequency and asymptotically approaches zero. Thus it avoids the negative values of response exhibited by the equally-weighted running mean. Although zero response is theoretically never reached, for practical purposes this smoothing function has a finite “cutoff frequency.” The “cutoff frequency” of a smoothing function will be defined as the lowest frequency at which the response reaches zero for all practical purposes and remains zero for all higher frequencies. Thus in cases like this where the response function approaches zero asymptotically the cutoff frequency can be taken as that frequency where the response drops to some arbitrarily chosen low value, say one per cent. The cutoff frequency of the normal curve smoothing function is controlled by the value of the parameter u in equation (5.10). The response of the normal curve smoothing function having a filtering interval of T is plotted in Fig. 2 as a dashed line for comparison with that of the equally-weighted running mean having the same filtering interval. The filtering interval of a normal curve smoothing function is taken to be 6u, for beyond 30-from the origin the normal curve ordinates have negligible value. The equations in this section may also be used t o compute the frequency response of an exponential smoothing function; namely, one having the analytic form:

(5.12) where X is the so-called time constant or lag coefficient. The area under this function is unity so that the mean of the series is unaffected by the smoothing. This type of smoothing is that which is performed by physical instruments which are viscous-damped and have constant X and by simple, two-element resistance-capacitance low-pass electrical filters. An example of an instrument which smooths in this manner is the simple mercury-in-glass thermometer. Since this smoothing function defined above is continuous and not even,6 equation (5.1)must be used for determining its frequency response. This equation gives

+

(5.13) R(f) = (1 h Z f 2 X “ ) - ’ and from (5.2),’ (5.14) IR(f)I = (1

- (2nifX)(1 + h 2 f Z X 2 ) - ’

+

47?f2X2)-%

This smoothing function has zero values for t > 0 since no physical instrument can take future variations into account in its smoothing. 7 Middleton and Spilhaus obtain this same result by means of differential equations [Ill.

361


Since the imaginary part of R(f) is not zero owing to the unsymmetrical distribution of weighting about the origin, the smoothed series has phase error which is computed from equation (5.3) and given below

4

(5.15)

=

tan-' (-27rfX)

The frequency response and phase shift for exponential smoothing is shown in Fig. 3 where X equals unity. Notice that this frequency response also approaches zero asymptotically. Smoothed data are often obtained by deliberately increasing the viscous damping of the measuring instrument [I21 or by use of simple resistance-capacitance electrical filters. The fact that this type of smoothing introduces phase error into the data suggests that a better smoothing

1.0

-+ c

6'0

r

Y !

0.5

30'

/

I

I

I

I

I

I

I

FREQUENCY

FIG. 3. Frequency response R ( f ) and phase shift function having a lag coefficient of unity.

I

I I .o

0.5

+ of

I

1

an exponential smoothing

procedure would be to obtain the d ata from the fastest response instrument available and to smooth these data later by means of smoothing functions having no phase error, such as the normal curve smoothing function. However, no matter how fast its response is, any physical instrument smooths the data to some extent. It is only necessary th a t the time constant of the instrument be short in comparison with the wavelength of variations in the data whose amplitudes and phases are desired to be recorded accurately. The results of smoothing a time series by the three methods described above are compared in Fig. 4 where the original unsmoothed series is shown as a solid line and equally-weighted running means of five consecutive observations are connected by a dotted line. The dashed line in this figure shows the series smoothed by a discrete-weighted approximation data interval; the of a normal curve smoothing function having u =

362


weights of this function are 0.03, 0.23, 0.48, 0.23, and 0.03 in that order. The frequency responses of the running means and the normal curve smoothing function are given by the dotted and dashed lines, respectively, in Fig. 2 when the filtering interval is taken to be 5 data intervals. The dot-dashed line in Fig. 4 represents the series smoothed by a discreteweighted approximation of an exponential smoothing function having X = 2.5 data intervals. The frequency response of this exponential filter versus frequency in cycles per data interval is obtained from Fig. 3 by multiplying the values on the abscissa by X-’ = 0.4. The filtering intervals and u’s and X’s of these three smoothing functions were chosen so that the degree of smoothing of each of the three methods would be about

I

0

5

I

10 TIME

IN

I

I

I

15 20 25 30 DATA INTERVALS AFTER ARBITRARY ORIGIN

I

35

-

40

FIG.4. Time series (solid line) and the same series smoothed by means of an equallyweighted running mean (dotted curve), by a normal curve smoothing function (dashed curve) and by exponential smoothing (dot-dashed curve).

the same. Fig. 4 illustrates the unfortunate polarity reversals effected by equally-weighted running means at some frequencies (for example, at point A ) and the phase shift produced by exponential smoothing (at points B and C).The normal curve smoothing does not exhibit either of these two shortcomings. The frequency response of many other smoothing and filtering functions may be determined from the equations in this section. When the exact response function of a particular discrete-weighted smoothing or filtering function is desired, equation (5.4) or (5.6) must be used. However, as illustrated in this section, equations (5.1) and (5.5) may be used for determining estimates of the response of such functions having a relatively large number of weights, provided the envelopes of these


363

weights can be expressed in simple integratable analytic forms. If the smoothing or filtering function is intrinsically continuous (as is the case with exponential smoothing by an electrical filter, for example), either equation (5.1) or (5.5) will be required for computing the exact frequency response function. Other methods for computing frequency response functions will be discussed later in this paper. 6. DESIGN OF SMOOTHING FUNCTIONS AND FILTERS WITH SPECIFIED FREQUENCY RESPONSE

The procedure in the last section may be reversed and a smoothing or filtering function w(t) having a specified frequency response function of R(f) may be obtained by solving the integral equation (5.1). This solution is8

For a n even response function R(f), equation (6.1) reduces t o (6.2)

w(t) = 2

lom R(f) cos (27rft) df

These equations may be recognized as those for Fourier transforms of

R (f)*

An example of the use of equation (6.2) is the determination of the smoothing function having a flat response of unity out t o some cutoff frequency fc and zero response beyond; namely,

(6.3) R(f)= Use of equation (6.2) gives (6.4)

w(t) = 2

( 0:1

OSflfC f

> fc

f cos (27rft) df =

(Tt1-1

sin (2Tjct)

This smoothing function is a damped wave extending forward and backward in time from the origin. However, because the damping is rather slow, this function will often be impractical t o use, since it will extend over so much of the series to be smoothed. The function can, of 8A wave of negative frequency is merely one of the corresponding positive frequency with reversed polarity. Ordinarily frequency is not thought of as having negative values; the above definition is given only to clarify equations such as (6.1) where frequency is allowed to take on negative values. From the above it is clear that a t least the absolute value of all frequency response functions must always be even functions, for it would be contradictory for a filter to have a different response to a negative frequency than to the corresponding positive one. When l R ( f ) (is not even, w(l) will generally be a complex function (composed of both real and imaginary parts) and therefore will have no physical meaning as a filtering function.

364

J. LEITH HOLLOWAY, JR. I

1

I

I

I

I

I

I

I

1.2

I .o

\

0.8

-c

0.6

LT

0.4

I

I I \

.

i \

*.

0.2

'i. \

0.0 I

I

I f

c

2fc

FREQUENCY

FIG.5. Theoretical frequency response of a smoothing function having the shape of the damped wave defined by equation (6.4)(dashed line), and the actual response of this function truncated beyond the first negative lobes on either side of the central positive lobe (solid curve). The dotted curve is the response of the normal curve 1 smoothing function having u = -. 3fc

course, be taken to be zero (truncated) at some convenient distance on each side of the origin, but this alters the response in an undesirable way, and the closer t o the origin it is truncated, the less desirable the response becomes. For example, the frequency response of this function truncated at t = kl/fc is shown in Fig. 5 as a solid line. This condition specifies two negative lobes on each side of the central positive weights. The desired frequency response specified by (6.3) is shown in this figure as a


365

dashed line. Notice that the truncation causes the actual frequency response of the smoothing function to differ considerably from the desired response a t most frequencies. In fact, there is actually amplification a t intermediate frequencies and undesirable negative response at certain higher frequencies. For comparison, the frequency response of a normal 1 3fc

curve smoothing function having u = - is also shown in Fig. 5 as a dotted line. The filtering intervals of these two smoothing functions are essentially the same. It is seen that the cutoff frequency of this normal curve smoothing function as defined in the last section is lower than that of this truncated version of the function designed to have sharp cutoff characteristics. It should be mentioned here that in performing mathematical filtering it is tacitly assumed that the periodicities present at the time for which the filtered variable in the time series is being estimated are unchanged in amplitude and phase during the filtering interval. Thus, i t is advisable to have this filtering interval as short as possible so as to have this assumption reasonably justified. Smoothing functions having negative weights beyond the positive central values do stretch this assumption rather far owing t o their longer filtering intervals for given cutoff frequencies. 7.

HIGH-PASS A N D

BAND-PASS FILTERING FUNCTIONS

Earlier in this paper electrical band-pass and high-pass filters were described. The same type of frequency separation can be accomplished numerically by a modification of the smoothing procedures. If smoothed values are subtracted from the corresponding values in the original unsmoothed time series, only high frequencies will remain; thus, this operation is equivalent to high-pass filtering. If well-smoothed values in a time series are subtracted from values smoothed to a lesser extent, only intermediate frequencies will remain, for the high frequencies will have been smoothed out and the low frequencies will have been subtracted out of the original series; this operation then is equivalent t o band-pass filtering. Therefore, by use of these methods the oscillations in a time series can be separated into three time series each containing a particular band of frequencies-high, intermediate, and low. For example, let xt represent the observation in a time series at time t , and f t and Zt be the smoothed values computed by normal curve smoothing functions having u = % day and u = 5 days, respectively (see equation (5.10) for a definition of u). Only low frequencies of the original data will appear in the time series of Zt values. The intermediate frequencies will appear in the series resulting from the subtraction of 3,from I t , and

366


only the high frequencies will remain in the series computed by subtracting & from zt. The frequency response of the intermediate frequency band-pass filter is given by R(f)z-2 =

(7.1)

R(f)z - R ( f h

where the R(f)'s are frequency responses and the subscripts indicate the filters to which they correspond. The frequency response of the high-pass filter is

R(fL-3 = 1 - R(nz

(7.2)

The frequency responses of these filters in the above example are shown in Fig. 6. Notice that there is considerable overlap in the response

I'

1

1

1

1

1

1

I

1

PERIOD

IN

I

I

l

l

1

1

I

DAYS

FIG.6. Frequency responses of a low-pass filter (solid curve), a band-pass filter (dashed curve), and a high-pass filter (dotted curve) generated by two normal curve smoothing functions having u 36 day and u = 5 days, respectively. =i

curves of the three filters, and the transition between the appearance of a wave in the output of one filter and in the next filter is smooth with a uniform change of the frequency of the wave. If & and Zt had been computed by equally-weighted running means instead of by normal curve smoothing functions, this transition would have been less smooth and the response of each filter would have been more irregular owing to the negative response characteristics of running means to certain frequencies. To illustrate this filtering technique a series of twice daily barometric pressures a t the Washington National Airport for spring 1956 are filtered by means of the filters described above. The pressures a t the National Airport at 0100 and 1300 EST are plotted in Fig. 7(A), and a solid line is drawn connecting them. These values are smoothed by the normal curve smoothing function having u = 5 days, and the resulting smoothed series is represented by a dashed line in Fig. 7(A). The weights for a discretevalued approximation of this smoothing function are given in Table I. The

367

BMOOTHING AND FILTERING

process of smoothing individual half-day observations by this smoothing function would be laborious. However, it is only necessary to compute thesezwell-smoothed values for every tenth observation (every fifth day) becauseathe resulting smoothed series contains no waves of shorter period thanltenrdays. Intermediate smoothed values needed as the subtrahends of the band-pass filter can be obtained by graphical interpolation. The series resulting from the band-pass and high-pass filtering are shown in

W

f

I

29.0

I

I

I

I

30

5

10

I

I

I

\-I

-0.5

I

15

20

25 APRIL 1956

15 MAY 1956

20

25

FIG.7. Illustration of low-pass, band-pass, and high-pass filtering of station pressures a t the Washington National Airport. A. I n this the original pressures are indicated by the solid line and the output of the low-pass filter is dashed. B. I n this the solid line is the output of the high-pass filter and the dashed line tha t of the band-pass filter.

Fig. 7(B) as dashed and solid lines, respectively. The weights of the normal curve smoothing function having CT = 4% day used in computing the lesser smoothed values are given in Table I. The low-pass filtered series in Fig. 7(A) appears to have a low amplitude wave of about 30 days period. Since the band-pass filter has appreciable response to waves of this period, this wave also appears weakly in the output of this filter along with waves of much shorter period. Likewise there are also waves which appear in the output of both the band-pass and high-pass filters. Thus, the partition of the waves according to period is not perfect, but it is adequate to facilitate greatly the analysis of complicated fluctuations in data such as are exhibited by the original pressure series in Fig. 7(A).

368


TABLE I. Weights for discrete-valued approximations of normal curve smoothing functions having = 5 days and 35 day.* (I

u =

5 days

0.001 0.001 0.001 0.001 0.002 0.002 0.003 0.004 0.004 0.005 0.007 0.008 0.010 0.011 0.013 0.015 0.017 0.020 0.022 0.024 0.027 0.029 0.031 0.033 0.035 0.037 0.038 0.039 0.040 0.040 (Principal weight) 0.040 0.039 0.038

o =

35 day

0.004 0.054 0.242 0.400 (Principal weight) 0.242 0.054 0.004

t?tC.t

* Values are rounded t o a few significant places for convenience in computation.

t Rest of the weights of this smoothing function are identical with the ones above the principal weight but in reverse order.

Similar sets of filters with different frequency ranges could be designed for studying other scales of atmospheric phenomena such as atmospheric turbulence. For example, Panofsky [13] used this type of filtering for making a crude spectral analysis of wind fluctuations at the Brookhaven National Laboratory. If desired, the operation of smoothing and subtracting may be com-

369


bined into a single filtering function. The appropriate weights for the high-pass filter are the negative of the weights for the associated smoothing function except for the principal weight which is one minus the principal weight of the smoothing function. For example, a high-pass filter generated by the subtraction of values smoothed by a smoothing function 34, and Ks in that order (where 36 is the having weights of ?46, +34, -%, and principal weight) would have the weights: -).is, -%S. Notice that the sum of the weights of a high-pass filter such as this will be zero because it does not pass the mean of the original series (a component of zero frequency). Therefore, the output of this high-pass filter will be a series having a mean of zero. Two smoothing functions are used in the derivation of the band-pass filter. The weights for the band-pass filtering function are merely the weights of the smoothing function having the lowest cutoff frequency subtracted from the corresponding weights of the other smoothing function, This filter also does not pass the mean of the original time series.

x, x,

-x,

8. ELEMENTARY SMOOTHING AND FILTERING FUNCTIONS

A useful approach t o the design of smoothing functions and other mathematical filters is t o build up filtering functions from elementary filtering functions consisting of only three weights each. The two outer weights of the elementary filtering functions are made equal,g and the sum of the weights is made equal to unity for smoothing functions and zero for filters not passing the mean of the original series. The frequency response of an elementary smoothing or filtering function is obtained from equation (8.1) below, which is derived from equation (5.6) (8.1)

R(f)= WO

+2

~ cos 1 (2~jAt)

where wo is the central (and principal) weight, w1 is the value of each of the two outer weights, and At is the data interval. If one elementary filtering function provides insufficient filtering, a time series may be successively1° smoothed or filtered by this same elementary filtering function until the desired smoothing or filtering is 9 Each elementary filtering function is made even (namely, wP1 = wl)so that it will produce no phase error and so that more complicated filters built up from a combination of these elementary filters will also be even and therefore produce no phase error. 10 By “successive” filtering here is meant the repeated filtering of the entire series by the same or different filter-not the normal application of the filter to the series centered on each successive observation which is required for computing the filtered series.

370


accomplished. Also different elementary filtering functions may be used on succeeding filterings. It can be shown that the frequency response of the sequential application of these elementary filtering functions is the product of all the responses of the individual elementary filtering functions used; namely, M

R(fh

- .

R(f)M By use of this equation it is possible to specify certain features of the resultant frequency response R ( f ) R desired at given frequencies and t o solve for the various weights required in the elementary filtering functions to be used to give this response. This type of procedure has been successfully used by Shuman [14]. Once the elementary filtering functions have been designed, a composite filtering function can be generated which will circumvent the operation of sequential application of each elementary filtering function and give the resultant filtered series in one step. This is accomplished by first computing the cumulative cross-products of the weights of the first and second elementary filtering functions at various lags. Then the resulting series of weights is multiplied in a similar manner by the third elementary filtering function and so on until all the elementary filtering functions have been multiplied as many times as they would have been used in the sequential filtering operation. The final resulting series of weights is the composite filtering function. For example, when two three-weight elementary filtering functions having principal weights of wo and WOand outer weights of w1 and WI, respectively, are combined, the composite filtering functions will have weights of wlWI, wowl wlW0, 2wlW1 WOWO,wowl wlWo, and wlWl in that order with the principal weight being 2wlW1 wowo. After combining a number of elementary filtering functions, often many of the outer weights of the composite functions will become insignificantly small and may be neglected. However, it will be necessary to adjust the remaining weights in the composite function to insure that their sum is correct (namely, unity in the case of a smoothing function). A method of smoothing that has been discussed by Brooks and Carruthers [15] is to compute running means of pairs of observations in a time series and in turn to take running means of pairs of these smoothed values and so on until the time series is smoothed sufficiently. This procedure may be considered as the repeated application of a two-weight elementary smoothing function having weights of one-half each. Composite smoothing functions generated solely from this elementary smooth=

+

+

*

R(f)2

*

*

*

+

+

371


ing function will have weights proportional to the familiar symmetrical binominal coefficients; namely, the coefficients of the expansion of (P 4)". For example, the composite smoothing function generated from two of these two-weight elementary smoothing functions has weights of g, 36, and in that order, these weights being proportional to the coefficients (1, 2, and 1) of the second power expansion of ( p q ) . This particular three-weight elementary smoothing function will be referred to hereafter as the elementary binomial smoothing function. It is well known that the envelope of the coefficients in the expansion of ( p q) approaches the shape of the normal probability curve as a limit a s the exponent increases. Therefore, the above procedure of taking repeated running means of pairs of observations in a time series approximates the use of a normal curve smoothing function when the number of successive smoothings is large; this corresponds t o the use of a large number of weights in the composite smoothing function. The frequency response of the two-weight elementary smoothing function may be computed from equation (8.1) by setting wg = 0 and w1 = $5. However, the data interval t o be used in this equation, At', will obviously be one-half the actual data interval At of the time series. This substitution into equation (8.1) gives

+

+

+

(8.3)

R(f)

=

0 f 2(0.5) cos (27rfAt') = cos (.fat)

From equation (8.2) the frequency response of the elementary binomial smoothing function will be the square of the result in equation (8.3); namely,

(8.4)

R ( f ) = cos2 ( r f A t )

The above result can also be determined directly from equation (8.1) by In general, the binomial smoothing funcsetting wo = $5 and w1 = q)" has a tion having weights proportional to the coefficients of ( p frequency response of

x.

(8.5)

R(f) =

+

COS"

(rfAt)

The derivation of equation (8.5) assumes that the sum of the weights of the binomial smoothing function is made equal to unity a s is done with all smoothing functions considered in this paper. Some workers smooth time series by successive use of running means of more than two terms each. Brooks and Carruthers [15] give the following general expression for the frequency response of the operation of M

372


successive smoothings by the equally-weighted running means of N terms each

With N set equal to two, this expression is equivalent t o equation (8.5).

9. DESIGN OF INVERSE SMOOTHING FUNCTIONS An application of the use of elementary filtering functions to the design of filters is the computation of weights for inverse smoothing functions. A composite inverse smoothing function can be generated from elementary inverse smoothing functions in the same way th a t composite smoothing functions can be generated from elementary smoothing functions; that is, cumulative cross-products of the weights of the inverse smoothing functions involved are computed for various lags. Unfortunately, it is not possible t o design a three-weight inverse smoothing function which has exactly the inverse or reciprocal frequency response of the elementary binomial smoothing function. If this were possible, the design of inverse smoothing functions for correcting for binomial and normal curve type smoothing would be facilitated. However, the three-weight elementary inverse smoothing function having a central weight of 2 and outer weights of -0.5 has a frequency response which is roughly the reciprocal of the response of the elementary %, and )/4). The binomial smoothing function (having weights of frequency response of this inverse smoothing function as computed by equation (8.1) is

x,

(9.1)

R(f)

=

2 - cos ( 2 ~ f A t )

This frequency response equals the reciprocal of the response of the elementary binomial smoothing function a t f = 0 where R ( f ) = 1 and a t 1 1 f = - where R ( f ) = 2. However, the response is three at f = - as 4At 2At compared with a value of infinity for the reciprocal of the zero response of the elementary binomial smoothing function a t this frequency. It is impossible to restore a wave to a time series by inverse smoothing if this wave was completely smoothed out previously. Some residual amplitude must remain in the smoothed series at each frequency t o be restored, and this residual amplitude has t o be greater than the noise for equalization t o be effected with relative freedom from errors owing to noise. For example, rounding of smoothed values would introduce a type of random error, called " quantizing error," into the smoothed series. High-frequency noise is amplified in addition to the true high-frequency


373

fluctuations in the time series, and this noise can “drown o u t” not orlly true high frequencies but also all other components besides if the amplification at these high frequencies is too great. Overemphasis of noise or random nonsignificant variation in the time series t o be equalized creates a type of mathematical instability which plagues many attempts a t inverse smoothing. Thus, in designing inverse smoothing functions a compromise must be made between having a stable output and having a response nearly equal to the inverse of that of the original smoothing. 1 An amplification of three a t f = - instead of infinity in equation (9.1)

2At

is a n example of just such a compromise. By combining two of the above elementary inverse smoothing functions (having weights of -0.5, +2.0, and -0.5) a composite function can be designed t o reverse the smoothing by a binomial smoothing function generated from two elementary binomial smoothing functions; namely, the function having weights of H.~‘s,3.8, and W S .Although this inverse smoothing function would provide the correct restoration of 1 waves of frequency equal to -, it would overemphasize oscillations of

x, x,

4At

some of the lower frequencies by as much as 40%. I n order to eliminate this excessive overemphasis of low frequencies the composite inverse smoothing function should instead be generatjed from two elementary three-weight functions each having a central weight wo of 1.7 and outer weights w1 of -0.35. These weights were obtained empirically. This composite function would have the five weights: +0.12, -1.19, f3.14, -1.19, and +0.12 (with $3.14 being the principal weight), and its frequency response would be computed according t o equation (8.2)from the square of the right-hand side of equation (8.1); namely,

(9.2)

R(f)

=

+ +

+

4 ~ 0 cos ~ 1(2~fAt) 4201~COS’ (2~fAt) 2.89 - 2.38 cos (2~jAt) 0.49 C O S ~(2~fAt)

= WO’

At low frequencies, this response is approximately the inverse of that of the binomial smoothing function considered above, The frequency response of the latter is

(9.3)

R(f)

=

c0s4 (nfAt)

from equations (8.2)and (8.4)or (8.5). These frequency responses are compared in Fig. 8 to show the extent to which this type of inverse smoothing can restore the original balance of frequencies in the smoothed time series. A curve representing the product of the frequency response of the inverse smoothing function considered and that of the binomial smoothing function a t each frequency is also included in this figure.

374


Ideally this product should be a straight line of unit response. Departure from this ideal product shows the extent to which this form of inverse smoothing fails to compensate for the previous binomial smoothing, Figure 9 showing three curves illustrates how well inverse smoothing

5

wa > 3 0

z

W 3

0 W

I

0.I

0.2

0.3

0.4

0.5

FREQUENCY IN CYCLES PER DATA INTERVAL

FIG.8. Frequency response of a five-weight binomial smoothing function, R ( f ) s , and that of a n inverse smoothing function designed to reverse this binomial smoothing, R ( f ) r . The product of these two responses is also shown.

performs on an actual time series. The solid line in this figure is the original time series; the dashed line is this series smoothed by the five-weight binomial smoothing function considered above; and the dotted line is the smoothed series inversely smoothed using the five-weight function described above. The departure of the dotted line from the solid line indicates the limitations to applying inverse smoothing t o this type of

375


smoothed series. At point A in Fig. 9 where the original series contains primarily low frequencies there is nearly perfect restoration of the original series by the inverse smoothing. On the other hand, the inverse smoothing does not restore the high-frequency fluctuation at B in the original series because this component was completely smoothed out by the previous binomial smoothing. The examples in this section illustrate how inverse smoothing functions may be designed t o compensate for two types of binomial smoothing. These procedures may also be applied to the problem of reversing many other forms of smoothing. However, serious error can be made when inverse smoothing is applied to a series which has been smoothed by a method having appreciable negative response beyond the first cutoff frequency as is the case with equally-weighted running means. AS

0

5

10 15 20 TIME I N DATA I N T E R V A L S A F T E R A R B I T R A R Y ORIGIN

25

FIG.9. Time series (solid line), same series smoothed by a five-weight binomia smoothing function (dashed line), and the smoothed series inversely smoothed (dotted line).

has been mentioned in Section 5 such a smoothing function will introduce spurious fluctuations into the smoothed output, and these fluctuations will generally be amplified by the inverse smoothing process. A good way of remedying this condition is to smooth the series again by the same smoothing method applied originally. This will restore the original polarity t o all the waves in the series. Then this double smoothing may be partially reversed by an inverse smoothing function similar to those described earlier in this section. In this case inverse smoothing can a t best only partially restore the original series, for no procedure can restore waves previously completely smoothed out. Smoothing methods having negative responses generally completely smooth out waves of a number of relatively low frequencies. The fact that smoothing by equally-weighted running means is not very reversible is another good argument against using this type of smoothing.

376

J. LEITH HOLLOWI’AY, JR.

The concept of reversibility occurs in the field of thermodynamics in connection with entropy change. The more reversible a thermodynamical process is the less entropy is increased during this process. The concept of entropy, which is a measure of the disorder or randomness in a system, has been extended to statistical analysis by information theory where the negative of entropy change is taken as a measure of the information change occurring in statistical data. The foregoing implies that the less reversible a smoothing or filtering process is the more entropy is created, and therefore the more information is lost in the operation. No entropy would be created, and thus no information would be lost by perfectly reversible smoothing or filtering. In order for smoothing to be perfectly reversible several conditions must be fulfilled. First, the original smoothing cannot have completely smoothed out any wave of finite frequency. This is true in the case of exponential and normal curve smoothing. Second, the smoothed series must be continuous-not a series of discrete values. Third, the series must be completely free from noise. Finally, waves of all frequencies must remain unchanged in phase and amplitude during the filtering interval of the original smoothing function. This last condition is fulfilled in general only when the filtering interval is infinitesimal as is the case when the series is smoothed by the derivative method described in Section 11 of this paper. These last two conditions are so stringent that for all practical purposes no smoothing operation is completely reversible, although some (for example exponential and normal curve smoothing) are more reversible than others. 10. DESIGNOF PRE-EMPHASIS FILTERS

It may be desirable to amplify high frequencies in a time series more than the low frequencies for reasons other than for correcting for previous smoothing. For example, in statistical spectral analysis it is often desirable to pre-emphasize high-frequency components before computing the autocorrelation function of a series. This type of high-frequency preemphasis has been named “pre-whitening” by Tukey because it tends to equalize amplitudes at all frequencies [16]. This pre-emphasis prevents a great deal of instability in making spectral estimates when originally the amplitudes of waves of low frequency in a series are much greater than those of high frequency. After the spectral estimates have been obtained, they are then corrected by a factor which is the inverse of the frequency response of the pre-emphasis operation applied to the original series.11 “ T h e square of the frequency response is required for making this correction when the relative variance rather than the relative amplitude is computed in the spectral analysis.


377

An inverse smoothing function similar to those described in the last section may be used for this type of pre-emphasis. However, in spectral analysis work a simpler two-weight filter is generally used which has a principal weight w o of unity and a weight wP1 of minus b. By means of this two-weight operator the pre-emphasized variable xi is computed from time series values xt and xt--l as follows:

(10.1)

~ t = ’

~t

-

brt.t-1

where the subscripts refer to the times of the observations. The weight b is generally made equal to 0.75 in studies involving atmospheric turbulence. This choice of b makes the sum of the weights of this filter equal t o one-quarter. Therefore, the mean of the resulting pre-emphasized series will be only one-fourth of that of the original series. However, in spectral analysis work the original series is usually made up of deviations from a mean and therefore has a mean of zero. Consequently, the effect of the pre-emphasis on the mean of the series is of no concern. Because this two-weight pre-emphasis filter is not a n even function, equations (5.2) and (5.4) must be employed for determining its frequency response; namely,

(10.2)

R ( f ) = [I - b cos (-2rfAt)I

and

IR(f)J = [ l - 2b cos (2rfat)

+ i[-b

sin (-2rfAt)l

+ b2]’$

This pre-emphasis filter also introduces phase error which may be computed from equation (5.3). However, this error is of no significance unless co- and quadrature spectra are t o be computed between the preemphasized series and another series which has either not been preemphasized or has been done so differently. A special case of this type of pre-emphasis is where b in equation (10.1) equals unity [17]. This filter is a finite difference approximation of the derivative of a time series. The response of this filter is given by equation (10.2) ;namely,

(10.3)

IR(f)I = [2 - 2

COB

(2rfAt)l”

=

2 sin (.!At)

When rfAt is small,

IW)I = %!At

(10.4)

The phase shift of this filter would be given by equation (5.3) as follows:

(10.5)

$I = tan-’ [1 = ?r/2

- rfA2

= =

90°(1

- 2fAt)

tan-’[cot (rfAt)]

378


I n the limit as At goes t o zero, 4 becomes 90" which is the phase shift accomplished by differentiating any sine curve. An exactly opposite type of pre-emphasis also used in spectral analysis work consists of taking consecutive means instead of consecutive instantaneous values of a variable which is continuously recorded such a s wind [16, 181. These means are usually obtained by eye from a graphical record. This procedure emphasizes low frequencies and is used to reduce errors resulting from '' aliasing "-the process by which high-frequency waves appear as lower frequencies in a time series having a data interval too long to portray these shorter wavelengths. The spectral estimates obtained from the series generated in this manner are corrected by the inverse of the frequency response of the equally-weighted mean (computed by equation (5.9)). 11. FILTERING BY MEANSOF DERIVATIVES OF TIMESERIES

Additional insight into the operation of smoothing and filtering may be gained by considering what effects are produced on the time series by adding specified fractions of time derivatives of the series to the series itself. Consider a sine wave of angular frequency w = 27rf, amplitude a, and phase 4. A time series having only this one component would be defined as s(t) = u sin (wt

(11.1)

+ 4)

The second, fourth, sixth, and 2nth time derivaties of this series would be:

+

(11.2)

+ +

~ ( t =) d2x/dt2 ~ ~ = -aw2 sin (wt 4) z(t)$"= d4x/dt4 = aw4 sin (wt 4) x(t)"i = dsx/dta = - a m 6 sin (wt $1 ~ ( t ) ~=" dZnx/dt2n= (-l)nuw2n sin (wt

+ 4)

From the derivatives given above one can form a power series which defines a new modified time series as follows: (11.3)

+

+

+

Z(t) = ~ ( t ) k z ~ ( t ) ~k 4~~ ( t ) ~ '

*

..

+

k2n~(t)zn

Substituting values in equations (11.1) and (11.2) into equation (11.3) one obtains (11.4) Z(t) = [ l

- kzw2 + k4w4 - k~w' +

*

*

+ (- l)nkZnwzn]a sin (wt + 4)

Notice that the power series in brackets in equation (11.4) is actually the frequency response for the operation defined in equation (1 1.3) since it is the factor by which the original sine wave is multiplied in order to obtain the new filtered wave; namely,

379


(11.5)

R(u) = 1 - kzw2 f k4w4 - k6we f

*

*

.

+ (-1)"ICZnUzn

If in equation (11.5) (11.6)

kz

=

k4

=

c2, c4/2!= c4/Z,

ks = c 6 / 3 ! = c S / 6 , kzn = c Z n / n !

this frequency response reduces to exp ( -czwz) which is the response of a normal curve smoothing function having u = c ~, Thus it follows that the addition of these fractions of each even time derivative to the original series is equivalent to normal curve smoothing of the series. The technique described here would be an alternate method for designing a filter with specified frequency response. The k's in equation (11.5) can be determined so that this power series would be equal to the desired frequency response. Many desired frequency responses may be expressed as power series, and in such cases the determination of the k's would amount merely to equating the coefficients in two power series. However, in practical application of this method only a small number of derivatives can be used, so that the actual frequency response of the derived filter is not exactly the same as the desired frequency response expanded into the complete power series. Furthermore, if the derivatives are computed by the method of finite differences, additional discrepancies between the actual and the desired frequency response will result owing t o the tendency for the finite difference method t o underestimate derivatives. This method is especially valuable for the design of inverse smoothing functions. For example, suppose that the original smoothing had a frequency response of (11.7)

R(w) = exp ( -c2wz)

The frequency response of the inverse smoothing function required t o reverse this smoothing should be the reciprocal of the above R(w) or (11.8)

R'(w) = [R(w)]-' = exp

(c2w2) = 1

Equating coefficients in equation (11.8) the power series in equation (11.5) one the k's: kz = -c2, ks = (11.9) kq = c 4 / 2 , kzn =

+ c2w2+ (2!)-'c4w4 +...

+ (n!)--1cZnw2n

with the corresponding ones in obtains the following values of

-ce/6, (-l)n~zn/n!

These k's are identical with those in (11.6) except that alternate ones have negative signs. When these values of k are substituted into equation (11.3), the required inverse smoothing operator is obtained.

380


That derivatives can be used for inverse smoothing could have been inferred from the fact that smoothing itself is a form of integration over time and t,hat tlhe inverse of integration is differentiation. The most straightforward application of differentiation to inverse smoothing is that of the equalization of a time series smoothed by an exponential process such as is performed by an instrument with a constant lag coefficient. n For such an instrument the following well-known differential equation holds: (11.10)

(z - 2)

= X(dz/dt)

or

=

z - X(dz/dt)

where Z is the smoothed value of the variable x in the time series and X is the lag coefficient. If equation (11.10) is solved for 2, one obtains the so-called inverted lag equation after McDonald [19] (11.11)

2 =

z

+ X(dZ/dt)

From equation (11.11) it follows that perfect restoration of the original time series is theoretically possible given the smoothed values without noise, by differentiating the smoothed values with respect to time, multiplying this derivative by X, and adding this result to the original smoothed values. A finite difference form of equation (11.11) is (11.12)

zt =

Zt

+ X(Zt -

&-I)

=

(1

+ X)Zt - XZt-1

where X is in units of At, the data interval. Since the time derivative in equation (11.12) is estimated by the method of finite differences, it is necessary that the data interval be a small fraction of the lag coefficient, say smaller than one-fifth, in order for the results to be accurate. If the output of the instrument is electrical, the above equalization may be done by means of a resistance-capacitance network as shown by Hall [8]. This circuit performs the required differentiation electrically. A practical limitation to this method is that the lag coefficient is seldom constant but is a complicated function of many variables, Secondly, the record must be relatively free from noise for accurate results to be obtained by this procedure. 12. SPACESMOOTHING AND FILTERING

So far in this paper only smoothing and filtering of time series has been dealt with, The methods of time smoothing and filtering may, nevertheless, apply equally as well to space smoothing and filtering in one dimension, except, that the terms “wavelength” and ‘(wave number” (number of waves in a given distance) are used instead of period and frequency, 14

See Section 6 for a further discussion of this type of smoothing.

381


respectively. Furthermore, these concepts may be extended t o space smoothing and filtering in two dimensions. In space smoothing in one dimension, the term wave number response may be substituted for frequency response. However, with two-dimensional space smoothing the concept of wave number or wavelength is not very meaningful unless the waves are essentially one-dimensional. The term “scale size” is perhaps better t o use in this case. Scale size will be defined a s the average 2 81

1 81

-1 3

-1

-1

3

3

-1 4 -1 4

1 -

1 -

9

9

1 9

-1 4 1 4

1 -

16 2 16 1 16

-1 9

2 -

16 4 16 2 16

8 81

1 81

1 -

16 2 16 1 16

81

I -

8 -

l5 81

8 81

2 81

2 -

8 81

8

sl

81

1 9

3 9

1 9

2 -

1 81

81

81

2 81

8 81

1 81

2 -

81

1 81

1 256

4 256

6 256

4 256

1 256

4 256

16 256

24 256

16 256

4 256

6 256

24 256

36 256

24 256

6 256

4 256

16 256

24 256

16 256

4 256

4 6 4 1 256 256 256 256 256 FIG.10. Generation of composite space smoothing functions from the elementary space smoothing functions in the first column. The functions in the second column result from one iteration of those in the first; those in the third column, from one iteration of those in the second. 1 -

distance between adjacent centers of high values or between adjacent centers of low values. Space smoothing will decrease the ranges between values in high and low centers of small scale without much affecting these ranges between large-scale centers. Space smoothing and filtering functions are generally isotropic; namely, they are functions only of the radial distance from the origin. The function will usually have a maximum at the origin and decrease with radial distance outward from this origin. The scale response of twodimensional smoothing and filtering functions will be comparable t o the

382


wavelength response of a one-dimensional filter which is a cross-section of this two-dimensional filter. Discrete-weighted space filters will have weights proportional t o the ordinates of continuous two-dimensional space filtering functions. The sum of these weights is made equal to unity

(A)

FIQ.11 (A,B,C,D). Space-smoothed Northern Hemisphere sea-level pressure maps. for smoothing functions and zero for filters not passing the mean of the field. As with most one-dimensional filters, the central weight of a twodimensional filter is the principal weight. If no weight is centrally located, the space-smoothed value corresponds to the location of the center of gravity of the filter’s weights. It is conceivable that nonisotropic space smoothing may be desirable for some purposes. For example, one may wish t o suppress north-south


383

fluctuations more than those in the east-west direction. This could be accomplished with a space smoothing function whose weights decrease to zero faster in the east-west direction than t o the north and south. Composite space smoothing and filtering functions may be generated

(B) FIG.11. (Continued).

from elementary two-dimensional functions just as composite onedimensional smoothing and filtering functions were built up from elementary one-dimensional functions earlier in this paper. Such elementary functions could consist of either an array of three weights of one-third each at the corners of an equilateral triangle or a n array of four weights of one-fourth each at the corners of a square. Iterative cumulative multiplication of these elementary space functions a t various relative positions creates composite smoothing functions whose weights approach the circularly-symmetric bi-variate normal distribution (with standard de-

384


viation the same in all directions from the origin) as is illustrated in Fig. 10. The bi-variate normal smoothing function has the same desirable characteristics in two dimensions that the normal curve smoothing function has in one dimension. One such desirable property is that the bi-

(C) FIG. 11. (Continued).

variate normal smoothing function will not reverse polarities of fluctuations of any scale whereas, for example, an equally-weighted space smoothing function will reverse the polarity of features at some scale sizes. The familiar Fjgrtoft method of space smoothing is merely the single application of the square elementary space smoothing function described above [20]. Therefore, by smoothing twice by the Fjgrtoft method a first approximation is made to circular bi-variate normal smoothing illustrated in Fig. 10.


385

The use of space smoothing is illustrated in Fig. 11 showing four Northern Hemisphere sea-level pressure maps smoothed by a space smoothing function very similar to the one in the upper right of Fig. 10. The distance between the grid points on the triangular grid used averaged

(D) FIG. 11. (Continued).

about 500 miles. This smoothing almost completely attenuates features having a scale size of about 1500 miles, but it retains 4000-mile features at about 75% of their original amplitude. These smoothed maps thus display only large-scale pressure systems which are of particular interest to extended weather forecasters. The small-scale features of the pressure field 'can- be-isolated by-subtracting the smoothed map from the original unsmoothed pressures. This operation constitutes the space analog of the high-pass filter of time series terminology. An example of this process is shown in Fig. 12 where the

386


FIG.12. Original North American sen-level pressures (left) and small-scale pressures (right) obtained by subtracting the space-smoothed map from the original map.

small-scale patterns of the January 1, 1953 North American surface map are isolated by subtraction of the smoothed map for this date obtained previously. This procedure essentially eliminates systems of greater extent than about 4000 miles. The original map is also shown in Fig. 12 for comparison. The filtered map has the same general appearance as the original map but gives slightly more emphasis t o certain smallerscale features. The fact that these maps look very similar suggests that in viewing a weather map we naturally concentrate more on small-scale features than on the large-scale ones and thus do high-pass filtering in our “mind’s eye.” ACKNOWLEDGMENT8 The writer wishes to acknowledge the helpful suggestions and assistance given during the preparation of this paper by the following people: Messrs. F. Hall, G. W. Brier, I. Enger, R. A. McCormick, and J. E. Caskey, Jr., of the U.S. Weather Bureau, Col. P. D. Thompson of the Joint Numerical Weather Prediction Unit and Dr. Max A. Woodbury of New York University.

SMOOTHING A N D FILTERING

387

FIG. 12. (Continued). LIST OF SYMBOLS amplitude of wave weight in pre-emphasis filter constant frequency cutoff frequency square root of minus one imaginary part of y summation variable n t h coefficient number of times a series is smoothed or filtered number of weights in smoothing or filtering function after principal weight number of terms in a n equally-weighted running mean number of weights in smoothing or filtering function before principal weight; arbitrary exponent arbitrary variables ratio of the mean of the filtered series to that of the original series frequency response function; frequency response when it is a pure real quantity absolute value of frequency response function; frequency response when R(f) is complex


resultant frequency response frequency response of filter y frequency response as a function of W , the angular frequency real part of y filtering interval time data interval kth weight of a smoothing or filtering function principal weight of a smoothing or filtering function smoothing or filtering function unsmoothed or unfiltered discrete variable a t time t smoothed or filtered discrete variable a t time t well-smoothed discrete variable a t time t pre-emphasized discrete variable a t time t continuous function of time smoothed continuous function of time ith time derivative of x ( t ) lag coefficient normal curve dispersion parameter phase angle angular velocity or angular frequency = 2 ~ f REFERENCES 1. Carslaw, H. S. (1930). “Introduction to the Theory of Fourier’s Series and Integrals.” Macmillan, New York. 2. Bracewell, R. N. (1955). Correcting for Gaussian aerial smoothing. AustraZian J . Physics 8, 54-60. 3. Bracewell, R. N. (1955). A method of correcting the broadening of X-ray line profiles. Australian J . Physics 8, 61-67. 4. Bracewell, R. N. (1955). Correcting for running means by successive substitutions. Australian J . Physics 8,329-334. 5. BraceweIl, R. N. (1955). Simple graphical method of correcting for instrumental broadening. J . Opt. SOC.Amer. 46, 873-876. 6. Bracewell, R. N., and Roberts, J. A. (1954). Aerial smoothing in radio astronomy. Australian J . Physics 7, 615-640. 7. Burr, E. J. (1955). Sharpening of observational data in two dimensions. A u s tralian J . Physics 8, 30-53. 8. Hall, F. (1950). Communication theory applied to meteorological measurements. J . Meteorol. 7, 121-129. 9. Kovasznay, L. S. G., and Joseph, H. M. (1953). Processing of two-dimensional patterns by scanning techniques. Science 118, 475-477. 10. Kovasznay, L. S. G., and Joseph, H. M. (1955). Image Processing. PTOC.I.R.E. 43, 560-570. 11. Middleton, W. E. K., and Spilhaus, A. F. (1953). “Meteorological Instruments,” 3rd rev. ed. Univ. of Toronto Press, Toronto. 12. Amble, 0. (1953). A smoothing technique for pressure maps. Bull. Am. Meteorol. SOC.34, 293-297. 13. Panofsky, H. A. (1953). The variation of the turbulence spectrum with height under superadiabatic conditions. Quart. J. Roy. Meteorol. Sac. 79, 150-153. 14. Shuman, F. G.(1955). A method of designing finite-difference smoothing operators


15.

16. 17. 18.

19. 20.

389

to meet specifications. Technical Memorandum No. 7, Joint Numerical Weather Prediction Unit, Washington, D. C. To be published in Monthly Weather Rev. 86. Brooks, C. E. P., and Carruthers, N. (1953). “Handbook of Statistical Methods in Meteorology,” H.M.S.O., London. Gifford, F., Jr. (1955). A simultaneous Lagrangian-Eulerian turbulence experiment. Monthly Weather Rev. 83, 293-301. Dedebant, G., and Machado, E. A. M. (1955). Effectos de ciertos 6ltros sobre la correlati6n. Meteoros 6, 163-176. Griffith, H. L., Panofsky, H. A., and Van der Hoven, I. (1956). Power-spectrum analysis over large ranges of frequency. J. Meleorol. 13, 279-282. McDonald, J. E. (1952). Lag effects in the measurement of turbulent temperature fluctuations. Scientific Report No. 1, Contract No. AF19(122)-440, Iowa State College, Ames, Iowa. FjBtoft, R. (1952). On a numerical method of integrating the barotropic vorticity equation. Tellus 4, 179-194. GENERALREFERENCES

Berry, F. A., Haggard, W. H., and Wolff, P. M. (1954). Description of Contour Patterns at 500 mb. Bureau of Aeronautics Project AROWA, U. s. Naval Air Station, Norfolk, Va. Goldman, S. (1953). “Information Theory.” Prentice Hall, New York. Jeffreys, H., and Jeffreys, B. (1950). “Methods in Mathematical Physics.” Cambridge Univ. Press, London and New York. Wiener, N. (1949). “Extrapolation, Interpolation, and Smoothing of Stationary Time Series.” Wiley, New York.


EARTH TIDES

.

Paul J Melchior* Obrervotoire Royal de Belgique. Uccle. Brussels. Belgium

Page 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 2 Static Theory of the Tides . . ................................ 394 2.1. Calculation of the Attra ................................ 394 2.2. Deformations of the Level Surfaces Caused by Luni-Solar Effects . . . . . . 396 2.3. The Three Kinds of Tides According to Laplace ..................... 397 .......................... 399 2.4. Numerical Values . . . . . 3 . Definition of Love’s Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 401 4 . Study of the Amplitude of Oceanic Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Effect Due to the Deformations of the Crust . . . . . . . . . . . . . . . . . . . . . . . . 401 402 4.2. Application to Oceanic Tides of Long Period., . . . . . . . . . . . . . . . . . . . . . . 4.3. Application to Short Period Tides in Lakes . . . . . . . . . . . . . . . . . . . . . . . . . 402 5 Periodical Deflections of the Vertical with Respect to the Crust . . . . . . . . . . . 403 5.1. Effect Caused by Deformations of the Crust ........................ 403 5.2. The Horizontal Pendulu ......................... 404 5.3. Observations Obtained from Lar r Levels . . . . . . . . . . . . . . . . 5.4. Numerical Results of the ......................... 409 5.5. Discussion of the Observa t Effect of the Oceanic Tides 411 5.6. Method of Numerical Evaluation of the Indirect Effects . . . . . . . . . . . . . . 411 5.7. Empirical Method of Separation of the Two Effects . . . . . . . . . . . . . . . . . . 414 5.8. Effect of the Tides on High Precision Leveling Operations . . . . . . . . . . . . 417 6. Measurement of E1ast.ieTensions and Cubic Dilatations Due to Deformations

. .

.

6.2. The Sassa Exte

8.4. Results of the Observations., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Observations Made with Clock Pendulums . . . . . . . . .

429

* Reporter General for earth tides of the International Associations of Geodesy and Seismology . 391

392

PAUL J. MELCHIOR

Page 9. The Role of the Geologic Structure of the Crust in the Indirect Effects. . . . 432 9.1. Oceanic Effects. ...... 9.2.Atmospheric Effects.. ..................................... 434 10. Theory of Elastic Deformations of the Earth.. 10.1. Conclusions Drawn from the Obse .............................................. 435 ............................................... 435 10.3.Herglotz Theory. . . . . . . . . . . . . . . . 435 10.4.Note on the Relatio . . . . . . . . . . . . . 437 10.5.Dynamic Effects of . . . . . . . . . . . . . . . . . . . 438 Earth. . . . . . . . . . . . . 439 12. Program of the International Geophysical Year.. . . . . . . . . . . . . . . . . . . . . . . . . 440 List of Symbols.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 References. . . .. .... ........................... 441

1. INTRODUCTION The concept of an earth not completely rigid but deformable came into being in the beginning of the nineteenth century. Simultaneously, astronomers began to suspect the existence of a variation of the latitudes (Brioschi a t Naples) and to study variations in the direction of the vertical and local deformations of the earth’s crust. The experiments made by d’Abbadie, beginning in 1837, on pools of mercury revealed some rather irregular variations which, near the Gulf of Gascogne in particular, seemed to correlate with the oceanic tide, although in a quite complex manner. Furthermore, classical theoretical considerations predicted the existence of small periodic variations of the vertical as well as small periodic displacements of the pole. The discrepancies, which will be discussed later, between this theory and the observations in both of these phenomena only indicate the existence of elastic deformations of the earth due to the influence of underlying disturbing forces. It will be necessary to consider these elastic deformations in the calculations in order to obtain results conforming to the observation. The static theory of the tide shows immediately that the direction of the vertical and the intensity of gravity do not remain constant but vary under the influence of the luni-solar attraction. Kelvin directed attention to the effect of the deformations of the earth; one is not able, in effect, to imagine it to be infinitely rigid and consequently has to admit that it ought to be deformed under the influence of a perturbation of the potential, similar to the ocean layer but in a lesser degree. The amplitude observed at the surface of the earth for all the phenomena dependent on this potential (oceanic tides, deviations of the vertical variations in the intensity of gravity) will thus be affected by the deformation of this surface, where all our measurements are made.

EARTH TIDES

393

Thus, the oceanic tides are observed by comparison with bench marks reputedly “fixed,” established on the “terra firma.” These bench marks would be fixed if the earth were perfectly rigid and the amplitudeobserved would be equal to that calculated. However, if the crust itself rises as a result of the action of the disturbing potential, the measured amplitude will be the difference between the tide of the ocean and that of the crust. One can apply the same reasoning to other manifestations of the phenomenon. For more than half a century, observations of all kinds have confirmed the views of Kelvin and, when compared with the data of seismology and of the motion of the pole, permit one to study the elastic properties of the earth with increasing precision. Before considering the analysis and the interpretation of the results of these observations, it is desirable to investigate whether it is possible to apply the static theory to tides of the “solid” earth. This theory, proposed initially by Newton and Laplace in order to attempt an explanation of the phenomena of oceanic tides, is based upon the hypothesis that at each instant the sea takes the equilibrium position which corresponds to the instantaneous distribution of the vertical (taking into account here the influence of the displacement of the mass of water itself). However, it is quickly apparent that this theory does not give a quantitative representation of the phenomenon, in particular for short period oscillations (semidiurnal and diurnal). Indeed, the liquid particles, not being rigidly connected, because of inertia exceed their equilibrium position and oscillations are initiated. Darwin, Kelvin, and Love have taken this effect into account and constructed a dynamic theory of the tides. This theory was still not entirely satisfactory and did not explain the large dissimilarities that the phenomenon presents from one ocean basin to another. A better approximation to reality is furnished by the theory of Harris who introduced the consideration of ocean basins where certain tidal oscillations may or may not be able to enter, in resonance with characteristic periods determined by the geometric form of the basin. In the case of the solid earth, the rigid connections (we will see below that the moduli of rigidity are of the order of that of steel and larger) between the molecules do not allow currents, and the particles are only displaced a few decimeters and equilibrium is thus rapidly attained. It is easy to show that the periods of tidal oscillations are long compared to the period of free oscillation of a liquid earth having the dimensions of the earth (1.5 hr), thus the phenomena of resonance will not be able to develop (cf. [l]). This can be seen by considering that the crust is comparable to a very deep ocean, which leads to a very short free period (function of fi).

394

PAUL J. MELCHIOR

On the other hand, a seismic impulse takes about 20 min to be propagated along one diameter. Compared with these periods, the semidiurnal period of the tides may be considered long, and the phenomena which are its manifestation may be treated by the static theory. We shall now develop the essential elements of this theory in order to compare the theoretical data with the results of observations which are described below.

2. STATICTHEORY OF

THE

TIDES

1.1. Calculation of the Attraction

Let L be the disturbing body of mass m (the mass of the earth being taken as unity) ; A , a given point a t the surface of the earth; and G, its center of gravity (Fig. 1). The attraction is exerted on G and A in the LGA plane. We separate the components parallel to the axes OX and O Y :

a

FIG.1.

where f / a 2 = g (acceleration of gravity). The relative attracting force acting on A is the resultant of these two forces which can be written

A pendulum suspended at A , when in equilibrium, would make an angle e with AG given by

395

EARTH TIDES

From Fig. 1 we notice that

(

7r2= 1 + 2 % o s z r a sin 2' = sin z -sin z cos z r a cosz' = cos z - -sin2z r

+

Thus, we obtain

and, neglecting small terms of an order greater than the third power of a/r, we obtain

(2.7) I n order to study the intensity of gravity, it suffices to consider equations (2.3) using values of r2/r" and cos 2' given by (2.5). Again neglecting terms of the 4th order these give:

The change in gravity is thus (2.9)

dg

=

Y t

gm -

(1 - 3 cos2 z )

=

a f m - (1 - 3 cos2 z ) r3

The expressions (2.7) and (2.9) of the horizontal and vertical components of the perturbing force are derived from a potential (2.10)

w2

fm a2 2 r

= - 7(3

cos2 z - 1)

This may be seen by forming (2.11)

1 aw2 - -~

g adz

and

dW2 aa

-

The quantity within the parentheses in (2.10) is the surface zonal harmonic function of the second order if one takes the direction between the center of the earth and of the disturbing body as the axis.

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PAUL J. MELCHIOR

9.8. Deformations of the Level Surfaces Caused by Luni-Solar Efects

Alevel surface is defined as a surface normal to the direction of gravity at all points; that is, normal to the resultant of the gravitational force due to the terrestrial mass and the centrifugal force caused by rotation of the earth about its instantaneous axis. One notes that such a surface is equipotential and its equation may be written

v = C(h)

(2.12)

where h is the distance to a level surface chosen as reference. Helmert has called these surfaces level spheroids (these are not ellipsoids but irregular surfaces differing from a simple geometric surface because of the heterogeneous distribution of densities in the earth’s crust). The name “geoid” has been given to the level spheroid corresponding to the mean surface of the oceans, and its equation is

vo = C(0) Thus, the acceleration of gravity a t a point P may be expressed by g=

(2.13)

av -ah

where h represents the altitude of point P above sea level. Let us now consider the effect on the level spheroids, and in particular, on the geoid, of a perturbation caused by the luni-solar attraction whose potential will be W z (an harmonic function of second order). The equipotential surface passing through P is deformed; let P be the amplitude of the radial deformation (height of the tide). The potential of surface h f is

+

and the equation of the surface becomes (2.14)

Using in this relation equations (2.12) and (2.13),it follows that (2.15)

which is the theoretical height of the observed tides.

EARTH TIDES

397

d.3. The Three K i n d s of Tides According to Laplace

Beginning with expression (2.10) for the potential of the tides, one can separate the phenomenon into several distinct types of oscillations. In order to do this it suffices to introduce the well-known trigonometric relation (2.16)

cos z

=

sin cp sin 6

+ cos I$ cos 6 cos (0 - A)

giving the zenithal distance from the heavenly body as a function of its declination 6, its hour-angle e and the coordinates of the location (lati-

Fra. 2. The three types of tides according to Leplace.

398

PAUL J. MELCHIOR

tude 4, longitude X). One thus obtains cos2 4 cost 6 cos 2(0

- A)

+ sin 24 sin 26 cos (0 - A) + 3 (,sin2 4 - -k)

(sin2 6

-

k))

The three terms contained in the parentheses are surface spherical harmonic functions. Their significance is demonstrated in Fig. 2. (A) The first of these functions has as nodal lines (lines where the function is zero) only the meridians located a t 45" on either side of the instantaneous meridian of the body. These lines divide the sphere into four sections where the function is alternately positive and negative: regions where W is positive are those of the high tides (f > 0); the negative regions are those of the low tides (f < 0). This function is called the 8ectorial function, the period of the tides to which this corresponds is semidiurnal, and the amplitude is maximum a t the equator when the declination of the disturbing body is zero. These are Laplace's tides of the third type. (B) The second ,function has as nodal lines a-meridian (90";from the meridian of the disturbing body) and a parallel, the equator. This is a tesseral function. The regions into which it divides the sphere change in TABLE I. Principal short period terms of the potential which generates the tides. ~

Waves

Mean value of coefficient.

Relative amplitude to Mz

~~

~

~

-~

Hourly displacement

Semidiurnal terms

M z semidiurnal lunar N t large lunar elliptic L p small lunar elliptic Sz semidiurnal solar Kz semidiurnal luni-solar Diurnal terms

diurnal lunar diurnal luni-solar P diurnal solar & I large lunar elliptic; JIsmall lunar elliptic 00 second-order diurnal lunar 01

K1

0.45426 0.08796 0.0126 0.21137 0.05752

1 0.194 0.028 0.465 0.127

28'98410 28'43973 29'52848 30"00000 30'08214

0.18856 0.2655 0,08775 0.0365 0.0149 0.0081

0.4151 0.5845 0.1932 0.0803 0.0328 0.0178

13'94304 15"04107 14"95893 13"39866 15'58544 16'13910

General coefficient for semidiurnal terms: General coefficient for diurnal terms:

$4(:)'

ag cost

+ = r cos*+

sin 2+ = sin 24

399

EARTH TIDES

sign with the declination of the disturbing body. The period of the corresponding tides is diurnal and the amplitude is maximum at f45” when the declination of the body is maximum. It is zero at the equator and the poles. These are Laplace’s tides of the second type. (C) The third function depends only on the latitude. It is a zonal function whose nodal lines are the parallels and -35’ 15’. Since it is a function of the sine squared of the declination of the disturbing body only, its period will be about fourteen days in the case of the moon and six months in the case of the sun. These are Laplace’s tides of the first type. The level surface will be depressed about 28 cm a t the pole and elevated about 14 cm at:the equator. The effect of this permanent tide is to increase the constant flattening of the earth. Table I gives the characteristics of the principal waves of types (A) and (B).

+

2.4. Numerical Values

Introducing in the precedingformulas, (2.7) and (2.9), the ratios of the masses of the moon and sun to that of the earth: mL = 1181.45

m,

=

333,432

and expressing the distances in terrestrial radii: a/rL

=

1/60.27

u/r,

=

1/23,400

(cosec 1”

=

206,265)

we obtain for the amplitudes of deflections of the vertical and variations in the intensity of gravity for the moon for the sun

e = 0.0174” sin 22 e = 0.0080’’ sin 22

dg = 0.168 mgal dg = 0.075 mgal

Using the formula (2.15) we are also able t o calculate the tides of the geoid, which become

This equation gives a maximum elevation of 35.6 cm and a maximum depression of 17.8 cm, hence a total amplitude of 53.4 cm. At the time of a new or full moon, the lunar effect (54 cm) and solar effect (25 cm) combine to give a maximum tide of 79 cm. These results would apply to the oceans if sea water had zero density and no viscosity, if no continental barriers or islands existed, if the solid portion of the earth were a niveau. surface, and, finally, if no oscillations were produced in the liquid mass.

400

PAUL J. MELCHIOR

3. DEFINITION OF LOVE’SNUMBERS To characterize the various aspects of the tide of the solid earth, Love [3] introduced two dimensionless numbers which bear his name. Shida has shown that a third number is necessary to obtain a complete representation of the phenomenon. Their significance, as we are going to show, is very simple and each type of elastic deformation can be represented by a combination of these numbers, which, in turn, are bound to the distribution of moduli of rigidity and densities in the earth by rather complex differential equations which have been established by Herglotz. We shall return later to this theoretical aspect. The guiding idea of Love’s theory is that, the perturbation potential being represented with sufficient accuracy by a spherical harmonic function of the second order, all the perturbations produced on the earth by this potential may be represented by the same harmonic function multiplied by the proper numerical coefficient for each aspect of the phenomenon. This coefficient is a simple algebraic combination of Love’s numbers. The radial displacement U and the cubical dilatation D,produced a t all points in the solid by forces deriving from the potential of the second order, Wz, may be expressed in the form

where r is the distance from the point considered to the center of the earth. The coefficients considered depend only on the radial distance. One may indeed presume that below the level of isostatic compensation there exists hydrostatic equilibrium and a symmetrical distribution of the densities, moduli of the rigidity, and compressibility around the center. Likewise, the potential due to the variation of density accompanying the cubical dilatation and the surface displacement of material can be expressed in the form

v

K(T)W* Moreover, the horizontal displacements in the meridian and in the prime vertical will be expressed as functions of the derivatives of Wz with respect to the latitude and the longitude: (3.3)

=

(positive directions of displacement are those of increasing tj and A).

EARTH TIDES

401

For the surface of the earth, where the observations are made, one defines H(a) = h (3.5) K(a) = k L(a) = 1 as the three Love numbers: h represents the ratio of the height of the terrestrial tide to the height of the corresponding static oceanic tide at the surface; k represents the ratio of the additional potential produced by this deformation to the deforming potential; and I represents the ratio of the horizontal displacement of the crust to that of the corresponding static oceanic tide. The observations of the earth tides relate to six aspects of the phenomenon : 1. Reduction of amplitude of oceanic and lake tides 2. Deflections of the vertical with respect to the crust 3. Elastic tensions and cubical dilatations in the crust 4. Deflection of the vertical with respect to the axis of the earth 5. Variations of the intensity of gravity 6. Changes in the speed of rotation of the earth due to zonal tides. In approaching successively the examination of the results of each type of observation, we shall establish first how one can introduce the effect of the deformations of the earth in the theoretical prediction with the help of particularly simple combinations of Love’s numbers.

4. STUDY OF

THE

AMPLITUDE OF OCEANIC TIDES

4.1. Efect Due to the Deformations of the Crust

This phenomenon was the first method of approach and measure of earth tides suggested by Kelvin. This method has only historical interest because indirect effects distort the results. It is impossible to eliminate these effects through the calculations. Following the definitions of Love’s numbers, the total perturbing potential is evidently (1 k)Wz and the static oceanic tide will have an effective height (1 k)Wz/g. But the tide observed on the tide gauges is the difference between the tide of the ocean and that of the crust, hWz/g, to which the tide gauges are attached. The amplitude observed for the oceanic tide will therefore be

+

+

(=

(1

wz = y w2 + k - h) I7 J !

402

PAUL J. MELCHIOR

4.2. Application to Oceanic Tides of Long Period

The comparison of observations and the theory should, therefore, permit us to deduce the factor (1 k - h ) , but that will only be practically possible for tidal oscillations of long period. These, because of the slowness of the displacements and the presence of continental barriers, which prevent the formation of currents, are little affected by the inertia, little damped and thus obey the static theory. Darwin, in 1881, analyzed the records from 14 ports in Europe and in the Indies covering 33 years of observations; later Schweydar analyzed those of 43 ports distributed over the world and covering 194 years of observations. Their results are the following:

+

Darwin Schweydar

Fortnightly tides 0.675 c! 0.084m.e. 0.6265 f 0.043

y = y =

y = y =

Monthly tides 0 680 k 0.387 m.e. 0.6053 f 0.102

Meanwhile, some very serious objections can be raised against this procedure of evaluation of the factor y. One has neglected, in the theoretical predictions of the long period tides, the gravitational effect exerted on the water by the liquid layer raised by the tide itself. One can easily estimate this “secondary” effect in the case of an ocean covering the whole earth, which is assumed to be homogeneous. We find that the heights observed should be reduced by 10% and also the numerical values of the coefficient y to which they are related. But this correction is very uncertain, because the oceans cover only of the earth and are irregularly distributed. In addition, the presence of continental barriers modify the amplitude and phase of theoretical tides. Finally, some static tides of the second type (which result from the Coriolis force) can also exist, even in restricted oceanic areas appearing as permanent currents that neither the continental barriers nor friction seem able to impede. They are therefore superposed on the ordinary static tide and only observations will establish to what an extent they affect the phenomena. It is interesting to note that notwithstanding all these theoretical objections, the numerical values obtained by this method for the coefficient y are in quite good accord with those provided by the other methods which are free from these objections.

N

4.9. Application to Short Period Tides in Lakes In 1925, Proudman suggested attempting to obtain a value of y from the study of tides in narrow seas or large lakes. Grace attempted the test

EARTHITIDES

403

for the Red Sea but did not obtain a significant determination (the factor found being negative). The reason for this can be attributed to the connection of that sea with the Indian Ocean. On the contrary, study of tides of Lake Baikal and Lake Tanganyika gave satisfactory results. ~

Lake Baikal [4]? Bay of Pestchanaia von Sterneck wave M z wave K 1 Grace (semidiurnal tide) Aksentcheva (semidiurnal tide) Bay of Tankoi Aksentcheva (semidiurnal tide) Lake Tanganyika-Albertville [5] Melchior wave M P

y = 0.52

0.73 0.54 0.72

x = -3" +4"

0.55 y =

0.56

x = +go

' x = phase shift.

5. PERIODICAL DEFLECTIONS OF THE

THE VERTICAL WITH RESPECTTO CRUST

6.1. Efect Caused by Deformations of the Crust

The deflections are affected by the addition of the potential kW,, and also by the deformations of the ground with reference to which they are measured. Love's numbers again permit a simple expression for the proportion by which the amplitude of the deflection of a pendulum will be modified. The problem is to measure the angle between the vertical and the normal to the deformed surface. From the force acting on the pendulum

one must subtract this effect caused by the deformation, with the result that one will observe 1 aw, 1 IG - h a-w-2 g aae -Ti=

+

The deformation of the crust has the same sense as the force and thus offsets a part of its effect. We find the same coefficient, 1 k - h, as that for the amplitude of the oceanic tides. This could be expected since, in practice, the static tides result uniquely from deflections of the vertical. The ratio of the observed amplitude, for the deflections of the vertical, to the calculated static theoretical amplitude will give directly the factor y.

+

404

PAUL J. MELCHIOR

Two very different types of instruments are able to measure these deflections: horizontal pendulums and large water levels. One can easily see that the difference of potential, a t the two ends of a sufficiently long tube of water, ought to produce slight oscillations of the liquid which one could be able to detect with the aid of very sensitive measuring devices, which we will describe later. We shall first discuss briefly the elementary theory of the horizontal pendulum. 5.2. The Horizontal Pendulum

The deflections of the vertical are so small (0.02”) that a special instrument must be devised in order to record them with sufficient precision. If, for example, one should want to represent a variation of 0.02” by a displacement of 1cm a t the end of an ordinary pendulum, the length of this pendulum should be 103 km! In order to increase considerably the sensitivity of a pendulum to the small deflections of the vertical, a method of almost horizontal suspension of the boom has been developed. This concept dates back to the beginning of the nineteenth century. Subsequently, the idea was the object of much research by Zollner (1869) a t Leipzig, whose name is generally attributed to this type of instrument. The boom of the pendulum AG, of an average length of 20 cm, is weighed with a mass P of about 20 gm. The boom is supported by two wires (of quartz or of superinvar) in such a manner that it makes a very small angle with the horizontal (of the order of 1’). This type of suspension is such that only the component Mg sin i affects the boom of the pendulum, since Mg cos i is always perpendicular to it and its effect is sustained completely by the system of suspension. On the other hand, the angle that the component Mg sin i subtends with the boom is variable. This is the angle 0. It is necessary now to introduce g sin i in place of g in the well-known formula giving the period of oscillation of the pendulum. We have now (5.3)

:

T

= 2rJ-

L g sin i

Now, the sensitivity of the pendulum is proportional to T2, that is, to cosec i. The amplification of variations of inclination is very large and corresponds, as does the period of oscillation, to a fictitious increase in the length of the pendulum boom by a factor cosec i. I n practice, one carefully measures the periods of oscillation of the pendulum in two positions, in order to obtain exactly the value of sin i which will indicate the amplification of the variations t o be observed. A horizontal pendulum permits the study of the displacements of the

405

EARTH TIDES

vertical in a direction perpendicular to the direction of the boom at rest. If one desires to reconstruct the complete trajectory described by the base of the vertical, it will be necessary to utilize two horizontal pendulums preferably at right angles, in order to register the two components of movement (the pendulum directed toward Oz permits observation of the component following Oy and inversely). The amplification of the movements thus realized can be further increased optically by fixing to the boom of the pendulum a mirror which reflects a light beam on a cylinder rotated by a clock motor and carrying photographic paper. The difficulties encountered in the first models of the Zollner suspension system resulted from the poor quality of the wires which were available a t that time. Hengler had proposed nonspun silk or horsehair; Zollner used steel and torques were appreciable. Orlov used platinum wires.

>At & I

I

I

I

1 I I

I

I

I

I I

I

I

I

I

I' I

I

I

FIQ.3. Schematic representationof different types of horizontal pendulums.

Although similar in principle, several methods eliminated the use of suspension wires. E. von Rebeur Paschwitz achieved this result with the aid of two pins acting in two different directions on the boom of the instrument which had a weight of 42 gm. These instruments were, moreover, destined for additional research, e.g., the recording of seismic waves. For the latter, Milne and Shaw built an instrument where one of the suspension wires was replaced by a point. The different types of instruments are shown schematically in Fig. 3. Hecker perfected the design of von Rebeur Paschwitz by placing the pins in such a way that their axes and the direction of gravity, applied at the center of gravity, would be concurrent. Obviously, the difficult problem in this kind of suspension was to find the proper materials for the pins and their supporting plates. The makers generally used steel pins resting on cups of agate, but wearing of the pins created some serious difficulties. Also, for the horizontal pendulums, various workers have returned to the Zollner suspension, using wires of tunga@n, fused quartz (Ishimoto), superinvar (Nishimura-diameter 30 p ) ,

406

PAUL J. MELCHIOR

or strips of phosphorbronae (Schaffernicht). Use is primarily limited to instruments which are light and of small dimensions; for example, the apparatus used by Schaffernicht a t Marburg was built with an aluminum boom weighted with a mass of 30 gm. The numerous pendulums constructed by Ishimoto were made completely of quartz, including the frame (height of 20 cm, boom of 8.6 cm, and wires of 12 cm and 7 p

FIG.4. Horizontal pendulum of Tomaschek. radius) in order to render the thermal effects practically negligible. The latter instruments have been used in prospecting where it is not always possible to find locations protected from thermal variations. The suspension of Zollner presents several advantages in comparison with that of von Rebeur Paschwita: first in the simplicity of construction; also, according to OrIov, in greater stability of the zero position and the amplification factor; and in the fact that the period T is independent of the amplitude, which is not always the case for the von Rebeur Paschwitz. Finally, we note that Lettau has obtained even greater sensitivity by

EARTH TIDES

407

coupling two horizontal pendulums, in a manner shown schematically in Fig. 3. However, this arrangement is quite delicate, and it is not possible to “couple” any two horizontal pendulums in this manner. 6.3. Observations Obtained from Large Water Levels

As we have mentioned previously, the study of the reduction of the amplitude of the oceanic tides is greatly complicated by the irregular forms of the ocean basins and the existence of considerable perturbing effects that result therefrom. It is practically impossible to take account of these effects. Source

f

FIG.5. Michelson Interferometer on tube for tidal observations.

In 1914, Michelson and Gale proposed to observe the phenomenon in a specially constructed container (with a regular geometrical form) through microscopic measurements and, later, by measurements of interference effects [6]. They buried, to a depth of 1.80 meters, two tubes 150 meters long and 15 cm in diameter. These tubes were half-filled with water and oriented along the meridian and the prime vertical. The first series of experiments, designed to study variations of the level a t the two extremities of each tube, were executed by measuring, with the aid of a microscope, the distance separating the extremity of an immersed rod from its image obtained through total reflection in the liquid. More precise results were obtained in the course of a second series of observations using the interference method, with the aid of an apparatus shown schematically in Fig. 5, which was placed a t each end of the tubes.

408

PAUL J. MELCHIOR

A horizontal mirror is submerged in such a manner that it is covered by a half-millimeter thick layer of water, whose viscosity damps small perturbations. Variation of level will produce a modification of the optical path of the rays issuing from the source S, and one will observe a displacement of the interference fringes from the two light beams. The two beams are obtained a t the first of the parallel faced glass plates, L1, which is half-silvered on one of its faces and inclined a t 45’. The second plate, inclined a t 45”, is a compensating plate. It equalizes the thickness of glass traversed along the optical path of the two interfering light beams and serves, a t the same time, to seal the tube. The displacements of the fringes are recorded on a film transported a t a rate of 2 cm per hour. One is able to record the hours by means of automatic slits in the recorder which appear on the film in the form of a fine bright line. The observations covered one year (November 20, 1916 to November 20, 1917). The films were measured with a microscope, the displacement of the fringes being estimated to one-tenth. The difference in the movement a t the two ends of each tube give a measurement leading to the observed height of the tide. Instruments based on the same principle are going to be installed in a cave near Trieste, Italy, by Professor A. Marussi, in order to obtain observations during the International Geophysical Year. The theory of the tides in the tubes of water has been developed by F. R. Moulton. The authors have carried out a harmonic analysis on the theoretical calculated curve, as well as on the curve resulting from the observations, in order to eliminate distortion due to imperfect elimination of certain components. I n 1934 Egedal and Fjeldstad applied the same procedure in Europe, installing two tubes of water in a 125-meter tunnel a t Bergen, Norway [7]. Recording of the variations of level was obtained in a different manner. The apparatus used, which is called a “level variometer,” consists of a cylindrical vessel 14 cm in diameter and 7 cm high communicating with the water of the tube which partially fills it. I n order to prevent evaporation, the water is covered by a layer of Nujol, 1 to 2 cm thick on which floats a cylindrical glass float 5 cm in diameter and 1.7 cm high. Vertical movements of the float are transmitted though a vertical rod to a rotating mirror whose variations of inclination are observed in a small telescope. The same principle applies to the observation made by W. B. Zerbe in the basins situated a t Carderock (near Washington, D. C.) which were designed for testing of ship models (David Taylor Model Basin) [8]. These long and narrow basins are enclosed under an arched roof and are well suited t o observations of this type. The basin used by Xerbe is

EARTH TIDES

409

845 meters long, 16 meters wide, and 6.7 meters deep. It is oriented along direction W 16" N-E 16' S. One gauge is situated 17 meters from the west end and the other is 15 meters from the east end, thus, they are separated by a distance of 813 meters. Measurements made a t the two ends of the basin permit the elimination of changes in level caused by evaporation and thereby reduce observational errors. The gauges were read alternately each quarter of an hour and each time the small oscillations of the pointer were noted in the following manner: four readings of the minima and four readings of the maxima. The difficulties encountered in obtaining a precise leveling between various Danish islands lead Norlund to use since 1939 a new, hydrostatic procedure. This installation, resembling that of Michelson but considerably longer led, unquestionably, to observations of terrestrial tides. Two vertical glass tubes, graduated in millimeters, were mounted on the islands Fyn and Sjaelland and connected by a water-filled pipe, 18 km in length, submerged to 60 meters on the bottom of Store Belt. Because water surfaces in the two vertical tubes belong to the same level surface, one can thus observe differences of elevation between the different islands. The differences between the readings of the vertical tubes at the two terminal stations are not constant but change regularly with time. The semidiurnal oscillations show an amplitude of 2 to 3 mm and a period of 12 h 25 m. The great length of the pipe (18 km) aids considerably the observation, which gives a coefficient 1 k - h = 0.8.

+

6.4. Numerical Results of the Observations

Figure 6 represents the reconstruction of the trajectory described by the position of the vertical, obtained from records of two horizontal pendulums oriented along the meridian and the prime vertical a t Freiberg in Saxony. On January 9, 1912, the moon and sun being near to the quadrature, the amplitude of the movement was a minimum. On January 12 of the same year, the two bodies being about in opposition, the amplitude of the movement was obviously a maximum. Such records, extending over long periods of time, are analyzed following the procedures ordinarily applied to oceanic tides, and from them the amplitudes and phases of the various components are deduced (Table I). Figure 7 represents three of these components obtained a t Barim, Manchuria. The dashed curve is the theoretically predicted curve, while the solid curve has been deduced from the observations. The amplitude is thus reduced in the ratio (1 k - h) just as has been predicted in formula (5.2). Thus, a t each station one is able to deduce, in each direction (meridian

+

410

PAUL

J. MELCHIOR

West

South

South

West

10

-

0 0.005" 0.01"

January 9, 1912, Oh-January 10, 1912,2h

January 2, 1912, Oh-January 3, 1912,2h

FIQ.6. Reconstruction of the trajectory described by the base of the vertical from records of two horizontal pendulums at Freiberg (Saxony).

N

FIQ.7. Components Mp,

,632,

01

of the earth tide at Barim (Manchuria).

and prime vertical, in general), a value of the coefficient y and the corresponding phase for each one of the components that can be isolated from the observed curve by harmonic analysis. An exhaustive list was given in 1954, including all determinations (33 in number) that have been made up to that time since that of von Rebeur Paschwitz in 1892 [l]. This list contains the essential characteristics of the stations.

EARTH TIDES

41 1

From a series of observations recently made at Winsford, England, Tomaschek has obtained a value 0.72 [16]. J. Picha [17], analyzing the observations made by Cechura at Brkaov6 Hory, Czechoslovakia from 1935 to 1940, a t a depth of 1000 meters, obtained for the average of the oscillations MzNzS2 in the direction of the meridian and the prime vertical the values 0.701. We will refrain here from details about the values found for the coefficient y because they are of very unequal quality. The best values of this coefficient will be mentioned in the following paragraph. 6.6. Discussion of the Observations-the

Indirect Efect of the Oceanic Tides

The first statement that can be made is that there exists a systematic difference between values of the factor y along the meridian and along the prime vertical, the latter generally giving a higher value of y (approximately 20%). To our knowledge, Hecker first indicated in 1907 the origin of this disagreement, attributing it to “indirect effects” caused by the complex influence exerted on the vertical by the masses of water moving in nearby seas. This influence presents three aspects: (a) the attraction of the water masses on the vertical; (b) a variable deflection of the crust under the influence of the extra load of water imposed on it (this is the most important part of the indirect effect, cf. Fig. 8); (c) the variation of the potential caused by this supplementary deformation of the crust, an effect of smaller order but of opposite sense to the two preceding. T. Shida [9] calculated, for the first time, the effects (a) and (b) for the station a t Kamingamo (Kyoto). We have reproduced his very striking results in Fig. 8. Two methods, based on essentially different principles, permit the separation of the direct effect from the global indirect effect. We shall now describe these briefly. 6.6. Method of Numerical Evaluation of the Indirect Eflects

This method has been applied principally by Japanese workers. (a) Calculation of the attraction of the water

The distribution of ocean water varies with the tides and is represented on cotidal charts. Dividing the ocean surface into small sectors, of uniform water height, of which the station occupies the top, and which are limited by radii rl, r2 and azimuths 01, & (taken in a clockwise direction), one obtains the attraction that it exerts by the formulas:

412

PAUL J. MELCHIOR

(5.4)

Ax = ___ jPh (log r2 - log r,)(cos e2 - cos el) towards the east g sin 1“ A s = fPh (log r2 - log rl) (sin O2 - sin 0,) towards the south g sin 1” ~

( p being the density of the ocean water). In these formulas it is necessary to replace h by h cos (nt - 4) for each component of the tide.

(b) Calculation of the deformation of the crust caused by the burden of the water Already in 1878, J. Boussinesq had calculated the distortion of a plane elastic surface under the effect of an additional load and showed Kamigarno IS

s

s

FIG.8. Components Ms, 0,of the earth tide at Kamigamo, according to T. Shida. Curve a: observed deflections Curve b: calculated effect due to deformation of the crust Curve c: calculated effect due to attraction of the water Curve d: direct effect = a - (b -k c) Curve e: calculated effect, using the static theory

that the vertical displacement of the crust was, at each point, proportional to the gravitational potential of the load and a function of the elastic constants of the medium (A, p the Lam6 constants) [lo]. Writing the attraction in the form (5.5)

the deformation is written, following Boussinesq :

EARTH TIDES

413

if (5.7) (c) Variation of potential This effect is less than the two others and acts as if it were produced by a negative mass. Nishimura, who appears to be the only one to have taken it into consideration in the case of the tides, acknowledges that, in a first approximation, one is able to consider the effect to be proportional to A , and defines

v

(5.8)

- E A [Ill

=

The sum of the indirect effects is, thus, expressed by

I

(5.9)

=

(1

+ v - €)A

The Japanese authors T. Shida, R. Sekiguchi, R. Takahasi, E. Nishimura have attempted to take into account variations with depth of the elastic characteristics of the medium by observing that the depth p of the crust participating in the deformation depends upon the distance r to the disturbing mass. Since X and p are functions of p , one is able, here, to consider them as functions of r and write (5.10)

I

=

(1

+

y(r)

- €)A

In order to obtain empirically the numerical values of these coefficients, Nishimura has made observations simultaneously a t two Japanese stations: Aso and Kamigamo. The difference between the components M z (or 0,)obtained a t these two places by harmonic analysis of the observations is freed from the direct effect, which is practically the same, and contains only the indirect effects, which depend upon the position of the stations with respect to the surrounding seas. The interpretation of this difference with the aid of (5.10) permitted Nishimura to obtain the values (5.11)

v(r) =

12.6 r+3

-

E

= 0.5

r being expressed in degrees (distances computed in arcs). We have compared this experimental result with previous theoretical research due to L. Rosenhead [12]. That author, primarily interested in the movements of mobile air masses on the surface of the earth and their effect on the moments of inertia of the earth, attempted, in 1929, to evaluate theoretically the importance of the compensation A due to the deformation of the crust under the weight of these air masses.

414

PAUL J. MELCHIOR

Adapting the Herglotz theory of the elasticity of the earth to this particular case, he found 0.3 < B < 0.4 which is exactly the same order of magnitude as that found by Nishimura. Rosenhead adopted for the calculation a rigidity modulus of zero in the core and equal to 16.95 10" dynes/cm2 in the solid shell. In the present state of the theory of the elasticity of the earth and of the precision of the experimental results, one can consider this agreement as being quite remarkable. Serious objections have frequently been raised against this method and to its applications: (1) the theory of Boussinesq is related to the deformation of a plane surface and thus, is only valid in the case under consideration over a very small distance because of the curvature of the earth; (2) the rocks constituting the upper layers of the earth do not rigorously obey Hooke's law on which the theory of the elasticity is based; (3) the tides aTe poorly known on the open ocean and the pattern of cotidal lines presents many uncertainties, rendering the application of the method very uncertain; (4) we have not been able to take into account the heterogeneity of the geological structure of the crust (cf. [lo]).

-

5.7. Empirical Method of Separation of the Two Efects

This methodwas introduced by Corkan and Doodson [13]. It is based on an exceedingly probable hypothesis and presents the advantage of great simplicity while appearing extremely effective. One notices that, over vast oceanic expanses, the ratio of amplitudes of various oscillations of the same type (for example the semidiurnal components) remains essentially constant. This constant differs from that given by the static theory and can be deduced from tide gauge observations. Thus, for the region of Liverpool : A ( S 2 ) / A ( M 2 )= 0.320 (static theory 0.465) (5'12) A ( N 2 ) / A ( M z )= 0.190 (static theory 0.193) The fundamental hypothesis of the method acknowledges that this is valid for all of the phenomena resulting from the oceanic tides, and therefore in the indirect effects studied here. Let f = E cos ( A - b) a component given by the theoretical static terrestrial tide, f l = I cos ( A - L) the indirect effect resulting from the presence of this component in the ocean tides, and c0 = K cos ( A - K ) the same component deduced from the observations by harmonic analysis.

415

EARTH TIDES

Obviously, one can write To

=

rT

+

TI

from which one can deduce, by equating the coefficients of sin A and cos A , I COB i y E cos b = K cos K (5.13) I sin 1 y E sin b = K sin K

+ +

We know t ha t the phase lag of the static terrestrial tide is very small. We assume initially, as the first approximation, that it is zero: b=O and the equations become : (5.14)

I cos i f y E = K COS K I sin i = K sin K

By virtue of the fundamental hypothesis developed, we write amplitude of the oceanic tide ( M z ) amplitude of the oceanic tide (S2) (5.15) static amplitude (Sz) static amplitude ( M 2 )

I(Sz) - o.320 I(Mz) r E ( S z ) - 0.465 ~E(Mz)

and two analogous relations for the ratio of the oscillations N z / M 2 ,for which the numerical values are 0.190 and 0.193, respectively. To each value of the phase of the indirect effect L there corresponds, for each oscillation (Mz, SZ,N z ) , values of I and y B from which the values of I(M2) and yB(M2) can be deduced from the relations (5.15). I n order t o verify the hypothesis developed, one should find, for each oscillation, a suitable value of i which should lead to the same values of I ( M 2 ) and yB(M2). This is done graphically, with I ( M z ) as abscissa and y B ( M 2 )as ordinate. Each oscillation is represented by a characteristic curve. The three curves determine a triangle of which one takes the center of gravity (by weighting the curves in proportion to the amplitudes of the oscillations). Corkan has thus found as coordinates of the center of gravity for Bidston (Birkenhead-Liverpool) : (5.16)

I ( M z ) = 0.0593” E(M2) = 0.0059”

from which y = 0.77 results, with the restriction th a t b = 0 and th a t the hypothesis of Doodson-Corkan is correct. Corkan has attempted to solve the problem completely, th a t is t,o determine the phase b by calculating the values of y corresponding to various values of b for two different stations [14].

416

PAUL J. MELCHIOR

He analyzed, for this purpose, the results from Bergen following the same scheme as for those a t Bidston and obtained y = b =

0.65 8"

a somewhat surprising result since it was generally acknowledged that the phase b should be very small. However, according to Corkan himself, the station a t Bidston would not be very adequate for this kind of determination since the indirect effect is ten times greater than the direct effect, while at Bergen the two effects are of the same order of magnitude. Melchior has discussed Corkan's method [15]and has shown that its application is more difficult for a continental station than for a coastal station where one can pre-estimate the approximate phase of the indirect effects. He shows that the number of solutions offered by the method is rather variable, because the representative curves y = f(1)of each wave ( M I , 82,N 2 ) are hyperbolas with parallel transverse axes. On the other hand, it would be necessary to know a priori the exact role of each oceanic region in the indirect effects acting on the particular station, in order to weigh correctly the values observed for these relations in these oceans. Thus relations accepted for the north-south component and for the east-west component will in general be different. Finally, Melchior has obtained the following: Freiberg in Saxony Br6zov6 Hory Bidston

y y Y

= 0.704 (weight = 4) (observations of Schweydar [15]) = 0.709 (weight = 9) (observations of Cechura-Picha [17]) = 0.703 (weight = 8) (observations of Doodson-Corkan [13])

These values are in very good internal agreement and in satisfactory agreement with the results of measurements made in Japan. The most probable value of y would then actually be (5.17)

y =

0.706

a value which should be very close to the true value of the coefficient y. Zerbe [8] has obtained in America 0.742, while the results of studies by Japanese scientists seem to indicate a value a little smaller, undoubtedly in the neighborhood of 0.68. One can wonder if it is necessary to infer the existence of a regional heterogeneity in the crust or rather an incomplete elimination of the indirect effects resulting from the application af the formulas of Boussinesq and the cotidal charts. Furthermore, note that the phase b has not been reduced to zero a t any of the Japanese stations, the phase being very variable from one station to another and appreciably different between the NS and EW components.

417

EARTH TIDES

6.8. Effect of the Tides on High Precision Leveling Operations

The hydrostatic leveling operation accomplished by Norlund has attracted the attention of geodesists to the eventual advantage of applying a “tidal” correction to the raw results of precise leveling measurements. The Danish geodesists Jansen and Simonsen were the first to do this in a practical manner. Beginning with the classical formulas giving the perturbing action of a heavenly body on the vertical of the level surface, they have calculated the correction to be applied to the results of precise leveling recently carried out in Denmark. By way of example, and to establish an order of magnitude, the luni-solar correction was 2.49 mm for a leveling operation extending over 125 km. L. Jones has applied the same method to the Belgian base network near Tournai. He has observed : (1) That there is not necessarily an amelioration of the leveling results after luni-solar correction. (2) That the proposed method permits no definite conclusions; the experimental measurements, which would be investigated, would only be able to show that the precise levelings are in a position to contribute in the general study of tides. (3) That the study of the possible effect of the luni-solar action on the systematic errors is intimately connected with the study of all of the causes of systematic errors. The International Association of Geodesy has recommended, during its meeting a t Brussels in 1951, an exhaustive study of these problems. In this connection, the Instituto Geogrbfico y Catastral has undertaken, in Spain, a program of experimental leveling. The author believes, in agreement with the opinion of L. Jones, that the application of the classical formulas (provided with the coefficient y = 1 k - h ) appears extremely hazardous, especially in a region near the ocean. This is due to the importance of the indirect effects (at Bergen and at Winsford, for example, they are of the same importance as the direct effect, and at Liverpool, 10 times greater) whose phase can be very different from that of the direct effect. Furthermore, the local geologic structure in certain cases is able to induce flexures in an opposite sense to those which one would expect from the localization of the additional loads (cf. Section 10). It appears to us that one could simultaneously employ a horizontal pendulum during the course of a leveling operation. This horizontal pendulum would be oriented perpendicular to the direction of the survey and, thus, would give along this direction not only the direct effect and indirect effect but also the thermal effects on the crust, that is, perhaps the complete correction to be applied to the leveling. Such an operation appears possible with pendulums of the type of Nishimura or of Ishimoto.

+

418

PAUL J. MELCHIOR

6. MEASUREMENT OF ELASTIC TENSIONS AND CUBICDILATATIONS PRODUCED BY THE EARTH TIDES DUE TO DEFORMATIONS 6.1. Components of the Tension

The deformations resulting from the action of the luni-solar potential give rise in the earth to elastic tensions which one can measure, a t least a t the surface. Dependent on the amplitude of the deformation, they will constitute a new way of approach for the experimental determination of Love’s numbers. Using relations (3.4) and (3.5)the deformations of the earth’s surface may be written in the form along the meridian

aw,

i

following the prime vertical

The classical theory of elasticity is based on Hooke’s law according to which the tensional components are linked to those of the dilatation (derived from the deformation, u, Y, w)by linear relations. If in addition, one assumes restrictive conditions of isotropy, homogeneity, and incompressibility, the tensional components in the X Y plane are directly proportional to the corresponding dilatational components :

Y 3 = - au f - = - - av

dy

ax

1

aso + -1-asA = i ax u a6 ag sin 8 ~

a sin 0

. _a2wz _ aeax

and the tension, exerted in such a direction that the direction cosines are (m, n), will be proportional to (6.4)

+ ezn2+ yamn

elmZ

Measurement of these will allow direct determination of the number 1. 6.2. The Sassa Extensometer

The measurement of such weak stresses in the crust is not possible, but Professor Sassa of Kyoto University has arrived at a very elegant method of making them evident [18]. His instrument, simple and ingenious in concept, is the only one which actually measures the number

EARTH TIDES

419

I

directly. A superinvar wire (1.6 mm in diameter) is stretched almost horizontally between two fixed supports about 25 meters apart and a weight of 350 grams is suspended from its center. Variations in the distance between the supports resulting from deformations of the crust will cause variations in the tension of the wire as expressed by equations (6.3) and (6.4) and vertical oscillations of the weight. This up and down movement is converted into rotation of a mirror through a bifilar suspension consisting of two superinvar wires, 0.5 mm in diameter, under torque. Figure 9 shows the disposition of a part of the apparatus in a room a t the Makimine station (one can also see a pair of Nishimura horizontal

FIG. 9. Instrument room at Makimine, Japan: extensometer of Sassa and two horizontal pendulums of Nishimura ,

pendulums). At the bottom is one of the supporting pillars, in the background to the left the device carrying the bifilar suspension on which the weight and the mirror may be seen. The apparatus is calibrated by displacing one of the suspension points by a quantity measured with a micrometer. Since the observation rooms are at great depth ( p ) , variations in temperature are very small (of the order 0.2"C per year). As shown by the theory [equation (6.4)] measurements in three different directions, to determine the components el , €2, y3,will allow a complete study of the stresses a t the surface. A complete set of equipment has been installed a t Osaka-Yama. The first results published by Sassa, Ozawa, and Yoshikawa relate to three different stations in Japan. These are shown in Table 11. The mean error in the value of the quantity 1 is of the order of +0.011.

420

PAUL J. MELCHIOR

The results marked with asterisks have had the indirect effect evaluated by Boussinesq's method and removed, the others are the raw results. Ozawa has tried to calculate the amplitude of the indirect effects beginning with the observed results, for various values of 1. These results are only preliminary and observations are still in progress. Ozawa [18a] has published the first detailed results obtained with the aid of a group of 13 extensometers placed in three stations and following different orientations and inclinations (cf. Table 111). TABLE 11. Numerical value of 1 number obtained with extensometer. Station

&

x

Time interval

P

Mitsubishi 35" 40' 135" 47' -800 Ikuno 32" 38' 121" 24' -165 Makimine Osaka-Yama 34" 54' 135" 51' -150 Tune1

9/19/43-10/5/43 12/7/49-12/8/50 10/24/47-10/29/48 9/2/52-10/1/52 5/19/52-5/3/52

Azimuth of wire

1

X

E 3" S 0.051 -40"* N 57" W 0.025 +74O S 38" W 0 . 0 5 -61°* 0.035 S 76" W 0.091 - 5" 0.021 -12" S 2" W

Studying first the direct effect, the author has developed the calculations of the tension according to its six components, cubic dilatation, and horizontal areal tension B = eee e4@. In order to take into account the indirect effects, he has calculated the corresponding tensions from the expressions for displacements given by Boussinesq's solution. He was able to show that for this solution the horizontal areal tension, cubic dilatation, and radial tension which are due to indirect effects, are nil at the surface. The recording of each instrument over a period of a month was subjected to harmonic analysis. Since the direct effect is approximately the same at the three stations the observed indirect effects were compared with the calculated effects by the formulas deduced from Boussinesq's solution, by taking the difference between the observed effects M zfor each pair of stations. This comparison makes evident an anomaly of the station a t Suhara in relation to the other two which are in agreement. However, Suhara is right on the seacoast. The direct effects are studied only in the representative combinations at the surf ace :

+

radial tension (6.5)

err =

2 [4h + a a$ dr

2 horizontal areal tension Z = - (h - 31)W~a ag

cubic dilatation

421

EIARTH TIDES

If Boussinesq's solution is accepted these combinations are free of indirect effects. The results obtained for the different combinations of parameters are listed in Table 111. From these data the author concludes that the weighted mean is: h - 31 = 0.434

Taking h

=

+ 0.026

0.600, Ozawa then deduces that 1

=

0.055.

TABLE 111. Results obtained from a group of extensometers. Distance from ocean

$J

x

Osakayama

34'59.6' N

135O51.5' E

- 150 m

65 km

Kishu Suhara

33O51.7' N 34" 2.6' N

135'53.4' E 135'21.7' E

-100 m - 30 m to - 60m

15 km 50 m to 90m

Station

Station Osakayama

Wave

Mn 82

Ki 0 1

Kishu Suhara

M1 M2

P

Extensometers 3 horizontal 2 vertical 2 oblique 3 horizontal 3 horizontal

Horizontal areal tension h - 31

Radial tension ah' 4h

+

Cubic dilatation ah' 6h - 61

0.427 0.572 0.294 0.511 0.304 0.521

-0.246 -0.615 -0.371 -0.402 -

0.604 0.616 0.221 0.644 -

+

Benioff has installed extensometers a t Dalton Canyon (Pasadena) and Isabella (Kern County), and positive results have already been obtained. He proposes to make additional installations in South America ~91. 6.3. Efect of Cubic Dilatations on Wells

This result of earth tides is demonstrated by variations in the water level in wells which are too far from the ocean to have any relation to it. The effects have been noticed positively a t fifteen points in Europe,

422

PAUL J. MELCHIOR

Africa, and America [23]. We will cover here only the two cases which have been rigorously analyzed.: These are the Kiabukwa well (Belgian Congo) and the Turnhout boring (Belgium) [23] which are located a t 7" south latitude and 51" north. They present the two extreme cases with respect to tidal theory since the diurnal tides are zero at the equator and maximum at 45". In both cases the temperature of the water was very high (92" t o 102°C). It should be noted, before discussing these phenomena, that one cannot predict from theory the absolute amplitude of the tide since one does not know the volume of water confined in the dilated beds nor the porosity of the beds, both of which factors will affect the results. Under these conditions two tests make the tidal effects clear. First, the various oscillations must have amplitude ratios which are related by the static theory. Secondly, the phases of the oscillations must be 180" since the effect of compression will invert the tide; there will be a contraction of the beds a t low crustal tide and this contraction will raise the water level in the spring and cause a high tide in it. Table IV demonstrates that this is the case at Kiabukwa and Turnhout. TABLE IV. Tides in wells. Turnhout

Kiabukwa 4 = -7'47'

6 = +51° 19' Relative amplitude

Oscillation Phase Semidiurnal Ma 185'1 N2 176'6

Lz S2 Diurnal K1 01 &1

J1

001

Relative amplitude

Amplitude Static Amplitude Static cm Observed theory Phase cm Observed theory

-

-

-

0.198 0.073 0.524

0.194 0.028 0.465

242'4 209"3

1.48 0.24 0.04 0.68

0.154 0.025 0.455

- 183.7 0.194 170.8 0.028 235.1 0.465 182.6

161"6 151'2 178'5 238'9 179'5

1.41 1.20 0.23 0.09 0.08

0.952 0.810 0.157 0.063 0.057

1.460 At Kiabukwa the weak amplitudes 1.037 of the diurnal oscillations are not 0.201 significant, for want of data on the 0.082 atmospheric pressure 0.044

7.53 1.49 0.55 3.95

These phenomena constitute the most easily measured effect of earth tides since they involve quantities of the order of several centimeters. Perkeris' suggestion, of constructing an experimental dilatometer, deserves to be pursued. This is particularly interesting if one wishes to

423

EARTH TIDES

verify the recent theory of Jeffreys (see 1 0 . 5 ) which suggests that the diurnal oscillations are related to movements in the core. The diurnal oscillations should then have a different amplitude relation with the semidiurnal oscillations. The Turnhout observations appear to support a first experimental confirmation of this theory; the K 1 oscillation there has a slightly reduced amplitude nearly in the ratio predicted by Jeffreys theory [19]. K. Sperling [23a] has noticed a very sensitive effect of the earth tides on the yield of oil wells in Nienhagen, Germany (4 = 52'30'N, X = 1O"E). He made this clear for tides Mz and Sz. However, a rigorous harmonic analysis is not possible because priorities of production have not permitted recording of the data. Moreover, the level is often irregularly disturbed by accidents (fallen rock, paraffin, sands, etc.).

7. DEFLECTIONS OF THE VERTICAL WITH RESPECT TO THE EARTH

THE

AXIS OF

7.1. E$ect of Deformation of the Crust

The deflections of the vertical determined by horizontal pendulums or levels are related to the crust itself. With a horizontal pendulum the deviations of a light beam reflected by a mirror attached to a moving arm are compared on a recording drum with the trace of a light beam reflected from a mirror fixed to the instrument; i.e., to the deformed crust. The situation is quite different in the case of astronomical instruments. All deflections from the vertical a t a place must vary the astronomic coordinates, especially the latitude. This latter is determined by comparing the direction of the vertical measured by two levels (Talcott's method) or a mercury bath (Zenith Tube, Astrolabe) to the observed directions of a series of ((fixed" fundamental stars. In this case the deflections of the vertical are not fixed with respect to the crust but with respect to the direction of the axis of rotation of the earth in space. If the earth were perfectly rigid, there would be no difference between the results of the two methods of measurement, but it is clear that, if the earth deforms, the observed phenomena will be different for the two cases. We have expressed in (6.1) and (6.2) the respective components of the deviation of the vertical and the deformation of the surface in two directions chosen conveniently in the plane tangent to the surface. The deformation will also induce (Eq. 3.3) a supplementary perturbing potential, kW2, which will be added to Wz in the expressions (6.1) so that, finally, the deflection of the vertical with respect to the axis of the earth will be:

424

PAUL J. MELCHIOR

in the meridian

(1

(7.1) in the prime vertical (1

1 awz + k - 1) ag a0

1 aw, + k - 1) ___ ag sin 0 ax

The intervening factor is no longer.?

=

1

+ k - h but

It would appear, a t first glance, to be impossible that astronomical observations of the meridian having a precision of the very most a tenth of a second of arc would reveal such weak phenomena. Despite this objection, highly precise measurements made using instruments especially equipped for the accurate measurement of latitude and carried out methodically over the course of many years have made it possible to evaluate this phenomenon and to obtain a relatively accurate value of the coefficient, A, which, while not reaching the precision attained by other methods of observation of earth tides, is nevertheless in good accord with the value that theoretical research indicates. 7.2. Indirect Efects From the principles governing the methods of observation in meridian astronomy arises a property which is important in the consideration of the indirect effects. It is evident that deformations of the crust, due to the varying load of the ocean waters, cannot play a part since they do not affect the direction of the vertical, which is the only reference used, and which is defined by the level bubble or the mercury bath. The observed phenomenon is then composed only of the oscillations of the vertical resulting from direct luni-solar action and of “indirect” oscillations resulting from the attraction of the waters and the effect on the potential of the deformations of the crust, the last acting as partial compensation for the effect of the attraction of the water (about 40%) cf. Section 5.6). 7.9. The Zenith Telescope This type of instrument, utilized since 1899 in the various stations of the International Latitude Service, has a mounting especially adapted to the precise determination of latitude by the method of Horrebow and Talcott. This method consists of successive observations of two stars symmetric with respect to the zenith and passing the meridian a t a few minutes interval.

EARTH TIDES

425

In this manner the latitude is obtained by the relation SNorth

(7.3)

4=

+2

88outh

ZN - 28 $7-

The measurement is made using only a micrometer screw and without the need of a divided circle and this makes the method superior. Several authors have attempted a t various times, and with different data, to analyze the observations in relation to the hour angle and the

7 8 9 1 0 1 1 O h 1 2 3 4 5

FIG.10. Apparent variations of latitude due to earth tides at stations of the International Latitude Service.

declination of the moon. (The sun does not play an important part here since the observations of latitude are always made in such a manner as to be centered a t local midnight.) Their results are given in Table V, and for an example we reproduce the figure given by E. Nishimura relating to five stations in the northern hemisphere (Fig. 10). This last has taken the indirect effects, evaluated by the cotidal chart method and Boussinesq’s formulas, into consideration. 7.4. The Photographic Zenith Tube

Markowitz and Bestul have discussed the observations made in Washington with this instrument. They have taken into account the indirect effect due to the attraction of the waters but not that arising from varia-

426

PAUL J. MELCHIOR

TABLE^ V. Factor A. Results obtained by analysis of the observations of variations of latitude." Station

A

Mizusawa Carloforte Ukiah Gaithersburg Tschardjui Cincinnati La Plat,a Greenwich Washington Pulkovo Poltava Pino-Torinese Rabelsberg a

0.68 1.07 1.00 0.56 1.06 1.33 2.23 1.16 1.27 1.05 1.25 1.37 1.60

Thew data are the weighted means of numerous results of analysis by several authors.

tions in the potential due to flexing of the crust. They obtained a coefficient of 1.30, but if the compensation of 40% of the attraction, due to the variation in potential, is taken into account, the coefficient will be 1.27. 7.5. Value Concluded for A It is difficult to decide on a value for A since the results are disparate. A careful correction for the indirect effects would be necessary but the cotidal chart method is uncertain. A weighted mean of all the valid results of Table V is A = 1.13

(7.4)

(Tschardjui, Cincinnati, La Plata, Pino-Torinese, Babelsberg : weight

x,All the others: weight 1.)

It is generally accepted that A is between 1.1 and 1.2. 8. VARIATIONS IN

THE

INTENSITY OF GRAVITY

8.1, Effect of Crustal Deformation

The variations calculated in (2.15) are affected by elastic deformation of the globe with the resulting potential variations. With the help of Love's numbers the importance of this modification of the amplitude calculated from the classical theory may be evaluated. If V ois the initial potential, the gravitational acceleration will be

427

EARTH TIDES

The potential of the body under luni-solar action and deformed is (8.2)

V'

= w2

+ Wz' + r avo+ vo 7 & -

The variation of g will be that derived from the additional potential with respect to r, for (8.3)

91 - go = -

-+aT aw; (awz ar

+r$)

Here, Wz' represents the additional potential due to the deformation of the globe and is of the form R2/a3,while [ = h(Wz/g). Consequently,

and finally,

I t is this new combination of Love's numbers given as (8.5)

6 =1

+ h - 4gc

which allows us to determine the observations of the variations of g.

8.2,BiJilar Gravimeters The first effort to reveal this phenomenon was made by W. Schweydar in 1913. The observations, made a t Potsdam with a Schmidt bifilar gravimeter, gave a result ( 6 = 1.2) that recent observations tend to confirm. I n 1927, Tomaschek and Schaffernicht, having created a highly sensitive gravimeter, also of the bifilar type, made measurements at Marburg which gave quite a different result (6 < 1) that is even theoretically impossible. Ellenberger in Germany and T. Ichinohe in Japan have improved on these gravimeters by use of a double bifilar suspension giving a sensitivity which appears to be much greater, assuring a precision of better than a microgal. These instruments will be used during the International Geophysical Year. Apparatus of this type is not easily transportable.

: I. Prospecting Gravimeters

)

(:reat progress has been made in the development of portable grav' ters of high sensitivity. These are designed to disclose anomalies of le than 1 mgal and will measure tidal phenomena with amplitudes of 0. mgal.

428

PAUL J. MELCHIOR

3

FIQ.11. Askania Gravimeter GS.11.

Many variations are employed in the construction of the mounting, but the basic principle is the same (see Fig. 11). A mass is fixed a t the end of a beam making an angle a with the vertical and swinging about a horizontal axis. An inclined spring, also in the vertical plane, supports the beam so that rotation of the beam varies the pull of the spring. Thus,

EARTH TIDES

429

the moment of the force exerted by the spring will be equal to the couple due to gravity so that the moving assembly will be in equilibrium for all values of a and the sensitivity is great and constant over a large range. The addition of a supplementary spring allows compensation for gravitational variations and the shifting of the zero by a known amount in order to extend the range of the measurements (change of zone). Prospecting companies have taken the initiative in determining the factor (a), knowledge of which is necessary to correct their readings starting from tables or abacus giving the classical theoretical effect. However, these surveys cover too limited a period of time (15 days) to permit the harmonic analysis which would give the various components with the accuracy desirable for geophysical research. 8.4. Results of the Observations

It has meanwhile been verified that the value of 6 determined for the different isolated oscillations by harmonic analysis is almost systematically less than the value concluded from a global comparison with a calculated curve. W. D. Lambert explained this effect by recalling a statement of R. Harris: each perturbing phenomenon which is such that its origin increases the difference between the greatest maximum and the greatest minimum will be eliminated in harmonic analysis by separation of the various components. Harris evaluated the effect on the amplitude of a factor 1.02, but it has not been possible to find observations which would support this numerical coefficient. In the type of observation studied here it seems that it is a question of a coefficient of the same order. The study of the indirect effects acting on the acceleration of gravity has hardly been approached. This is due to the fact that their role is relatively smaller than in the study of the deflections of the vertical. They have manifested themselves clearly in certain stations near the ocean and have been noted by prospectors in the vicinity of the Gulf of Mexico. On the other hand, Bollo and Gougenheim believe that they do not have the effect, which might be presumed to be found in the form of a phase shift, a t La Chapelle Saint Laurent, a station very close to the coast. In the case of variations in the intensity of gravity, the indirect effects include the attraction due to water masses, the variation of distance to the center of the earth resulting from flexure of the crust, and the variation of the potential resulting from that flexure. The only solution, which is actually possible, is to take the mean of the results of a great number of stations dispersed over all regions of the earth. Harmonic analysis has been applied by A. J. Hoskinson to nearly

430

PAUL J. MELCHIOR

all of the results of the important field work of the Shell Company, and W. D. Lambert [20] calculating the vectorial mean for each oscilration concluded that: for Ma for Sz for K1 for O1

8

K

1.191 f 0.012 1.213 f 0.022 1.192 f 0.026 1.178 f 0.016

l"99 f O"95 4"92 f 1'41 -0234 f 0'86 O"70 f O"77

Following the views of Jeffreys and considering the semidiurnal and diurnal oscillations separately, W. D. Lambert accepted: 6 = 1.198 K = 2'97 for the semidiurnal oscillations for the diurnal oscillations 6 = 1.186 K = -0"22 The probable errors are too large for one to admit without reserve the existence of a difference between the two types of oscillations as experimentally established. In this regard it is interesting to compare the results obtained by two separate sets of apparatus functioning simultaneously at the same station (Houston, Edmonton, Los Angeles) or during different periods. The mean phase shifts obtained by W. D. Lambert result without doubt from an imperfect elimination of the indirect effects. The more important series of observations were recently made at Strasbourg (France) by Lecolazet with a revised North American gravimeter (improvements and automatic recording due t o the author). Eight months of observation were analyzed and the results are given in Table VI.

TABLE VI. Results of Global Analysis. Place of observation : Strasbourg X = 7'46'E). Original time: 6 A. M., G.M.T., Jan. 5, 1955.

($ = 48'35";

Wave

8

K1

1.204 f 0.006 1.178 f 0.006 1.150 f 0.017 1.20 f 0.03 1.16 f 0.13 1.16 f 0.09 1.6 f 0.2 1.211 f 0.005 1.217 f 0.013 1.27 f 0.05 1 . 2 1 f 0.02 1.4 f 0.2 1.3 f 0 . 2

01

P1 &I

MI J1

S1

M2 S2 K1 Na

LZ 2N1

Phase shift, in degrees

-

1.48 1.59 0.9 - 2.4 4.0 7.8 -22 0.16 - 2.95 - 2.9 1.9 1 1

f 0.27 f 0.27 f 0.9 f 1.5 f 6.5 f 4.3 f8 f 0.21 f 0.61 f 2.2 f 1.0 f 6 f 7

431

EARTH TIDES

One of the difficultiesin applying harmonic analysis to these observations arises from the drift that is generally significant for gravimeters (just as for horizontal pendulums), and particularly disturbances in the drift. Lassovsky has proposed a simple procedure to eliminate this consisting of noting the hours when the tides are theoretically zero and to mark the observed values of g a t these times. This series of values will give the drift simply [4].However, the methods of harmonic analysis generally eliminate linear drift on each interval of 24 hours; a method given by Lecolazet eliminates a parabolic drift. 8.6. Observations M a d e with Clock Pendulums

E. Brown and D. Brouwer first suggested studies on the existence of small variations in g during the course of a day using observatory clocks of high accuracy. This method of observation not only can yield a precision as high as t!hat of the gravimeters, if the observations are continued over several years, but also constitutes a remarkable test. Brown and Brouwer made continuous comparisons, on a Loomis chronograph of the operation of three Shortt clocks, whose rate is dependent on g, with a quartz clock based on the vibrations of a quartz crystal, which is independent of gravity. The phenomenon was clearly measured, the coefficient being 0.9 and the phase shift 17"30', a result which is not surprising when we see the scattering of gravimetric measurements. The problem has been taken up again recently by N. Stoyko [22] with four Leroy pendulums and one Shortt pendulum compared with a Belin tuning fork clock a t the Paris Observatory. Studying the terms of the sidereal day, for which the theoretical global amplitude is 0.0013 sec, he found for the coefficient 6 = 1.195 a value in excellent accord with that concluded from the field work of the Shell Company by W. D. Lambert. The results of Stoyko, detailed by year and pendulum and reproduced TABLE VII. Determination of factor 6 by pendulum clocks. 1940

1941

1942

1943

Average

44 Shortt 1185 Leroy 1228 Leroy 1229 Leroy 1372 Leroy

1.45 1.14 1.15 1.17 1.18

1.04 0.98 1.02 1.20 1.28

1.40 0.97 1.38 1.29 1.56

1.08 1.26 1.14 1.42 1.03

1.24 1.09 1.17 1.27 1.26

Average

1.22

1.10

1.32

1.19

1.21

Phase (0-C)

-0.8h

+O .9h

-1,lh

-0.4h

432

PAUL J. MELCHIOR

.

in Table VII, give us an idea of the precision which may be obtained in this type of measurement. P. Sollenberger and G. Clemence, having tried to determine a variation in the corrections of the pendulum clocks at Washington from observations made with a zenith tube, found in fact a combination of the effect affecting the observations themselves (with the factor A) and of the effect acting on the pendulum (factor 6) but one cannot determine what part of this belongs to each phenomenon (the coefficient found was 0.92). 9. THEROLE OF

THE

GEOLOGIC STRUCTURE OF INDIRECT EFFECTS

THE

CRUSTIN

THE

9.1. Oceanic Effects

Studies of the indirect effect show that it is necessary to consider the possibility that local flexures in the crust may have an important effect. If, in the general phenomenon of earth tides (direct effect), the whole earth participates in the deformation, and if, as a consequence, small local inequalities of structure do not have a measurable effect on the amplitude of the deformation, it will not be, a priori, the same in the case of local flexures due to oceanic or barometric loads where only the superficial crust floating on a viscous substratum participates in the deformation. Tomaschek has shown, in a penetrating discussion of the observations, that one must consider independent compartments more or less rigidly interconnected. The question becomes one of determining the limits of blocks capable of movements of their own. Nishimura has succeeded in demonstrating the effect of an active fault on the flexures of the superficial crust. His study is based on the fact that, near the ocean, the indirect effect is large compared to the direct effect, so that the behavior of the crust will give measurable information on the microstructure of the superficial beds. Nishimura [ll]placed six pairs of horizontal pendulums along the active fault of Beppu (3G016’N, 131”30’E, Kyushu) placed a t intervals from 400 to 1800 meters starting from the fault. Figure 12 shows, in one part (a) the observed effect at six points and, in the other part (b) the residue after elimination of the direct and indirect effects, the latter calculated by Boussinesq’s method. The points D and E show a very curious anomaly. But the region to the south of the fault ( T T )is mountainous and built of ancient volcanic rocks (point F ) , while the region to the north is an alluvial plain (points ABCD). An important subsiding movement of the plain has been detected which actually still continues according to the results of precise leveling. From this phenomenon Nishimura estimates that the fault behaves

433

EARTH TIDES

like the boundary of an elastic plane, with the result that certain neighboring points bend toward the fault rather than in the direction of the bay. This is shown in Fig. 12 by the arrows which indicate the time when high tide occurs in the bay. (The center of gravity of the block would be found to the north of the station and of the overloading water mass. The fault constitutes the border of this block.) Examining the records of the German stations a t Marburg, Pillnitz, Beuthen, and Berchtesgaden, Tomaschek considers that the Alpine chain is manifested by a predominance of north-south flexures. Similarly, studying the gravimetric observations which he has made in Great Britain (Peebles and Kirklington) the same author described these phenomena as follows [24].

0.005”

74 -.,

Bay of Beppu

la1

FIG.12A. Deflections observed a t Beppu. Fra. 12B. The same curves corrected for the direct effect and calculable indirect effects (TT = fault).

“The tectonic block to which these stations belong is the Caledonian structure, comprising the Welsh mountains and the Pennines. Kirklington is situated in the Eastern basin, whereas Peebles rests on the northern spur of the Pennines. Observations with horizontal pendulums performed at Winsford (53’ 12’ N, 02” 20’ W) show, as do the observations of Gnass in the Alps, that this tectonic structure acts as a sort of long rigid bar, which is more easily tilted round an axis parallel to its length than an axis perpendicular to it. Peebles is situated on the longitudinal axis, whereas Kirklington lies on an axis which is perpendicular to the arc Welsh Mountains-Pennines. We have therefore to expect a different response to the loading tides in such a sense that Kirklington should be more influenced than Peebles. The higher value of 6 in Kirklington indicates that the movement is opposite to the loading of the Irish sea, that is, that the axis of this movement coincides nearly with the region of greater vertical thickness of the mountains.”

434

PAUL J. MELCHIOR

This observation is similar to that of Nishimura. By the application of Corkan's method t o the observations a t Freiberg and BrBzov6 Hory, Melchior shows that the indirect effects diverge greatly, notwithstanding the closeness of the two stations (distance: 145 km). The representative ellipses of the wave Mz due to indirect effects have their major axes perpendicular and the direction of revolution of the base of the vertical is reversed [15]. I n the course of deducing the indirect effects from the difference between the observed effect and the theoretical direct effect-the value of the coefficient being taken as 0.72-Professor Tomaschek had previously pointed out an analogous divergence between the two stations a t Freiberg and Pillnitz, which are very close to each other (distance: 40 km). It is established, therefore, that Pillnitz and BrBzovB Hory behave in exactly the same way. This obviously suggests a certain independence of movement of the various continental regions, constituting as many independent blocks. Unfortunately such an interpretation remains qualitative. 9.2. Atmospheric Efects

One can readily imagine that moving air masses can affect the direction of the vertical and the gravitational intensity just as is the case with the mobile ocean masses. As early as 1882 this effect was suspected, and G. H. Darwin wrote that i t did not appear impossible that, a t some future date, when very precise tidal and barometric observations could be attained, an estimation could be made of the modulus of rigidity of the upper 500 mi of the earth's mass (see reference [24a]). Assuming that the upper crust has a rigidity.slightly larger than that of glass, this author has calculated for a variation of pressure of 50 mm of mercury an inclination of the surface of the lithosphere by 0.012". In a recent paper by Tomaschek [25] the question has been reopened with the observation of a perturbation of almost 0.05" in the records of the horizontal pendulums a t Winsford simultaneously with the passage of a strong atmospheric perturbation on the British Islands (warm front followed immediately by a cold front and a zone of low pressure). However, the inclination observed is much too strong, and in the opposite sense, to be that given by the formulas of Darwin. The effect would likely be associated with the tectonic structure and of the type described above. Tomaschek envisages blocks with diameters of the order of 1000 km and explains the observations by stating that the Winsford station, as well as the low pressure center (58"N), are to the south of the center of gravity of the block and that Winsford is raised with the southern part of the block.

EARTH TIDES

435

10. THEORY OF ELASTIC DEFORMATIONS OF THE EARTH 10.1. Conclusions Drawn from the Observations of Various Efects of Earth Tides One can sum up the results of all relative measurements of earth tides which we have just described as follows: =

6

=

A

=

1 1 1

+ k - h = 0.706 f 0.01 + h - 3k/2 = 1.20 f 0.02 + k - 1 = 1.13 f 0.10 1

=

0.05 f 0.03

The two first relations, which are the best established, give

k

=

0.188 & 0.06

h = 0.482 f 0.07

and, admitting the results of the extensometer measurements, we obtain A = 1.13

i-0.09

so t ha t the four methods of measuring earth tides give coherent results. 10.2. Law of Elasticity Observations of deformations of the globe are related to various phenomena: earthquakes, tides, movements of the instantaneous pole of rotation with respect to the globe, isostasy. The periods involved in earthquakes and tides permit treatment of the phenomena as purely elastic, the raising of Fenno-Scandia under isostasy must be treated aq a viscous phenomenon. Between these two extremes with great differences in period it is necessary to resort to elastico-viscous theories which are not entirely satisfactory and do not conform to laboratory experiences [I]. Their application t o the movement of the pole, the period of which is about 1.2 years, is not very good and this movement has still not been satisfactorily explained [26]. 10.3. Herglotz Theory We limit ourselves here to the essential elements of the theory of elasticity as applied to earth tides. This theory is obviously based on Hooke’s law (proportionality between stress and deformation) and its object is to express Love’s numbers, h, k, I , as a function of the elastic constants and the density of the earth’s interior. The first attempt was made by Kelvin who adopted, as a first approxi-

436

PAUL J. MELCHIOR

mation, a globe which was homogeneous in density and modulus of elasticity and perfectly incompressible. Some simple and frequently reproduced formulas resulted:

but these formulas will not explain the observed phenomena since y = 0.73 will givep = 10 * 10” while 6 = 1.18 willgivep = 1.5 * loll cgs. Herglotz advanced the theory considerably by making the density and the modulus of rigidity functions of the distance, T , to the center of mass, the material remaining perfectly incompressible. He thus arrived at a differential equation of the sixth order which can only be integrated for certain particular cases. It is interesting to note that this equation is only a generalization of Clairaut’s equation relating to the flattening of concentric liquid layers in rotation, and it will revert to this equation if we set p equal to zero. This remark explains why forms of the type of Roche have been adopted for the distribution of modulus of rigidity and density within the earth in attempts to solve the Herglotz equation. However, these laws are only speculative and have no relation to the internal structure of the earth. Some applications have been made using Wiechert’s model by Jeffreys and by Rosenhead, assuming that the core and the mantle are homogeneous in density and elasticity. Finally, a very remarkable study has been made by H. Takeuchi [27] who developed the Herglotz equations by considering the compressibility of the earth. He succeeded numerically integrating for different models with the following characteristics: Model 1 : Bullen’s second density law (published in 1940). Compressibility and rigidity from seismic wave velocities of Gutenberg, Richter, and Wadati; zero rigidity in the core. Model 2: Bullen’s first density law (1936). Compressibility and rigidity from Jeffreys’ seismic wave velocities from 0 to 500 km, discontinuity a t 500 km, then Gutenberg’s and Wadati’s velocities. Rigidity zero in the core. Models 3, 4, 5, 6: The same constitution for the mantle as in Model 2, but in the core (from 2900 km to center) : X = 11 * 1OI2 Model Model Model Model

3 4 5 6

p = p =

lo7 lo9

p = loll p =

1013

437

EARTH TIDES

The numerical results are as shown in TabIe VIII. Takeuchi states that the results are essentially constant for core rigidities between 0 and lo9. The results from the most recent observations indicate that the best value which may be expected is between lo9 and 10". This table shows principally, in our opinion, that the tidal observations, disturbed as they are by the indirect effects, and the motion of the pole (whose period appears as variable), are not known with sufficient precision to permit accurate deductions on the internal constitution of the earth. TABLE VITI. Theoretical values of Love's numbers for several earth models. Model

k h klh 1 Y

8 A Pcore

1

2

3

4

5

6

0.290 0.587 0.494 0,068 0.703 1.152 1.22 0

0.281 0.606 0.464 0.082 0.675 1.188 1.20 0

0.275 0.601 0.457 0.081 0.674 1.189 1.19 10'

0.275 0.600 0.458 0.081 0.675 1.188 1.19

0.243 0.530 0.458 0.083 0.713 1.167 1.16 10"

0.055 0.109 0.504 0.092 0.946 1.127 1.96 10'8

109

The work of Takeuchi has advanced the theory beyond the observations and makes better observations most desirable. 20.4. Note on the Relation between the Numbers h and k Kelvin obtained, as stated, a simple relation between Love's numbers, 3h/5 but by assuming that the earth is homogeneous from the standpoint of density and rigidity. Melchior [29] has found that it is possible to reach a simple relation between these numbers without this hypothesis. This is done by graphical integration based on Bullen's density distribution but keeping the hypothesis that the elastic deformations are homothetic with respect to the center of the earth. It is shown that in all casea k / h is less than 0.6, and the value established by this author is

k

=

(10.2)

k

= /2 l/h

Developing this argument G. Jobert [30] has shown that if homothety of the deformations is abandoned, one can no longer determine k / h but the value W is an upper limit of the relation. This condition is an interesting criterion and it can be shown that, among all the efforts to integrate the Herglotz equation, only those of Boaga (trinomial law of densities,

438

PAUL J. MELCHIOR

Roche type of rigidity law) and those of Takeuchi fulfill this condition

I %.jh

(10.3) (cf. Table V). 10.6. Dynamic Efects of the Core

A serious objection has been raised to developments based on the assumption of a fluid core. The inertia of the fluid has been neglected in the theories, postulating that rigidities must be high; thus it is incorrect to state then that the rigidity is zero in the core. Jeff reys, who developed this argument, arrived at very significant conclusions. It is known, from Poincard and Hough, that if the core were fluid and the crust completely rigid the Eulerian period would not lengthen but shorten (to about 270 days). Jeffreys has shown that the elasticity of the mantle reduces the movements of the core and that agreement can be restored between the data of pole motion, tides, and seismology by assuming a modulus of rigidity for the mantle of 19 10l1 dynes/cm2. However, a new difficulty arises; the fluidity of the core has an influence on the luni-solar nutation. The theoretical amplitude calculated for the nutation by classical mechanics is 9.2272” while the values given by observations are systematically less, the mean being 9.2109”. The fluidity of the core reduces the theoretical value but in such large proportions that the result this time is too low [as]. Jeffreys and Vicente [19] demonstrated moreover, for the case of a fluid core, that the diurnal tides do not follow the static law. The diurnal oscillations, which may be represented by tesseral harmonic functions, affect the position of the pole of inertia and induce movements in the core. The dynamic theory applied to these waves by Jeff reys and Vicente leads to different values of Love’s numbers, according to the oscillations studied: Semidiurnal oscillations Diurnal oscillations K1

P 0 00

Y

s

0.704 0.714 0.676 0.658 0.695

1.152 1.183 1.209 1.221 1.196

The experimental confirmation of this important discovery will be sought during the International Geophysical Year by observations on the 45th parallel where the diurnal oscillations are a maximum. The variations of level observed in the Turnhout boring [23] appear to show a first confirmation, as previously mentioned. The question remains and, as can be seen, it is far from being resolved

EARTH TIDES

439

satisfactorily. A great number of phenomena are involved and the effect of each is a t the extreme limit of instrumental accuracy. The unknown constitution of the core of the earth is a great obstacle, and recent discoveries by Bullen with regard to it are of the greatest importance. The other part, the problem of the movement of the pole (and particularly the variations of the period), is not resolved in a satisfactory manner.

11. EFFECT OF EARTH TIDESON THE SPEEDOF ROTATION OF EARTH

THE

A new theme has become ripe for discussion. It deals with the influence of the earth tides on the earth’s rotation. As early as 1928, Prof. H. Jeffreys suggested the possibility of this influence and giving it a numerical evaluation [31]. It is essential to state first, th at these variations in speed result exclusively from the largest moment of inertia C, and this constitutes a sufficient approximation. Given th at only deformations of the zonal type are capable of affecting the value of C , we will only consider here long period tides. The process of very simple calculation used by Jeffreys consists of writing the additional potential engendered by the deformation first 2.s a fuiic,tion of the luni-solar potential and of the number k , then as a function of the variations of the moments of inertia, in order to identify the two expressions with the introduction of the condition of incompressibility. I n 1938, Andersson [32] applied Jeffreys’ method in a detailed fashion, introducing all the zonal waves with long periods. Integration shows that the longer the period, the more noticeable the cumulative effect. Since then, the techniques of time services have undergone a veritable revolution (quartz crystal clocks, the photographic zenith tube). The obvious increase of precision demonstrates the variations of the earth’s speed of rotation with considerable reliability. Professors Mintz and Munk [33] have tried to link the semiannual component of the change of speed of the earth’s rotation with the Ssa tide (semiannual tide). They have resumed the theoretical development along a line which seems to us less elegant than that of H. Jeffreys, for it introduces the number h in place of the number k (these authors consider the distribution of the heights of the earth tide a t every point of the globe). This necessitates further the somewhat disguised introduction of the relation k = 0.486h The coefficient in this equation is not yet known with sufficient precision. Thus, Mintz and Munk arrive a t apparently the same numerical

440

PAUL J. MELCHIOR

results as those obtained by the method of Jeffreys, Anderson, and Stoyko [34] but the latter retains the advantage of introducing the number k directly, leaving no room for any uncertainty concerning the value of the relation k/h. Mintz and Munk showed that the liquid core could be considered absent; their calculation came again to adopt lc = 0.30. Under these conditions the amplitudes of the different waves are, in milliseconds:

Mf (fortnight1y)Tl.0 M m (monthly) 1.1 fl (18.6 years) 197.0

Ssa (semiannual) Sa (annual)

5.8 2.3

One might have surmised that this effect would be too weak for actual detection. However, Dr. W. Markowitz demonstrated it in observations of the photographic zenith tube in Washington and Richmond during the years 1951-1954 [35]. He found the following terms, expressed in milliseconds:

Mf:f 1 . 2 sin 2L M m : + 0 . 7 sin U

+ 0 . 4 cos 2L = 1 . 3 sin 2(L + 9") - 0 . 2 cos U

=

0 . 7 sin (U - 16")

where L is the mean*longitude of the moon (13.6 days) and U is the mean anomaly of the moon (27.6 days). For the annual and semiannual terms, Markowitz has obtained amplitudes of 10 msec and 30 msec, respectively. The annual term is almost entirely due to meteorological effects but it is seen that the Ssa tide accounts for 60% of the observed semiannual effect. 12. PROGRAM OF THE INTERNATIONAL GEOPHYSICAL YEAR

Great developments in the observation of gravimetric tides are foreseen during the International Geophysical Year [ 191. The Commission of Gravimetry of the CSAGI (Special Committee for the International Geophysical Year) has considered that the time variation of g responds to the spirit in which the IGY was conceived and has recommended the organization of concerted measurements. These measurements will be made using horizontal pendulums, Michelson tubes, extensometers, and especially, gravimeters having a precision of at least 0.01 mgal and automatic registration. The uninterrupted duration of the measurements will in no case be less than 31 days, permitting the application of Doodson's method of harmonic analysis, which can be considered as the most selective. These measurements will be made simultaneously for groups of stations situated in related tectonic zones in order to be able to discuss the indirect effects on the basis of very detailed information.

EARTH TIDES

441

During this campaign, use will be made of gravimeters of recent construction such as the Askania gravimeter GS 11, which has a sensitivity of 0.01 mgal (Fig. 11), and special gravimeters constructed for the study of earth tides such as that of Ichinohe which has already been mentioned and that of LaCoste-Romberg [21] which has a precision of nearly 0.001 mgal. In addition a number of gravimeters of old types have been modified by specialists and will permit, without doubt, the attainment also of a microgal [21]. Thus, important results may be expected from this world-wide effort. LIST OF SYMBOLS

f universal gravitational constant 2

9 T

U

V m

D W2

t e d h k 1 y = l - k + h

A = l + k - l 6 = 1

+ h - 3k/2 E K K

I L

x,

P P

zenith distance of a n astronomical body (sun or moon) acceleration of gravity radial distance mean earth radius terrestrial potential mass of attracting body (sun or moon) cubical dilatation potential of the exterior forces amplitude of the static tide colatitude latitude first number of Love second number of Love third number of Love factor affecting the amplitude of the crustal tides and the deflections of the vertical with respect to the crust factor affecting the amplitude of the deflections of the vertical with respect to the axis of the earth factor affecting the amplitude of the variations of gravity amplitude of theoretical tide amplitude of observed tide phase of observed tide amplitude of indirect effects phase of indirect effect!s Lam& constants density of the interior of the earth

REFERENCES A practically complete bibliography of the discussion will be found in reference [I] as well as in the general reports on earth tides 121 (IUGG, Association of Geodesy). 1. Melchior, P. J. (1954). Les mardes terrestres. Obs. Roy. Belg. Monographies 4, 134 pp. 2. Lambert, W. D. Rapports g6nCraux sur les mar6es terrestres. Association Internationale de GkodBie, successive assemblies of the IUGG. 3. Love, A. E. H. (1911). “Some problems of Geodynamics.”

442

PAUL J. MELCHIOR

4. Aksentjeva, Z. N. (1948). Sur les marbes du Lac Ba’ikal. T r u d y Poltavskoi Gravirnetritcheskoi Obs. 2, 106-120. 5. Melchior, P. J. (1956). Sur I’effet des mardes terrestres dans les oscillations du niveau du Lac Tanganika A Albertville. Obs. Roy. Belg. Commun. 96, Sdr. Gdophys. 36. 6. Michelson, A. A., and Gale, H. G. (1919). The rigidity of the earth. Astrophys. J . 60, 330-345. 7. Egedal, J., and Fjeldstad, J. E. (1937). Observations of tidal motions of theearth’s crust made at the Geophysical Institute, Bergen. Geojys. Publzk. l l ( 1 4 ) . 8. Zerbe, W. B. (1949). The tide in the David Taylor Model Basin. Trans. Am. Geophys. U n . SO, 357-368. 9. Shida, T. (1912). Horizontal pendulum observations of the change of plumb line a t Kamigamo-Kyoto. Mern. Coll. Sci. and Eng. Kyoto I m p . Univ. 4, 23-174. 10. Boussinesq, J. (1878). Equilibre d’6lasticitb d’un sol isotrope sans pesanteur supportant diffBrents poids. Cornpt. rend. 86, 1261-1263. 11. Nishimura, E. (1950). On earth tides. Trans. Am. Geophys. U n . 31(3), 357-376. 12. Rosenhead, L. (1929). The annual variation of latitude. Monthly Not. Roy. Ast. SOC.Geophys. Suppl. 2, 140-170. 13. Doodson, A. T., and Corkan, R. H. (1934). Load tilt and body tilt at Bidston. Monthly Not. Roy. Ast. SOC. Geophys. Suppl. 9, 203-212. 14. Corkan, R. H. (1953). A determination of the earth tide from tilt observations a t two places. Monthly Not, Roy. Ast. SOC.Geophys. Suppl. 6, 431-441. 15. Melchior, P. J. (1957). Discussion du procbd6 de Corkan pour la sdparation des effets directs e t indirects dans les markes terrestres. Obs. Roy. Belg. Cornrnun. 116, S h . Gdophys. 40. 16. Tomaschek, R. (1954). Variations of the total vector of gravity at Winsford. Monthly Not. Roy. Ast. SOC.Geophys. Suppl. 6, 540-556. 17. Picha, J. (1956). Ergebnisse der Geaeitenbeobachtungen der festen Erdkruste in Brbzovb Hory in den Jahren 1936-1939. Trav. Znst. Gdophys. Acad. Tchdcosl. Sci. 42, Gcofys. Sbornik. 18. Sassa, K., Osawa, I., and Yoshikawa, S. (1952). Observation of tidal strain of the earth. Disaster Prevention Research Znst. Kyoto Univ. Bull. S(l), 1-3; Osawa, I. (1952). Observation of tidal strain of the earth by the extensometer. Disaster Prevention Research Znst. Kgoto Univ. Bull. S(2), 4-17. 18a. Ozawa, I. (1957). Study on elastic strain of the ground in earth tide. Disaster Prevention Research Inst. Kyoto Univ. Bull. 16. 19. Commission pour I’btude des marbes terrestres (1956). Rapports et recommandations, Rdunion de Paris 7 sept. 1956. Obs. Roy. Belg. Cornrnun. 100,Sdr. Gdophys. 36. 20. Lambert, W. D. (1951). Rapport gbnbral sur les marbes terrestres. Presented a t the AssemblCe GEnbrale de I’UGGI B Bruxelles, Association de GdodBsie. 21. Bulletin d’informations sur les mar6es terrestres (1956-1957). P. Melchior, ed. Observatoire Royal Belgique, TJccle. 22. Stoyko, N. (1946). L’attraction luni-solaire e t lea pendules. Bull. Ast. Paris 14, 1-36. 23. Melchior, P. J. (1956). Sur l’effet des marbes terrestres dans les variations de niveau observdes dans lea puits, en particulier au sondage de Turnhout (Belgique). Obs. Roy. Belg. Cornmun. 108, Sdr. Gdophys. 37. 23a. Sperling, K. (1953). Gibt es Gezeiteneinflusse im Erdblforderbetrieb? Erdiil u. Kohle 6(8), 446-449.

EARTH TIDES

443

24. Tomaschek, R. (1952). Harmonic analysis of tidal gravity experiments at Peebles and Kirklington. Monthly Not. Roy. Ast. SOC.Geophys. Suppl. 6, 286-302. 24a. Darwin, C. H. (1907). On variations in the vertical due to elasticity of the earth’s surface. Sci. Papers I, 448. Cambridge Univ. Press, London. 25. Tomaschek, R. (1953). Non-elastic tilt of the earth’s crust due to meteorological pressure distributions. Geojk. Pura A p p l . 26, 17-25. 26. Melchior, P. J. (1955). Sur l’amortissement du mouvement libre du p81e instantan6 de rotation B la surface de la Terre. Atti. accad. naz Lincei cl. sci. fis. mat. nut. Se?. [8]19(34), 137-142. 27. Takeuchi, H. (1950). On the earth tide of the compressible earth of variable density and elasticity. Trans. Am. Geophys. Un.31 (5), 651-689. 28. Jeffreys, H. (1949). Dynamic effects of a liquid core. Monthly Not. Roy. Ast. Soe. 109(6), 670-687; (1950). Ibid. 110(5), 460-466; Jeffreys, TI., and Vicente, R. 0. (1957). The theory of nutation and the variation of latitude. Monthly Not. Roy. Ast. SOC.117(2), 142-173. 29. Melchior, P. J. (1950). Sur l’influence de la loi de r6partition des densit6s B l’inthieur de la terre dans les variations Iuni-solaires de la gravit6 en un point. Geojis. Pura Appl. 16(3-4), 105-112. 30. Jobert, C. (1952). Mardes terrestres d’un globe fluide hCt6rogBne. Ann. gbphys. 8(1), 106-111. 31. Jeffreys, H. (1928). Possible tidal effects on accurate time keeping. Monthly Not. Roy. Ast. SOC.Geophys. Suppl. 2, 56-58. 32. Andersson, F. (1937). Berechnung der Variation der Tagesliinge infolge der Deformation der Erde durch fluterzeugende Kriifte. A l k . Mat. Ast. Fys. 26A, 1-34. 33. Mints, Y., and Munk, W. (1954). The effect of winds and bodily tides on the annual variation in the length of day. Monthly Not. Roy. Ast. SOC.Geophys. Suppl. 6(9), 566-578. 34. Stoyko, N. (1951). La variation de la vitesse de rotation de la Terre. Bull. Ast. 16(3); (1950). Sur l’influence de l’attraction luni-solaire et de la variation du rayon terrestre sur la rotation de la Terre. Compt. rend. 230, 620-622. 35. Markowitz, W. Private communication to the author.


AUTHOR INDEX Numbers in brackets are reference numbers and are included to assist in locating references in which the authors’ names are not mentioned in the text. Numbers in italics indicate the page on which the reference is listed. A Adel, A., 57, 58, 105, 106 Angstrom, A., 82, 107 Aitken, J., 65, 106 Aksentjeva, 2. N., 403[4], 431[4], 442 AlfvBn, H., 116, 122, 159, 170, 174, 176, 190[27], 211, 213 Almond, M., 278[103], 346 Amble, O., 361[12], 388 Anderson, F., 439, 443 1 Anderson, V. G., 81, 85[130], 107 Andrews, E. B., 225[19], 342 Aristotle, 110, 210 Armellini, G., 284, 347 Arnold, P. W., 58, 106 Arons, A. B., 201261, 103 Astapovich, I. S., 1141211, 211, 294[139], 32811651, 348, 349 Aubin, E., 46[60], 104

B Babcock, H. D., 174, 213 Babcock, H. W., 174, 213 Baer, F., 21[29], 103 Baker, G., 250, 344 Baldwin, R. B., 309, 310[147], 348 Barrett, E., 77, 107 Barringer, B., 220, 342 Barringer, D. M., 313[159], 314, 316, 325,

Bennett, W. H., 148, 149, 151, 812 Berry, F. A., 389 Berwerth, F., 335, 350 Biot, E., 33311841, 350 Birkeland, K., 111, 210 Birkhoff, G., 324[170], 349 Blaauw, A., 306, 348 Blackwelder, E., 317[163], 326, 349 Blackwell, D. E., 159, 182[53a], 189[53a], 212

Blanchard, D. C., 20[26], 75, 76[122], 103, 107

Blifford, I. H., 93, 108 Block, L., 177, 213 Bodenstein, M., 571871, 59[87], 105 Bornstein, R., 195[108],215 Boldyreff, A. W., 324, 34.9 Booker, H. B., 191, 214 Boothroyd, S. L., 283, 291, 347 Boussinesq, J., 412, 414[101, 442 Bowen, E. G., 270, 271, 272, 273, 346‘ Bowen, I. G., 51[74], 105 Bowley, H., 219, 34%’ Bracewell, R. N., 354[2, 3, 4, 5, 61, 388 Breuil, H., 333, 350 Brewer, A. W., 51, 52, 56, 105 Brezina, A., 219[4], 237[34],334, 341, S43, 350

Brodin, G., 77[126], 107 Brooks, C. E. P., 370, 371, 389 Brouwer, D., 269, 3.46 Brown, H. S., 241, 343 Buch, K., 46, 48, 104 348,349 Buddhue, J. D., 41[53], 104, 264, 265, Bartels, J., 168[64], 183[64], 190[64], 213 266, 3.46 Bates, D. R., 58, 106, 143, 193, 203, 211, Burkhardt, H., 19, 102 214 Behr, A., 159, 171151, 521, 182151, 521, Burr, E. J., 354[7], 388 Byers, H. R., 28, 103 189[51, 521, 212 445

446

AUTHOR INDEX

C Callendar, G. S., 45[571, 46[57], 47, 104 Canton, J., 110, 210 Capron, J. R., 111[8],210 Carmichael, R. D., 287[125], 347 Carruthers, N., 370, 371, 389 Carslaw, H. S., 352[1], 388 Cartailhac, E., 333, 360 Cauer, H., 18, 21, 66, 68, 71, 102, 106, 107 Chamberlain, J. W., 114[22],148[35], 193, 194, 195\105], 196[104], 198[105], 204, 205, 206, 211, 214, 216 Chambers, G. F., 333[183], 360 Chambers, L. A., 38[50], 69, 96[50], 104 Chant, C. A., 293, 348 Chapman, R. M., 61, 106 Chapman, S., 113, 147, 158, 160, 162, 164, 165, 166, 168, 169, 177, 183, 190, 221, 212, 21s

Cherwell, Lord, see Lindemann, F. A., 216

Cholak, C. E., 38[501, 69[50], 96[50], 104 Clegg, J. A., 282[112], 346 Code, A. D., 306[142], 348 Coghlan, H. H., 335[193], 360 Cohen, E., 338, 360 College, M., 85[137], 108 Corkan, R. H., 414, 415, 416, 442 Coste, J. H., 69, 70, 106 Courtier, G. B., 69, 70, 106 Cowling, T. G., 176, 213 Craig, R. A., 50[69], 106 Crooks, R. N., 72[120], 93[120], lOl[l20], 107

Crozier, W. D., 27[43], 103, 272, 346 Culbertson, J. L., 251[59], 344 Cunningham, A., 334[185], 360

D Daly, J. W., 258[65], 344 Darwin, G. H., 434, 443 DaubrBe, G. A., 332[178], 349 Davidson, I. A., 282[112], 346 Davies, J. G., 278[103], 346 Davis, L., 182, 213 DeBeck, H. O., 258[68], 346 Dedebant, G., 377[17], 389 De Mairan, J. J. D., 110, 210

Desch, C. H., 335, 360 Dessens, H., 5[5], 102 DeViolini, R., 294[140], 348 Dewey, F. P., 328, 329, 349 Dhar, N. R., 68, 106 Dodson, H. W., 115[23], 148[35], 211 Dodwell, G. F., 289[127], 347 Doherty, D. J., 5[2], 9[2], 101 Doodson, A. T., 414, 416, 442 Drischel, H., 82, 84, 85[131], 107 Dubin, M., 190, 205, 21.4 Dunbar, A,, 251[59], 344

E Eaton, J. H., 6511051, 106 Eaton, S. V., 65[105], 106 Edgar, J. L., 60, 96[98], 106 Edlund, E., 111, 210 Effenberger, E., 12, 102 Egedal, J., 408, 442 EgnBr, H., 21[32], 62, 63[102], 64, 66[32], 78[127], 88, 89[141], 90, 91, 92[141], 103, 106, 107, 108 Ehmert, A., 51, 53, 56, 106 Ehmert, H., 53, 106 Ellis, B. A., 69, 70, 107 Elsasser, H., 159, 171[51, 531, 182151, 531, 189[51, 531, 212 Elsasser, W. M., 185[84], 213 Emanuelsson, A., 88, 89, 90[141], 92, 108 Epstein, P. S., 290, 347 Eriksson, E., 48[63], 49, 62[102], 63[102], 64[102], 72, 82, 83[132], 84, 85[132], 87[1321, 88, 89[141], 9011411, 9211411, 104, 106, 107, 108

Esclangon, E., 222, 342 Evans, E. W., 2251191, 342

F Facy, L., 21, 76, 88, 103, 108 Farrington, 0. C., 225[20], 237[34], 242, 245,342, 343 Fath, E. A., 224, 342 Faucher, G. A., 43, 44, 10.4 Fenner, C., 289[1271, 347 Fenton, K. B., 182[75], 189[75], 213 Ferraro, V. C. A., 155, 158, 160, 162, 166, 168, 212, 213

447

AUTHOR INDEX

Finnegan, B. J., 237, 250, 343, 344 Fisher, E. M. R., 72[120], 93[120], 101 [120], 107 Fisher, W. J., 295[141], 348 Fjeldstad, J. E., 408, 442 FjZrtoft, R., 384, 389 Flach, E., 99, 108 Flohn, H., 19, 102 Fonselius, S., 49, 104 Foote, A. E., 313, 348 Foote, W. M., 250[55], 344 Forster, H., 18[23], 102 Foster, J. F., 251, 344 Fournier d’Albe, E. M., 22, 25, 103 Franklin, B., 110, 210

G Gale, H. G., 407, 442 Gartlein, C. W., 112, 211 Gerhard, E. R., 18, 65, 70[24], 102 Gifford, F., Jr., 376, 378[16], 589 Glavion, H., 40[52], 104 Gmelin, L., 62[101], 63[101], 65[106, 1071, 106

Gnevyshev, M. N., 149, 212 Goddard, R. H., 234, 343 Gotz, F. W. P., 49, 53, 54[81], 56, 104,

Harper, H. J., 85[137], 108 Harrison, J. M., 311, 348 Hart, H. C., 220, 342 Harvey, H. W., 76[125], 107 Hawkins, G. S., 273, 346 Haxel, O., 93, 108 Hedeman, E. R., 148[35], 211 Heiland, C. A., 258[66], 344 Henderson, E. P., 225[22], 246, 342, 344 Hettick, I., 87, 108 Hey, M. H., 225[22],235[30],311,3~2,343 Hogberg, L., 82, 107 Hoffleit, D., 264, 3.46 Hoffmeister, C., 269, 271, 3.46 Horan, J. R., 291, S47 Houghton, H., 87, 108 Hoyle, F., 159, 174, 177, 178, 181, 182, 212 Hulburt, E. O., 148, 190, 21.2, 214 Hutchinson, G. E., 45[58], 46[58], 47, 48 [58], 61[58], 67, 84, 104

I Isono, K., 24[381, 29, 103 Israel, H., 5[3], 6[3], 102

J

105

Gold, T., 186, 214 Goldberg, L., 57, 61[99], 67[111], 68, 106 Goldman, S., 389 Goldschmidt, V. M., 241, 343 Goldstein, E., 111, 210 Goody, R. M., 57, 58, 106 Gorham, E., 22, 87, 103 Gottlieb, M. B., 192[98],214 Goubau, F., 270, 345 Griffing, G., 193, 203, 214 Griffith, H. L., 378[18],389 Grogan, Robert M., 336[98], 350 Gustafson, P. E., 8311341, 107

H Eager, D., 308[145], 322, 348 Haggard, W. H., 389 Hall, F., 354[8], 380, 388 Harang, L., 193, 194, 214 Harkins, W. D., 240, 343

Jacobi, W., 15[19],102 Jacobs, M. B., 70[118], 107 Jakosky, J. J., 258[65], 344 Jeffreys, B., 389 Jeffreys, H., 389, 438, 439, 443 Jobert, G., 437, 443 Johnson, D. W., 225[19], S@ Jones, W. B., 225[23], 228, 342 Joseph, H. M., 354[9, 101, 388 Junge, C. E., 5[4], 6[8], 8[4, 81, 9141, 10 [ll],12[4, 8, 151, 15[17, 18, 191, 16 [17], 22, 25, 27[42], 29181, 30[8, 11, 491, 341111, 35[11], 60, 62, 64[111, 66, 69[11], 75[15], 79[8], 82[8], 83[1341, 93[37], 102, 103, 104, 107 Junner, N. R., 334[186], 360

K Kaiser, T. R., 273, 282[961, 346 KalIe, K., 92, 108

448

AUTHOR INDEX

Kasper, J. E., 192, 214 Kate, M., 69, 70, 71, 107 Katzman, J., 182[75],189[75],213 Kay, R. H., 51, 52, 56, 106 Kent, R. H., 290, 347 Khan, M. A. R., 33511941, 360 Kientsler, C. F., 20[26], 105 Kiepenheuer, K. O., 115[25],211 Kingsborough, E. K., 332, 360 Kinnicutt, P., 336, 360 Kohler, H., 15[16], 106 Koroleff, F., 491641, 104 Kovasznay, L. S. G., 354[9, lo], 388 Krinov, E. L., 265, 332, 346, 360 Krogh, M. E., 57, 106 Krumbein, W. C., 40[51], 104 Kulik, L. A., 224[17], 323[165],:342, S49 . i Kumai, M., 76, 107 Kuroiwa, D., 29, 74, 75, 104: Kvasha, L. G., 236, 237, 343

-

1 Lacroix, A., 218, 341 Lambert, W. D., 430, 441, 442 Landolt, H. H., 195[108],216 Landsberg, H. E., 5111, 11, 18[1], 19, 75 [l],80, 101, 107,272[90], 346 Landseer-Jones, R. C., 159, 212, 215 Langmuir, I., 79, 107 LaPaz, L., 222[12], 228[25], 235[29], 245 [44], 246[49], 250[49],251[12, 581, 253 [58], 254[60], 255[63], 257[58], 263 [72], 283[29], 284[119], 286[29, 1251, 288[29, 581, 290[29], 291[29], 310 [152], 311[154],316[63],319[164],323 [165, 1661, 324[167, 1691, 325[1741, s Q J 343,344,345,347,34, 349 Larson, T. E., 87, 108 Lebedinski, A. I., 178, 813 Leeflang, K. W. H., 88, 89, 92, 108 Lemstrom, K. S., 111, 204, 210 Leonard, F. C., 224[16], 236, 237, 238, 239, 294[140],337, 8-42, 34S, 348, 360 Liesegang, W., 84[136], 108 Lindemann, F. A., 112, 113, 148, 160, 210

Lippert, W., 15[19], 102 Lockhart, L. B., Jr., 93[144], 108

Lodge, J. P., 21[29], 22, 26, 103 Lomonosov, M. V., 110, 210 Love, A. E. H., 400, 441 Lovell, A. C. B., 234[28], 278, 282[1041, 343,346 Lowell, P., 293, 347 Lust, R., 188[87, 881, 214 Lynch, D. E., 101[151], 108

M MacCarthy, G. R., 223, 342 McDonald, J. E., 380, 389 McDougal, D. P., 324[170], 349 Machado, E. A. M., 377[17], 589 McKinley, D. W. R., 220, 281, 282, 283 [llsl, '4'J 346, 34r Maclaren, M., 334[186], 360 McMaster, K. N., 21[28], 103 McMath, R. R., 61[99], 106 Mallery, G., 331, 349 Malmfors, K. G., 177, 213 Maria, H. B., 190, $14 Markowitz, Pi.,440, 443 Martyn, D. F., 170, 213 Mason, B. J., 10[13], 11[13], 20, 21, 102, 103, 273, 346 Mason, M., 258[661, 344 Maxwell, J. C., 163, 2 f 3 May, K. R., 5[6], 102 Meen, V. B., 254[61], 309, 344 Meetham, A. R., 95[146], 96[146], 97 [146], 98[146], 100, 108 Meinel, A. B., 112, 114[22], 194, 198, 211, 816

Melcbior, P. J., 393[1], 403, 410[11, 416, 422[23], 434, 435[1, 261, 437, 438 ~31,441,449,443 Mellor, J. W., 335[190],360 Meredith, L. H., 192, 214 Merrill, G. P., 240, 245, 310[149], 315, 317,343, 348 Meunier, S., 218, 324[2], 341 Meyer, P., 182, 215 Michelson, A. A., 407, $42 Middleton, W. E. K., 360, 388 Migeotte, M. V., 59, 61, 67, 68, 106 Miller, A. M., 223[15], 348 Miller, L. E.,-57[89], 106

449

AUTHOR INDEX

Millman, P. M., 220, 266, 281, 282, 309, 342, 345,346,348 Milne, E. A., 115, 148, 211 Milton, J. F., 38[501, 69[50], 96[50], lo4 Minta, Y., 439, 443 Mohler, 0. C., 61, 106 Moissan, H., 313, 348 Moore, D. J., 10[13], 11[13],22, 25, 102, 103 Morgan, W. W., 306, 348 Mueller, E. A., 57, 106 Munk, W., 439, 443 Munta, A., 46[60], lo4

Petukhov, V. A., 142, 811 Picha, J., 411, 416, 442 Pickering, W. H., 293, 348 Plantb, G., 111, 210 Plass, G. N., 49, 10.4 Poincar6, H., 112, 122, 210 Preston, F. W., 225[22], 342 Price, W. C., 64[103], 106 Prior, G. T., 225[22], 235[30], 311, 3.42, 343 Pugh, E. M., 289[128],324[128, 1701,347, 349 Puiseux, P., 311, 348

N

Q

Neumann, H. R., 29[46], 90, 103 Neven, L., 59, 106 Newton, H. A., 332[178], 349 Nichols, H. W., 225[21], 266[21], 3.42 Nielsen, A. V., 285, 347 Nishimura, E., 432, 442 Noddack, I., 241, 343 Noddack, W., 241, 343 Nolan, P. J., 5[2], 9, 101 Nordenskiold, N. A. E., 265, 3.45

0 opik, E. J., 234[28], 248, 266, 273, 275, 279, 280, 281,282, 283, 284,292[133], 293,314,324,343,344,346,346,347, 349 OI’, A. I., 149, 212 Olivier, C. P., 291, 347 Omholt, A., 192, 195, 197, 199, 201, 203, 214,215 O’Neil, R. R., 289, 347 Osawa, I., 418[18], 442 Otake, T., 29, 74, 10.4 Ozawa, I., 420, 442 P Paetaold, H. K., 50[68], 105 Paneth, F. A., 60, 96[98], 106, 246, 257,

344

Panofsky, H. A., 368, 378[18], 388, 389 Parker, E. N., 182[74], 189, 113,214 Penndorf, R., 42, 43[55], 10.4

Quitmann, E., 71, 107

R Ram, A., 68, 106 Randolph, J. R., 225[22], 342 Raphael, M., 33311821, 350 Rau, W., 21, 103 Ray, E. C., 168[65], 213 Regener, E., 50[71], 51, 105 Regener, V. H., 511741, 53, 54, 56, 106 RBmusat, A., 219, S42 Renzetti, N. A., 56[76], 96[76], 981761, 106 Reynolds, W. C., 59, 60, 106 Richard, T. A., 335, 336, 350 Richards, E. H., 84, 108 Rinehart, J. S., 289, 324, 347, 349 Roberts, J. A., 354[6], 388 Rogers, A., 317, 349 Rose, D. C., 182[75], 189[75], 213 Rose, G., 2371341, 343 Rosenhead, L., 413, 442 Rosenstock, H. B., 93[144], 108 Rossby, C. G., 21[32], 66[32], 90, 91, 103 Rostoker, N., 324, 349 Russell, E. J., 84, 108 Russell, H. N., 240, 343

S Sagalyn, R. C., 43, 44, lo4 Sassa, K., 418, 44% Schaefer, V. J., 41, lo4

450

AUTHOR INDEX

Schmidt, A., 168, 21s Schulz, L., 5[3], 6[31, 102 Schumann, G., 93, 108 Schuster, A,, 113, 147, 183, 211, 213 Scott, William, 310, S48 Seaton, M. J., 192, 214, 273[96], 282[961, 346 Seely, B. K., 27, 10s Sheikh, A. G., 334[188], 960 Shida, T., 411, 4 2 Shirl Herr, 258, S46 Shklovski, I. S., 203, 216 Shuman, F. G., 370, 588 Siedentopf, H., 159, 171[51, 521, 182[51, 521, 189[51, 521, 212 Silberrad, C. A., 229[26], 231, 295[261, 349 Simpson, E. S., 219, S42 Simpson, G. C., 24, 10s Simpson, J. A., 182, 189[751, 21s Singer, S. F., 186, 193, 214 Slichter, L. B., 258[66], 344 Slobod, R. L., 57, 106 Slocum, G., 47, 48, 104 Sloss, L. L., 40[511, 104 Smith, J. L., 225[19], S42 Sorby, H. C., 338, 560 Sowerby, J., 335,560 Spencer, L. J., 246, 316, 344, 349 Sperling, K., 423, -4.42 Spilhaus, A. F., 360, 588 Spitzer, L., 116, 125, 126, 148, 151, 211 Stair, R., 501701, 106 Stearn, N. H., 258[66, 681, S44, 346 Stenz, E., 246, S44 Stewart, B., 111, 210 Stewart, N. G., 72[120], 93, 101[1201, 107 Stewart, R. H., 2541611, 3091611, 344 Stomgren, E., 284, 347 Stormer, C., 128, 133, 134, 135, 137, 140, 145, 211 Stoney, G. J., 267, 346 Storey, L. R. O., 159, 1711541, 182[541, 189[541, 212 Stoyko, N., 431, 440, 442, 4-63 Suess, H. E., 242, 343 Swanson, C. O., 258[681, 346 Swindel, G. W., 2251231, 228, S42 Swings, P., 112, 210

T Takeuchi, H., 436, 44s Taylor, G., 32411701, 349 Teichert, F., 54, 55, 56, 106 Theodorsen, T., 261, S@ Thomas, L. H., 148, 211 Thomson, E., 204, 216 Tomascheck, R., 411, 433, 434, 442, 443 Tonks, L., 149, 212 Turner, J. S., 75[123], 79, 107 Twomey, S., 16[20], 21, 27, 102, 103

U Udden, J. A., 220, 342 Urey, H. C., 241, 242, 309,338, 343, 348

v Van Allen, J. A., 192, 214 Van de Hulst, H. C., 234[281, 343 Van der Hoven, I., 378[18], 389 Van Orstrand, C. E., 328, 329, S49 Vegard, L., 112, 143, 147, 150, 193, 210, 211,214 Vestine, E. H., 183, 184, 185, 215, 270, 546 V'iunov, B. F., 190, $14 Volz, F., 10, 52, 53, 54[811, 56, 102, 106 von Fellenberg, T., 66, 106 von Heine-Geldern, R., 289[128], 324 [1W, 34'7 von Lasaulx, A., 265, 346 von Niessl, G., 276, 546

W Waerme, X., 49[64], 104 Wainwright, G. A., 335, 360 Walshaw, C. D., 57, 58, 106 Washington, H. S., 245, 544 Wasiutynski, J., 310, 548 Watson, F. G., 225[21], 262[71], 266, 283, 338, S/,8, 346, 560 Weaver, J. H., 287[125], S47 Wegener, A., 220, 342 Welander, P., 49, lo4 Wempe, J., 9191, 10% Wheeler, L. B., 27[43], 103

451

AUTHOR INDEX Whipple, F. L., 194[106], 214, 266, 273, 275, 276, 282, 285[120], 286, 29111241, $461 346,347 Whitford, A. E., 306[142], 348 Wiener, N., 389 Wiens, G., 323[1651, 949 Wilson, C. H., 2581651, 344 Wisman, F. O., 262[70], 346 Witherspoon, A. E., 58, 106 Wolff, P. M., S89 Woodcock, A. H., 6/71, 10, 12[12], 20, 22, 23, 25, 28, 75, 76[122], 102, 103, 107 Woodward, R. S., 267, 346 Wright, H. L., 24, 103 Wiirm, K., 112, 210 Wulf, 0. R., 184, dl9

Wylie, C . C., 285, 286, 287, 293, 349[1681, 9d7> 'd9

Y Yamamoto, G., 29, 74, 104 Yoshikawa, S., 418[18], 442

Z Zaslavskii, I. I., 245[47], 344 ZavaritskiI, A. N., 236, 237, 343 Zenncck, J., 270, 345 Zenzen, N., 218[1], 941 Zerbe, W. B., 408, 416, 442 Zimmer, G. F., 335[189], 336[189], 360 Zimmerman, W., 312, 348

SUBJECT INDEX A Achondrites, see Meteorites, achondritic Aerosols, 3 Aitken, 4, 11, 17 attaching of, to cloud droplets, 75 composition of, 19 nature of, 17 origin of, 17 sources of, 18 as condensation nuclei, 73 continental, 29 vertical distribution of, 41 definition of, 3 giant, 4, 6, 12 growth of, with relative humidity, 15, 16 curves for, 16 large, 4 natural, 3 definition of, 3 physical constitution of, 14 size distribution of, 4, 5 curves for, 8, 10, 13 limits of, 12 methods of determination of, 5 volume distribution of, 5, 10, 13 removal of, from atmosphere, 72 by fallout, 72 by impaction, 72 by washout, 72 sea-salt, 20 concentration of, 24-29 distribution of, 90 formation of, 20 role of, with respect to visibility, 24 size distribution of, 22 curves for, 23, 26 source of, 20 vertical mixing over land of, 91 washout through precipitation of, 90 Air pollution, 94, also see Polluted air Aitken particle, see Aerosols, Aitken 452

Ammonia, formation and role in soil, 63, also see under Trace gases Aurora(e), 109-215 electrical nature of, 111 excitation of hydrogen in, 196 main features of, 115 primary electrons in, 192 theory(ies) of, 109 Alfvbn’s, 174 electronic orbits in, 176 Bennett and H u Iburt’s .?elf-focused stream, 148 Chapman and Ferraro’s, 158 cylindrical sheet problem, 166 neutral ionized stream, 158 discharge, 204 Hoyle’s, 177 Lebedinski’s, 178 Lemstrom’s, 111 Maris and Hulburt’s ultraviolet light, 190 Martyn’s, 170 Parker’s, 189 Singer’s shock wave, 186 Stormer’s, 112, 113, 127-146 criticisms and modifications of, 147 equations of motion for meridian plane, 130 forbidden regions in the threedimensional problem, 134 V’iunov and Dubin’s meteor, 190 Wulf and Vestine’s dynamo, 183 typical display of, 116 Auroral arcs, 193 velocity dispersion of incident protons in, 202 Auroral excitation, 191, 194 luminosity curves, 194, 197 role of electrons in, 191 theories of, 191 Auroral forms, 143, 146 homogeneous arcs, 146

453

SUBJECT INDEX

rayed arcs, 146 single rays, 146 Auroral luminosity, 193 incident proton theory of arcs, 193 Vegard’s & Harang’s incident electron theory, 193 Zenith profile, 198 Auroral reflection, 191 Booker’s theory of, 191 Auroral zones, 115, 141

C Carbon dioxide, see Trace gases, atmospheric Carbon dioxide cycle, 46 Carbon monoxide, see Polluted air, main components of Also see Trace gases, atmospheric Charged particle(s), 116 motion of, 116 in dipole field, 128 in inhomogeneous magnetic field, 119 in monopole field, 122 in three dimensions, 136 in uniform electric and magnetic fields, 118 in uniform magnetic field, 116, 117 Stormer’s trajectories in equatorial plane, 131 motion of cylindrical stream of, 161 motion of plane slab of, 160 Chondrites, see Meteorites, chondritic Cloud droplets, 74, 75 size of, 75 Condensation nuclei, 73 size of, 75 distribution of, in clouds and fog, 74 Crustal deformation, effect(s) of, 401 on amplitude of tides, 401 long period, 402 short period in lakes, 402 on deflection of the vertical, 403 on intensity of gravity, variations in, 426

E Earth, elastic constants, see Elastic constants of the earth

moment of inertia, increase of, due to infall of meteoritic material, 267 Earth tides, 391 effect@) of, 417, 418 elastic measurement of, 418 on high precision leveling, 417 on speed of earth’s rotation, 439 on water level in wells, 421 indirect effect of oceanic tides on, 411 empirical method of separation of, 414 numerical evaluation of, 411 measurement of, from large water levels, 407 numerical results of observations of, 409 static theory of, 394 Elastic constants of the Earth, 435 derivation from earth tides, 435 Herglota’ theory, 435 Electric currents between the sun and the earth, 146

F Filtering functions, 354 band pass, 365 elementary, 369 high pass, 365 low pass, 367 Filters, 355 mathematical, see Mathematical filters pre-emphasis, 376 specified frequency response, 363 design of, 363 Formaldehyde, see Trace gases, atmospheric Frequency response(s), 355 negative, physical meaning of, 358 undesirable, suppression of, 358

G Geologic structure of the crust, role in indirect effects, 432 Gravity, variations in, 428 effect of crust.al deformation on, 426 measurements of, 427-432 bifilar gravity meters, 427 clock pendulums, 431

454

SUBJECT INDEX

recovered, 235 mineralogy of, 236, 238, 239 recovery index, 250, 251 siderites, 237 H siderolites, 237 auperficially buried, search for, 253 Halogens, see Trace gases, atmospheric worship of, 334 Herglotz’ equation, 436 Meteoritic abundances, 240, 243, 244 Takeuchi’s development of, 437 Meteoritic accretion, terrestrial, 240 Herglotz theory, 435 Meteoritic artifacts, 335, 336 Horizontal pendulum, 404 Meteoritic crater(s), 307 Hydrogen sulfide, see Trace gases, atmosage of, 323 pheric Barringer, 310, 311 Hyperbolic velocity problem, see MeteBrenham, 326 oritic velocities, hyperbolic diffusion of NiO in soil in, 326 Chubb, 308,311 I criteria for recognition of, 308 Odessa, 310, 319 Indirect effect, see Earth tides, indirect Podkamennaya Tunguska, 323 effect of ocean tides on Meteoritic dust, 264, 269 Intensity of gravity, see Gravity, variainfall of, 269 tions of light streaks due to, 269 Inverse smoothing functions, see Bmoothnoctilucent clouds due to, 269 ing functions, inverse rate of accretion of, 264 Meteoritic falls, 218 1 effects of, 218 ballistic headwave, 222, 223, 224 Level surfaces, deformation of, by lunidamage and injury, 225 solar effects, 396 ionization, 220 Lorents force, 116 seismic, 222 Love’s numbers, 400, 437 sound, 220 frequency of, as function of right ascenM sion of moon, 294 Meteoritic hits, 228 Magnetic storms, 164, 186, 189 probability of, 228 initial phase of, 164 on built-over target area, 230 Parker’s theory of, 189 on human targets, 228 sudden commencement of, 186 on rockets, 234 Meteoritic iron in the earth, vertical disMathematical filter, 356 frequency response of, 356 tribution, 255 Meteorite(s), 217 Meteoritic material, detection and recovachondritic, 237, 338 ery of, 252 aerolites, 237, 245 Meteoritic orbits, 273-278 ballistic potential of, 273 elliptic, 278 chondritic, 337 hyperbolic, 273, 274 classification criteria for, 336 parabolic, 281 contraterrene, 323 Stromgren type problems, 284 deeply buried, detectors for, 258 Meteoritic petroglyphs, 330, 332 effects of fall of, 219 Meteoritic pictographs, 329 mass of, crater-producing, 323 Meteoritic showers, 246, 247 prospecting gravity meters, 427 results of observations, 429

455

SUBJECT INDEX

relation with rainfall peaks, 272 strewn fields of, 246 Meteoritic velocities, 273 elliptic, 277, 278 hyperbolic, 273, 274, 282 B-processes, 306 comet question, 283 non-visual determination of, 286 black smoke method, 290 coma method, 289 inverse acceleration method, 286 radar detection of, 278 limitations of, 278, 281 Meteoroids, 218 Methane, see Trace gases, atmospheric Michelson interferometer on tube for tidal observations, 407

N Nitric oxide, see Trace gases, atmospheric Nitrogen dioxide, see Trace gases, atmospheric Nitrous oxide, see Trace gases, atmospheric

0 Ozone, see Trace gases, atmospheric

P Photographic zenith tube, 425 Pinch effect, hydromagnetic, 151 Pollutants, 95 concentration of, 96, 97 degree of, trends with time, 97 industrial origin, 98 Polluted air, 94 main components of, 94 carbon monoxide, 96 sulfur dioxide, 96 Pollution centers, area of influence around, 98

R Rain d r o m 79. 80 pH v a k e of individual, 80

size of, 80 correlation of, with chloride concentration, 75, 79 Rain water, 81, 88 continental components of, 81 ammonia and nitrogen trioxide, 82 change with latitude of, 82 long time variation of, 83 calcium, 86, 87 sulfur tetroxide, 85 change with latitude of, 85 long term variation of, 85 source of, 86 maritime components in, 88 chlorine, 88 concentration of, 89, 90 magnesium, 88 potassium, 88 sodium, 88 p H values of, 77 Ring current, theories of, 142, 168, 172 Chapman and Ferraro’s, 168 Martyn’s, 172 Stormer’s, 142

S Sassa extensometer, 418 Smog, cause of, 95 Smoke, composition of, 95 Smoothing, 351 binomial, 371 compensation for, 375 by resistance-capacitance electrical filters, 361 by viscous damping of measuring instrument, 361 exponential, 361 phase error due to, 361 Smoothing functions, 354 design of, 363 elementary, 369 frequency response of, 355 inverse, 372 design of, 372 frequency response of, 373 normal curve, 359, 361 Solar streams, 180 nature of, 181 Space filtering, 380

456

SUBJECT INDEX

Space smoothing, 380 Stormer’s theory of aurora(e), see Aurora(e), theories of Stormer’s trajections, 132, 133 forbidden regions, 134 Stromgren-type problem, 284 Sulphur dioxide, see Polluted air, main components of Also see Trace gases, atmospheric

T Tektites, 218 Theodorsen bomb detector, 261 Tides of the earth, see Earth tides Tides of the geoid, 399 Time series, 351 derivatives of, as means of filtering, 378 equalization of, 353 filtering of, 351 inverse smoothing of, 353 pre-emphasis of, 353 smoothing of, 351 Trace gases, atmospheric, 44-71 absorption of, in cloud droplets, 76 ammonia, 59, 61 soil as a source of, 84 carbon dioxide, 45 role of, in radiation balance of earth’s atmosphere, 49 secular changes in concentration of, 47 carbon monoxide, 67

concentration, decrease during precipitation, 81 formaldehyde, 67 hydrogen sulfides, 65 methane, 67 nitric oxide, 59 nitrogen dioxide, 59 nitrous oxide, 57 origin of, 58 ozone, 49 decomposition of, 52 distribution, 50 latitudinal, 50 stratospheric, 50 tropospheric, 51, 54 removal of, from atmosphere, 72 by escape into space, 73 by precipitation, 73 sulfur dioxide, 64 units for concentration of, 44

V Variation in gravity, see Gravity, variation in

W Wash out processes for cleansing of troposphere, 94 Wasiutynski’s theory of lunar craters, 310

Z Zenith telescope, 424

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