Lunar gravimetry

Lunar Gravimetry

This is Volume 35 in INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by WILLIAM L. D O N N A complete list of the books in this series appears at the end of this volume.

Lunar G ravi metry

M. U. SAGITOV

B. BODRl

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow, USSR

Department of Geophysics Eotvos University Budapest, Hungary

V. S. NAZARENKO

Kh. G. TADZHIDINOV

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow. USSR

Department of Gravimetry Sternberg State Astronomical Institute University of Moscow Moscow, USSR

1986

ACADEMIC PRESS

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Copyright @ 1986 by ACADEMIC PRESS INC. (LONDON) LTD 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

B r i t i s h Library Cataloguing in Publication D a t a Lunar gravimetry. 1. Moon-Gravity I. Sagitov, M. U. 523.3’1 QB581 ISBN 0-12-61 4660-8

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Preface

Gravimetry is the science dealing with the gravitational fields of the Earth and other planets and permitting studies, based on this knowledge, into their figures and internal structure. Investigating the structural characteristics and the features of variations of the planetary gravity fields, gravimetry is also instrumental in computing the trajectories of rockets and artificial satellites, as well as in inertial navigation. The branch of gravimetry specifically concerned with the Moon is known as lunar gravimetry. Like other natural sciences, gravimetry is based on measurement, and its development is closely interrelated with the enhancement of measurement accuracy. Under terrestrial conditions, gravity is now measured with a maximum accuracy of about lop6cm s - ~ ,while its gradients are measured to within lo-" s - ~ .The advent of artificial satellites has been marked by the development of new methods for studying the gravitational fields of the Earth and planets. The anomalous part of the gravitational field was first defined from observations of perturbations in the motion of satellites. The gravitational field of the Moon has become better understood with the introduction of methods designed solely for investigation of the lunar field. For example, the line-of-sight components of the force of gravitational attraction, which are often used as basic quantities for characterizing the gravity field of the Moon, are obtained through Doppler tracking of artificial lunar satellites (ALS). Some of the lower harmonics of the gravitational field have been derived from observations of the Moon's physical librations. Our knowledge of the gravitational field on the Moon's far side is less certain. Only a few single gravity measurements have been carried out directly on the Moon's surface. Moreover, the lunar gravitational field differs from the terrestrial one not only in that the gravity on the Moon is six

vi

Preface

times lower, in absolute terms, as that on the Earth, but also in the spectrum of gravity anomalies. Some series expansions achieved for the Earth’s gravitational field are not applicable to the Moon because of the lack of clearly defined flattening of the latter and the corresponding predominant gravitational field harmonic. Yet, in spite of all the differences, it should be emphasized that the physical and mathematical principles underlying the theory and methods of terrestrial gravimetry remain valid also in lunar gravimetry. Gravimetric studies of the Moon expand our knowledge of the terrestrial gravitational field. Results have been obtained on the Moon with the aid of gravimetric methods, which have later been found to be applicable to the Earth. The active gravimetric studies of the Moon carried out over the last 15 years have yielded a wealth of data. The need has arisen to summarize and generalize this material to some extent, and this is what we have attempted to do. Not all chapters are equal in scope, mainly because some problems of lunar gravimetry are better investigated than others, but also because we have chosen to concentrate on some aspects more than others. We have included many tables, some of which are summaries compiled for the book from other sources, which may help the reader to better comprehend the subject. In some instances, we have derived mean values by averaging a great number of previously found ones. Following the old Chinese saying that “one picture replaces ten thousand words”, we have included numerous illustrations. As part of his contribution to the book, M. U. Sagitov has drawn on his lunar gravimetry lectures delivered over a number of years at the astronomical division of the physics department of Moscow University. The present monograph differs from that published by M.U.S. in 1979 under the same title in that it includes a new chapter dealing with time variations of gravity and their use in studies into the Moon’s structure. Many findings concerning the Moon’s gravitational field have been revised and updated. Some of the previously published text has been excluded (theory of sample functions, orbital motion of the Moon, Moon’s origin and evolution). The first three chapters have been written by M. U. Sagitov, V. S.Nazarenko and Kh. G. Tadzhidinov, the author of the fourth chapter is B. Bodri. We are grateful to Dr Y. Nakamura (University of Texas, Galveston) for providing numerical listings of the parameters of the four radially heterogeneous lunar models investigated in Chapter 4. Moscow and Budapest June 1986

M. U. Sagitov B. Bodri V. S. Nazarenko Kh. G. Tadzhidinov

Contents

Preface

V

List of Symbols

ix

Chapter One The Gravitational Field of the Moon: Methods for Its Determination The Law of Universal Gravitation and Various Gravitational Constants Problems Solvable from the Gravitational Field Lunar Gravitational and TidalLCentrifugal Potentials: Gravity Quantities Characterizing the Lunar Gravitational Field Comparison of Methods Used for Studying the Gravitational Fields of the Moon and the Earth 1.6 Elements of ALS Orbits and the Concept of Perturbations in Motion as a Result of Irregularities of the Lunar Gravitational Field 1.7 Methods for Determining the Harmonic Coefficients of the Gravitational Field from ALS Tracking Data 1.8 Determination of Line-of-Sight Accelerations due to the Earth’s Rotation and Orbital Motions of the Moon and ALS 1.9 Determination of the Gravitational Field from Variations in the Line-of-Sight Velocity of the Circumlunar Satellite 1.10 Gravity Measurement Concepts and Requirements of Lunar Gravimeters 1.11 Direct Gravity Measurements on the Moon’s Surface 1.12 Studying Second Derivatives of the Lunar Gravity Potential 1.13 Generalized Model of the Lunar Gravitational Field

1.1 1.2 1.3 1.4 1.5

Chapter Two 2.1 2.2

1

5 7 14 17 19

25 31 41 49 54 59 73

Normal and Anomalous Gravitational Fields of the Moon

Structure of the Gravitational Field and Its Role in the Evolution of the Moon Expansion of Lunar Gravity Potential Derivatives in Spherical Functions

99 101

viii

Conrents

Expansion for Gravity Selenoid Figure of the Normal Moon Distribution of Normal Gravity Surfaces of Equal Gravity and Equal Radial Gravity Gradient Anomalies of the Lunar Gravitational Field Relation between the Coefficients of Expansion of Different Parameters of the Moon’s Gravitational Field and Figure 2.10 Moon’s Relief and Gravitational Field

2.3 2.4 2.5 2.6 2.7 2.8 2.9

Chapter Three

4.1 4.2 4.3 4.4 4.5 4.6

131 134

Spatial Variations in the Lunar Gravitational Field and Their Use in Studying the Figure and Internal Structure of the Moon

3.1 Covariance Analysis of the Moon’s Gravitational Field 3.2 Degree Variances and Covariance Functions of Various Characteristics of the Gravitational Field and Figure of the Moon 3.3 Degree Variances for Horizontal Lunar Attraction Components 3.4 Comparative Analysis of the Lunar and Terrestrial Gravitational Fields 3.5 Comparison of the Gravitational Fields of Terrestrial Planets 3.6 The Selenocentric Constant of Gravitation: The Mass and Mean Density of the Moon 3.7 Centres of the Moon’s Figure and Mass 3.8 General Geometric Figure of the Moon 3.9 Dynamic Figure of the Moon 3.10 Hydrostatic Equilibrium Figure of the Moon 3.1 1 Gravimetric Figure and Distribution of Plumb-Line Deflections on the Moon 3.12 General Comments on the Moon’s Internal Structure 3.13 Density Irregularity Parameter 3.14 Density Model of the Moon 3.15 Variations in Gravity, Its Radial Gradient, and Pressure with Depth 3.16 Mascons

Chapter Four

105 108 111 118 122 125

147 153 156 158 171 173 180 186 190 200 206 201 211 218 224 228

Inconstant Lunar Gravity

Periodic and Secular Variations of Lunar Gravity Tidal Potential on the Moon Brief Theory of Solid Tides Tidal Deformation of the Moon Tides of the Anelastic Moon Tidal Friction and Secular Variations of Lunar Gravity

239 242 25 1 259 272 276

References

28 1

index

289

List of Symbols

G a

r R

Newton’s constant of gravitation planetary radius radius vector distance between the Moon’s centre and that of the tide raising body planetary mass symbols referring respectively to the Moon, Earth and Sun zenithal distance and its mean value, respectively order of spherical harmonic function Legendre polynomials surface spherical harmonic function selenographic coordinates colatitude = 90” - cp selenocentric equatorial coordinates Moon’s mean longitude mean longitude of lunar perigee mean longitude of the node of the Moon’s orbit Sun’s mean longitude mean longitude of solar perigee gravity deviations of the vertical density undisturbed density time shear modulus, Lam6 constant bulk modulus

List of Symbols

X

displacement vector components of the stress tensor components of the strain tensor Kronecker delta hydrostatic pressure tidal potential hydrostatic gravitational potential gravitational potential due to tidal distortion gravitational potential of the deformed planet = Vo Love numbers frequency of the perturbing force imaginary unit, (- 1)1/2 phase lag of planetary tides quality factor

+ Vt

Chapter One

The Gravitational Field of the Moon: Methods for Its Determination

1.1 The Law of Universal Gravitation and Various Gravitational Constants

The basic postulate of gravimetry is the law of universal gravitation, stating that two particles of matter attract each other with a force F having a magnitude proportional to the product of their masses m and . m land inversely proportional to the square of the distance r between them: F=G-

mm 1 r2

(1.1.1)

where G is the constant of gravitation, or gravitational constant. In mechanics, force is defined from Newton's second law as the product of the mass m l by acceleration a: F

= mla

(1.1.2)

Force has the following dimensions: [F] = [M][L][7'--2, M being mass, L, length and T,time. For the force F in (1.1.1) to have the same dimensions as in (1.1.2), the dimensions of G must be

G

= [M]-1[L]3[T]-2

2

Lunar Gravimetry

The mass rnl in (1.1.1) and (1.1.2) exhibits different properties. In the former formula, it possesses the property of gravitational interaction, while in the latter, it is a measure of inertia. It is assumed that the gravitational mass (in the law of gravitation) is identical with the inertial mass (in Newton’s second law). This is consistent with the equivalence principle experimentally verified by L. Eotvos, J. Renner, V. Dicke, and V. B. Braginsky. As was shown by Braginsky, this principle holds to within 10- 1 2 . The question as to further enhancement of this accuracy remains open. It would be appropriate at this juncture to trace the history of the discovery of the law of gravitation (Sagitov, 1969), which is intimately linked with that of exploration of the Moon. The law of universal gravitation, as it is still used today, was first formulated by Isaac Newton (1643-1727) in his fundamental work “Philosophiae Naturalis Principia Mathematica” (Mathematical Principles of Natural Philosophy) published in 1687. Other scientists before Newton also tried to establish the rules governing the attraction between bodies, but their concepts were purely speculative and incomplete and definitely cannot be regarded as laws. Proceeding from the hypotheses of his precursors, Newton had arrived at a precise statement of the law, followed by its analytical description. Newton claimed priority, firstly for having derived an analytical expression representing changes in gravity, and secondly, for having proved the identity of the gravity exerted by the Earth with the attraction of the planets, just as that of all other bodies, to one another. According to his own statements in his letters to Edmond Halley (1 656-1 742), Newton established that gravitation varies with distance as far back as 1666. The lack of reliable data as regards the Earth’s radius and the distance to the Moon, which were necessary along with knowledge of the Earth’s gravity and the Moon’s period of revolution around the Earth to check numerically the law of universal gravitation, was the main reason why publication of Newton’s law was delayed. After Jean Picard (162s1682) had computed the radius of the Earth from his measurements of degree, Newton recalculated the Moon’s orbit and achieved an excellent fit of the precomputed positions of the Moon to observation results. This had provided practical proof of the new law. Newton then used it to explain the motion of planets, their satellites, comets, as well as tidal motions in the sea. The constant G of gravitation of (1.1.1) is used in different forms depending on the application of the law of gravitation. Since different systems of units are employed to measure mass, length, and time, the constant of gravitation may have different values. Each of these constants is known under a different name: Cavendish’s, Gaussian, Einstein, geocentric, selenocentric, heliocentric, and so on. Table 1.1 lists some gravitational constants, their numerical values, dimensions, and units of mass, length, and time measurement. All constants are defined in terms of Newton’s law of gravitation. In so far as

3

The Gravitational Field of the Moon

TABLE 1.1 Different constants of gravitation, their values and units of measurements

Constant Cavendish’s constant of gravitation Einstein constant of gravitation Gaussian constant of gravitation Geocentric constant of gravitation

Numerical value

Units of measurement Length Mass Time

(6.6742 f 0.0008) m x lo-” m3 kg-’ s - ~ (1.865 & 0.001) m x m kg-’ 0.017,202,098,95 a.u.

(398,603 1) 109 r n 3 s - 2 Selenocentric constant of gravitation (4902.7 f 0.1) x i09m3s-*

m

m

mass of the Sun mass of the Earth mass of the Moon

day S

S

certain astronomical problems are concerned, it is convenient to use, as the basic equation, the formula of the third law stated by Johannes Kepler (15711630), which, in the final analysis, is also derived from Newton’s law of gravitation. Kepler’s third law is used, in particular, to derive the selenocentric gravitational constant G M , from the motion of a satellite of the Moon. If this motion is due only to the gravity field of the Moon, which is assumed to be uniform and spherical, then the following relation exists between the period T of the satellite’s revolution around the Moon, the semimajor axis a . of the satellite’s orbit, and GM,: (1.1.3) This constant has significantly gained in importance after the launching of space vehicles toward and onto the Moon. It is used to calculate the trajectories of artificial lunar satellites (ALS) and spacecraft sent to other planets of the solar system, in studying the overall density distribution inside the Moon, in determining its dynamic figure, and for other purposes. Determined in a similar manner is the geocentric gravitational constant G M e which plays an important role in studying the motion of space vehicles and satellites in the gravitational field of the Earth. The Gaussian constant K of gravitation was initially derived assuming that the Sun’s mass is Ma = 1 and the semimajor axis of the Earth’s orbit around the Sun is Aa = l,,while the period T and relation ( M @+ Mu)/Ma were determined from observation. However, since the latter two quantities have been constantly refined during observations, the currently accepted value of K is that given in Table 1.1, whereas variations in T and ( M e + M t ) / M o are compensated by slight

4

Lunar Gravimetry

changes in A. The unit of distance becomes a derivative quantity corresponding to a definite, predetermined value of the Gaussian constant K of gravitation and is referred to as astronomical unit (AU). The tabulated numerical value of K was for the first time obtained by Karl Gauss (17771855) in 1809. However, Newton had derived it within a lower order of magnitude (K = 0.017,202,12) as early as 120 years before. The term “Gaussian constant” has taken root out of respect to Gauss for his having introduced the law of gravitation into celestial mechanics, rather than for priority in deriving this constant. The Einstein constant of gravitation enjoys currency in theoretical physics. Paradoxical as it may seem, the most “Earthbound” constant of gravitation derived on the assumption that mass, length, and time are expressed in the universally adopted metric system is known only in the roughest approximation. It is known as Cavendish’s constant G of gravitation. It is determined experimentally,by measuring the force of mutual attraction of test bodies spaced an exactly known distance apart (the masses of these bodies are also known exactly) and calculating it with due account for the shape of these bodies and the distance between them. The first experiment in which the contant G of gravitation was determined was carried out by Henry Cavendish (1731-1810). The significance of Cavendish’s experiment is not restricted to numerical definition of this constant. More important by far was experimental corroboration of applicability of Newton’s law of gravitation, not only to celestial bodies, but also small terrestrial ones. The importance of this finding cannot be overestimated in view of the assumption made after the discovery of the law of universal gravitation to the effect that the force of mutual attraction reaches sizeable proportions only between celestial bodies. As regards terrestrial bodies, the consensus at that time was that their small size made it impossible to observe the attraction between them. This conclusion had been drawn after Newton’s miscalculation. Table 1.2 summarizes the results of recent measurements of Cavendish’s constant G of gravitation. In all experiments, a torsion balance TABLE 1.2 Determinations of Cavendish’s constant of gravitation

Workers, country, year of publication Hey1 and Chrzanowski (USA), 1942 Rouse, Parker, Beams et al. (USA), 1969 Renner (Hungary), 1970 Facie, Pontikis and Lucas (France), 1972 Sagitov, Milyukov, Monakhov et al. (USSR), 1978 Laser and Towler (USA), 1982

Value of constant G ( l o - ” m3kg-ls-’) 6.673 f 0.005 6.674 f 0.004 6.670 f 0.008 6.6714 f 0.0006 6.6745 & 0.0008 6.6726 f 0.0005

The Gravitational Field of the Moon

5

was used. In experiments based on the dynamic method, the force attracting test bodies to each other was determined from the torsional oscillation period, whereas in rotational experiments the torsional system was made to rotate about the axis of the torsion wire for more accurate measurements of attraction. Earlier definitions of the constant of gravitation are reviewed in the monograph by Stegena and Sagitov (1979). Using Cavendish’s (G) and the selenocentric ( G M u ) constants of gravitation, one can determine the mass M u and mean density O, of the Moon. The determination of G M , , M , , and O, will be treated at greater length in Chapter 3.

1.2 Problems Solvable from the Gravitational Field

The complexity of the lunar gravitational field stems from the uneven density distribution within the Moon and non-uniformity of its figure. The gravitational field merely provides intermediate data on the Moon’s internal structure and figure. The final picture is much more complex and diverse. As far as selenophysical problems involving the internal structure and solved with the aid of the gravitational field are concerned, they can be listed in the following order of decreasing importance: (1) determination of the radially oriented planetary changes in the density of the lunar rocks; (2) investigation of density changes in a tangential direction; (3) determination of the Moon’s mass and mean density; (4) finding the difference in internal structure between the near and far sides of the Moon; ( 5 ) studying the departure of the Moon from hydrostatic equilibrium as well as estimation of the stresses inside the Moon; (6) estimation of the isostatic state of large lunar regions and craters; (7) definition of the boundary between the mantle and crust in the highlands and maria; (8) determination of the crustal structure in zones of transition from highlands to maria; and (9) examination of small-scale features of crater boundaries, rille regions, faults, and so on. Associated with the selenodetic aspects of gravitational field applications are such problems as: (a) determination of the selenocentric constant of gravitation; (b) more accurate location of the lunar centre of mass with respect to the geometric centre of the Moon; (c) determination of the distances between individual points on the physical surface and the lunar centre of mass; (d) construction, from satellite data, of the gravitational field model optimal from the standpoint of elimination of the systematic errors which are as high as 100 to 200 mGal in various models; (e) determination of plumb-line deflections on the Moon; (f) more accurate definition of the Moon’s figure and physical libration parameters; and (g) gravity distribution at different points on the physical surface or standard Moon.

6

Lunar Gravimetrv

We have already mentioned studies into spatial changes in the gravitational field. However, as is known, the gravitational field at every point on the Moon changes in the course of time. Secular changes occur on the Moon just as surely as on the Earth, although they seem to be extremely small. Even on the Earth there are no reliable ways of measuring them. As regards periodic gravity variations on the Moon, they are greater in magnitude than on the Earth, but since their period is longer, they are more difficult to measure. Periodic variations in gravity are used to determine the elastic properties of the lunar rocks. Moreover, knowledge of the lunar gravitational field is instrumental in calculating the trajectories of space vehicles in the neighbourhood of the Moon. So far, only problems solvable with the aid of the gravitational field have been mentioned. Many of these problems will be discussed in more detail in the following chapters. Even the above listing attests to their great diversity. Obviously, solution of different problems calls for different levels of knowledge about the gravitational field, different sets of its parameters (gravitational potential, gravity, horizontal gravity components, gradients of gravity, etc.), and different degrees of accuracy. Some of the above problems are solved using gravity data supplied by satellite observations. To solve all problems necessitates determination of gravity and its gradients directly on the lunar surface. To this end, gravimetric instruments mounted on modules performing “soft” landing on the Moon’s surface can be used. Of particular importance for gravimetric survey of the Moon are all kinds of lunar roving vehicles. The gravimetric surveying may involve area and profile measurements, and in some cases single measurements of absolute gravity must be taken. The selenodetic problems (b)-(d), namely, more accurate location of the lunar centre of mass and construction of a gravitational field model from satellite data, require absolute gravity determinations. It is necessary that the absolute measurement stations be spaced as widely apart as possible. The problem (c), or determination of the distances between individual points and the lunar centre, also calls for absolute gravity determinations. An accuracy of * 5 mGal in such measurements is quite adequate for solution of these problems at present and in the immediate future. Even measurements with an accuracy of f20mGal permit the position of the centre of mass to be determined to within f 1 0 0 m and the systematic errors in the lunar gravitational field models to be minimized. To solve the overwhelming majority of selenophysical and selenodetic problems, it is recommended to resort to relative gravity determinations, using for reference abs.olute gravity determinations at a few stations. For example, the terrestrial survey experience tells us that gravity determinations to within f 1-5 mGal are sufficient for solution of the selenophysical problems (1)-(6). Solution of

The Gravitational Field of the Moon

7

problem (9wefinition of small-scale details such as crater boundaries, rilles, and so on+alls for an accuracy within fractions of a mGal. The relative gravity determination range may span 500 mGal. In view of the ambiguity of solution of the inverse problem of gravimetry-determination of the anomalous mass distribution pattern from a known gravitational field-gravimetric methods should preferably be combined with other geophysical techniques (seismic, electrical sounding, etc.). If selenophysical problems do not go beyond local gravimetric surveys, most of selenodetic ones are difficult to solve without knowing the gravitational field of the entire Moon. It is precisely in the case of selenodetic problems that elimination of systematic errors in the gravitational field is highly desirable. It is well known that measurement of lunisolar tides is a difficult task even on the Earth, although the tidal period here is of a diurnal duration as opposed to the monthly period on the Moon. Studies into lunisolar tidal variations in gravity necessitate stationary gravimeters with a measurement range of about 3 mGal and an extremely high accuracy of fO.OO1 mGal (Chapter 4). The basic requirement of tidal gravimeters is stability of their zero-point. Measurements must be taken continuously, at least over a period of several months. The above figures are, of course, very approximate. In practice, different accuracies may be needed, although definitely no different than by a factor of two or three. Prediction is a risky affair. The validity of forecasts is affected by the leaps and bounds in the evolution of science and technology. This is especially true in space exploration, where scientific and technical advances are most spectacular.

1.3 Lunar Gravitational and Tidal-Centrifugal Potentials. Gravity

Let S be the surface of the Moon and R its volume (Fig. 1.1). We shall use a fixed-in-the-Moon system of rectangular coordinates X , Y, and Z . The origin 0 of the coordinates. corresponds to the lunar centre of mass. The axis Z is directed along the rotational axis of the Moon, and the axes X and Y are oriented in its equatorial plane so that the former will pass through the zeromeridian plane. The point P at which the gravitational field is examined has coordinates (x, y, z), while (5, q, c) are those of the mass element drn at the current point M of the lunar body. We assume that the point P accommodates unit mass; then, the force acting upon it is numerically equal to acceleration. This mass is acted upon by several accelerations of different origins: acceleration gl of lunar mass attraction, acceleration g2 of attraction of the masses of the Earth and other celestial bodies, and centrifugal acceleration g3.By gravity or, to be more precise, gravity acceleration g, is meant the

8

Lunar Gravimetry

Fig. 1.1. Coordinates, angles and distances used in deriving the lunar attraction potential.

resultant of all acting acceleration. The potential of lunar mass attraction is expressed as a triple integral over the Moon’s volume R:

(1.3.1) where drn = Q dR, dR((, q, [) is a volume element, and o((, u, [) is the volume element density. Let us now introduce a spherical coordinate system with the same origin (Fig. 1.1), where p, cp and 1stand, respectively, for the polar distance, latitude, and longitude of the point P at which the potential of lunar mass attraction is considered; p l , cpl, ill are the coordinates of the current point M of the lunar body. The distance between P and M is r

where

= (p2

+ pt - 2pp1 cos $)1’2

\I/ is the angle between the radius vectors

p and p l . Expand the

9

The Gravitational Field o f the Moon

quantity l/r in the integrand of (1.3.1) into a series of Legendre polynomials (1.3.2) where cos $ = sin cp sin cp, + cos cp cos cp, cos (1 - Al), and P,(cos $) is a Legendre polynomial of the nth order. Using the theorem of Legendre polynomial addition, one can express the polynomial with argument cos I(/ in terms of functions of ql,1 1 , cp, A: Pn(cos $) = f'no(sin cP)Pno(sin PI) +2 m=l

( n - m ) ! (cos m l cos mil, (n m ) !

+

+ sin mil sin m1,)

x P,,(sin

cp)P,,(sin cp,)

(1.3.3)

where P,,(sin cp) and P,,(sin cp,) are associated Legendre functions. Substituting the expansion for l/r (1.3.2) into (1.3.1) with due account for (1.3.3), we can write the potential V of attraction as VP, cp,

4=

~

P

2

n=O

y:(

(C.,cos

mil+ S,, sin ml)P,,(sin cp)

(1.3.4)

m=O

where R is the mean lunar radius. C,, and S,, are harmonic coefficients, sometimes referred to as Stokes coefficients. They have nothing to do with the coordinates of the observation stations but depend only on the density distribution a(pl, cpl, 2 , ) within the moon and its figure R:

c nm

- (n

2(n - m)! ~~[op;P..(sin + m)!MuR"

cp,) cos m l , dR

(1.3.5)

R

Snm

+

= (n 2(nm)! -m MuR" ) ! jj!op;P,,,,,(sin

cpl) sin mAl dR

n

where dR = pf cos cpl dcpl d l l dp, is a volume element in spherical coordinates. Consider now the potentials corresponding to the accelerations g2 and g3. The contribution of the Sun to tidal lunar gravity variations can be ignored because it is two orders of magnitude less than that of the Earth. We shall use

10

Lunar Gravimetry

t’

Earth

Fig. 1.2. Derivation of the tidal potential on the Moon.

the selenocentric equatorial spherical coordinate system (Fig. 1.2). Let P(p, cp, i) be the point at which the variations in the tidal potential, due to the Earth’s attraction, and variations in the centrifugal potential, due to the Moon’s rotation about its own axis and around the Earth, are examined. The Earth’s coordinates will be denoted (A, b,, l,). Having assumed the Earth to be a point body with mass M,, we can write the following expression for the potential V Q of terrestrial attraction at the point P:

where A = (A2 + p z - 2 p A cos z ~ ) ’ ’ is~ the distance between the Earth’s centre of mass and the point P , A is the distance between the terrestrial and lunar centres of mass, and z, is the angle with vertex at the origin of coordinates between the directions toward the point P and the Earth. This angle equals the zenith distance z, of the Earth at the point P. Let us expand

11

The Gravitational Field of the Moon

P e ( P ) into a series of Legendre polynomials: (1.3.6)

The potential of attraction at the lunar centre of mass is (1.3.7)

The difference of potentials (1.3.6) and (1.3.7) gives rise to a tide-generating potential on the Moon due to the effect of the Earth. Lacking the properties of an absolute solid, the Moon undergoes partial deformation which manifests itself in tides within the solid Moon. The term with spherical function P,(cos zQ) in (1.3.6) is excluded because the origin of the selected coordinate system coincides with the lunar centre of mass. Restricting ourselves to the second harmonic in expansion (1.3.6) in view of the smallness of the subsequent terms as a result of the rapidly decreasing relation @/A)”, we obtain V,(P)

=

P,(P)

-

Pe(0)= ___ G’@p2 P2(cos z e ) A3

(1.3.8)

Let the angle zQ be expressed in terms of angle $ which is the angle between the directions toward the point P and the mean position of the Earth, coinciding with the direction of the coordinate axis X . It is known that the angular coordinates of the Earth ( b e , l Q ) vary, from a lunar observer’s point of view, within -6”. Then, it is easy to find that COS’ ZQ = COS’

$ + F($,

~ p ,I ,

b e , I@)

where F($, cp, I , b e , l e )

%

2 cos cp(cos cp sin I sin le

+ cos I sin cp sin b Q )

is a time-dependent function because of the varying terrestrial coordinates (A, b e , lQ). Using the latter expression, we can rewrite (1.3.8) as

Using the addition theorem for Pzo(cos $) and bearing in mind that cos II/ = cos cp cos I , we shall write the final expression for the terrestrial tidal potential on the Moon:. VQ(P ) = GMQp2[ -2P2,(sin cp) 4A3 ~

+ Pzz(sin cp) cos 21 + 6F($, cp, I , be, re)] (1.3.10)

12

Lunar Gravimetry

Thus, the tidal potential on the Moon comprises two parts. The first, invariable part is independent of time, while the second part is timedependent. Both parts are dependent on the coordinates (cp, A) of the tidal phenomena observation station. The centrifugal acceleration potential due to the Moon's rotation is much smaller than that due to the Earth's rotation because of the low angular velocity of the Moon's rotation: w = 0.26616995 x lop5rad s-'. The centrifugal potential Q at the point P is

The value of w corresponds to a sidereal month which is determined according to Kepler's third law, from 0 2 =

~

GMQ A3

Use of this expression for w z gives the following formula for the centrifugal potential: (1.3.1 1) Addition of (1.3.10), in which the time-dependent part of the potential is omitted, and (1.3.11) gives the following expression for the tidal-centrifucal potential on the Moon: U ( P ) = ___ G M Q p [2 - 5Pzo(sin cp) 6A3

+

Pzz(sincp) cos 2 4 (1.3.12)

Thus, the tidalxentrifugal potential U is essentially the sum of the centrifugal (Q) and tidal ( W) potentials. It depends on the geocentric constant G M e of gravitation, the lunar point coordinates ( p , cp, A), and the distance A between the Earth and the Moon. As can be inferred from (1.3.12), the potential U is independent of the internal structure of the Moon, and therefore its definition does not seem to present any of the difficulties normally associated with poor knowledge of the lunar density distribution and figure, as was the case with the lunar potential of attraction (1.3.4).In fact, this is not so. Under the effect of the tidal-centrifugal potential, the Moon undergoes deformation: its mass is redistributed, and the coordinates of the observation station change. The result is uncertainty in the knowledge of the rigidity or susceptibility of the Moon to deformations (see Chapter 4). We have already mentioned at the beginning of this section that the gravity acceleration g is the vector sum of all accelerations acting at a given point.

13

The Gravitational Field of the Moon

The potential W of the gravity acceleration also equals the sum of potential V of acceleration due to lunar mass attraction and potential U of tidalcentrifugal acceleration: W ( P )=

V P , cp, A) + V(P,cp, A)

=-

P

(C,,,, cos mA n=O

+ S,,

sin mA)P,,(sin cp)

m=O

(1.3.13)

The full value of the gravity acceleration g(p, cp, A) will be found if the latter expression is differentiated with respect to the three orthogonal axes (p, cp, A), their squares are added, and the square root is taken: 1

(3"'"

(1.3.14)

Obviously, g is determined primarily by a W p p . Assuming the potential W(x,y, z) as a function of the coordinates x, y, and z and equating it to a constant, we obtain the following implicit equation of a surface: W(x,y , z) = const.

(1.3.15)

It exhibits some gravimetrically important properties. This is an equation for equipotential, or level, surface. By changing the value of the constant, we can obtain a family of level surfaces. Since the gravity acceleration component along a direction I is

the increment in the potential over an elementary distance dl can be determined using the formula d W = g dl cos (g?). If the direction of g coincides with 1, then cos (g?) = 1 and the increment d W is maximal. The equality of cos (g?) to zero indicates that no increment in the potential occurs in moving along the level surface. It should be emphasized that the equality of potentials on the level surface is far from implying that gravity is constant on the same surface. Also, if we constructed a surface of equal gravity, the potentials at different points on such a surface would not be equal.

14

Lunar Gravimetry

1.4 Quantities Characterizing the Lunar Gravitational Field

In addition to the such purely terrestrial characteristics of gravitational field as gravity potential, acceleration components along different coordinate axes, and gradients of these acceleration components, the Moon is also characterized by line-of-sight acceleration which is an acceleration component in the direction from the artificial lunar satellite to a terrestrial Doppler tracking station. It can be identified, to a sufficient degree of approximation, with the acceleration along the Earth-Moon line. Only in the central parts of the lunar disk do line-of-sight accelerations coincide with those oriented radially to the Moon. The acceleration component orthogonal with respect to the line-of-sight acceleration a W/ap, that is tangential to the spherical surface of the Moon, can be expressed as

where grp= (1/R)(aW/acp) is the latitudinal acceleration component, and g A = 1/(R cos cp)(aW/aA)is the longitudinal one. In view of the smallness of g1 in comparison with W/dp, in most problems the radial gradient a g / a p of gravity may be considered equal to a2 W/ap2.The line-of-sight acceleration in a rectangular coordinate system, selected such that the axis X with its origin in the Moon’s centre is directed earthward, is

a

An important characteristic of the gravitational field is the plumb-line deflection. The deflection component along the meridian is ( = 206265 (l/R)(aW/acp)

9

(in seconds of arc)

and that along the parallel is ( = 206265 (l’R)(a

w’aA)

9 cos cp

(in seconds of arc)

The full plumb-line deflection is 1 = ((2

+

t+)1/2

Not all of the above characteristics can be measured directly, yet all of them can be calculated, provided the distribution of one of them over a closed surface enveloping the Moon is given. Such calculations usually involve

15

The Gravitational Field of the Moon

methods employed to solve boundary-value problems in potential theory. The problem is stated as follows: all over a known closed surface S there exists a gravitational potential Ws(r,cp, A ) = f ( r , cp, A) generated by the masses randomly distributed within S. Find the potential or its derivatives at points in the space external with respect to S. A simple analytical solution of this problem exists only for an ellipsoid, a sphere and a plane. If the same potential function given on a sphere is determined at different points in the surrounding space (Dirichlet problem), the solution takes the form of Poisson's integral (Fig. 1.3): (1.4.1) S

In the case of the Moon, which is taken to be a sphere S of radius R, the following notation is used p, 0 and A are the sphkrical coordinates of the point P at which the potential W(p,(D, A) is determined, and W(R,cp, A) is the potential given at the current points M ( R , cp, A ) on the surface of the sphere, and r =' ( p 2 R 2 - 2Rp cos is the distance between the points P and

+

1'

Fig. 1.3. Angular coordinates and distances determining the Poisson integral equation for a sphere.

16

Lunar Gravimetry

M . The derivative of a potential of any order along the axes X , Y and 2 can A) if both parts of (1.4.1) are differentiated be obtained at the point P ( p , 0, along these directions: axm ayn

aZI =

11

W ( R , cp, A)K(R, cp,

A, p, a, A) dS

(1.4.2)

S

where =

[

r

am+n+l

axm ayn

azt

2 -RZ)] 4 a ~ r 3 xy==PpC Ocosasin S@COS~ A z=psin@

is the kernel of the transformation p = (XZ

+ y2 +

= C(X = (p2

z2)1/2

0’ + (Y - v)2 + ( z - 02 1 1 / 2

+ R 2 - 2Rp cos $)’/’

+ sin cp sin @ cos (A - A) And if we want to determine the gravity component a W/ap along the radial cos $ = cos cp cos @

direction p, the kernel is

a

K p = - a( p

p2 - R 2

4nRr3

)=

- p 2 ( R cos $ + p ) + R2(5p - 3R cos $) 4aRr

When components tangential to the spherical surface S (latitudinal g@and longitudinal gA) are determined, the kernels take the respective forms: K @ = - (a

pz - ~2

~ a @4nRr3 a K A = - - - (1

p2

RCOS@

aA

)=

3 ( p z - R Z )a~cos $

47cRr5

)=

- RZ

4nRr3

a@

3(pz - ~ z ) p a c o s +

~ ~ R ~ ’ c o saA@

Finally, here is another expression of the transformation kernel for calculating the second radial derivative a2 W/ap2of the potential: KPP

=

R2(5R2- 23p2 + 28Rp cos $ - 15R2 C O S ~$ - p2 COS’ $) 4aRr’

+ 2R cos $) + 2p3(p 4nRr’ Thus, calculation of a particular derivative of the lunar gravity potential boils down to integration of a given potential W ( R , cp, A) over a spherical surface with a respective kernel.

17

The GravitationalField of the Moon

In conclusion, we should like to point out an interesting possibility. If all derivatives of the gravity potential could be determined at a single point in space, this would be sufficient, in principle, to define the gravitational field within the entire space, the only prerequisite being that this space should not contain any masses that have generated the field. However, measurement of what amounts to third derivatives of the potential involves insuperable technical difficulties from the standpoint of the precision necessary to produce components of an appropriate instrument and ensuring stable conditions for the measurement. Table 1.3 presents some of the above-mentioned characteristics with their dimensions, full values, and the possible regional anomalies on the lunar surface. It also gives the corresponding terrestrial parameters for comparison. Note the much more pronounced anomaly of the Moon's gravitational field, as compared to that of the Earth. 1.5 Comparison of, Methods Used for Studying the Gravitational Fields of the Moon and the Earth

Investigations of the lunar gravitational field began with the launching of artificial lunar satellites (ALS). Interestingly, the chronological sequence in which the methods for determining the gravitational fields of the Earth and the Moon were used has been different. In the case of the Earth, satellite methods started being used when its gravitational field had already been studied in general with the aid of gravimeters and pendulum instruments. TABLE 1.3

Some characteristicsof the lunar and terrestrial gravitational fields Magnitude on the surface (possible anomaly) Characteristics; dimensions

Unit of measurement

Moon

Earth

2824 ( f12)

626,37 (k70)

Gravity acceleration potential,

K CLY, C T Z

10' cmz s - ~

Gravity acceleration, g; [ L ] ,

crl-2

Radial gravity acceleration gradient, dg/dp; [ f l - 2 Attraction components tangential to the surface, W,,W,;

cLIcrl-2 Plumb-line deflection components, t, rl

ems-' = 1 mGal 162,700 (k500) 980,600 (f300) s2 = 1 eotvos(E)

1870 (k500)

3086 ( f 100)

cm s - ~= 1 mGal

(k600)

( k 300)

* 500

+ 60

seconds of arc (")

18

Lunar Gravirnetry

Each method, satellite and gravimetric, has its advantages and drawbacks. The former provides more reliable means for determining the gravitational field harmonics of lower orders, whereas the latter is more suitable for higher harmonics. A practical approach has been to combine both methods with recourse to astronomical and geodetic data. As regards the Moon, satellite methods were the first to be used in investigations of its gravitational field, if we do not count some evaluations of changes in gravity with the selenocentric latitude and longitude, based on the lunar moments of inertia established by astronomical observations (Grushinsky and Sagitov, 1962). The first detailed definition of the lunar gravitational field, in the form of expansion into a series of spherical functions, was carried out by Akim (1966) using data supplied by the artificial lunar satellite Luna 10. Data from some American spacecraft, including Lunar Orbiters 1-5, Explorers 35 and 49, Apollos 14-17, and Apollo 15 and 16 subsatellites, as well as the Soviet spacecraft Luna 10 and 24, have widened our knowledge of the Moon's gravitational field. Direct gravity measurements on the lunar surface have just begun. So far, only four measurements with the aid of gravimetric instruments installed on lunar probes that have performed soft landing on the Moon (Apollos 11, 12, 14 and 17) are known to have been carried out. Gravity profile measurements on the Moon's surface were conducted with a special gravimeter mounted on the American lunar roving vehicle delivered by Apollo 17. Although ALS will continue to play a predominant role in investigations of the overall gravitational field of the Moon, we must not underestimate direct gravity measurements using soft-landing lunar vehicles and probes. The advantages of satellite methods for investigating the gravitational field of the Moon, as compared to that of the Earth, are as follows: (1) The flight of ALS can be watched from the same terrestrial tracking

station all the time it is not behind the Moon. Two tracking stations located at longitudes 180" apart will permit uninterrupted tracking. (2) Since the Moon rotates more slowly than the Earth, its gravitational effect on the ALS orbit is averaged to a lesser extent. (3) The dense and time-dependent atmosphere of the Earth produces a strong perturbing effect on Earth-orbiting satellites. The virtual absence of atmosphere on the Moon means that the motion of ALS is unperturbed by this factor. (4) Owing to the lack of atmosphere on the Moon, its satellites may orbit very close to the lunar surface. The pericentres of some Apollo spacecraft used to study the gravitational field of the Moon were at distances of only 15-20 km from the lunar surface. The shortest distance at which satellites may orbit the Earth is about 200 km. Low-

The Gravitational Field of the Moon

19

orbiting spacecraft are more prone to the effect of gravitational field anomalies. ( 5 ) The ratio of gravitational field anomalies to the overall field is greater for the Moon than for the Earth, which is why ALS are more susceptible to the perturbing effect than those orbiting the Earth. Of course, there are some factors that make investigation of the lunar gravitational field more difficult. These are, primarily, the highly complex procedure for launching ALS and their prohibitive cost. From the technical standpoint, tracking also imposes some limitations. When a spacecraft swings around the far side of the Moon, it escapes tracking and the efficiency of deriving the lunar gravitational field suffers. This drawback is inherent in the currently used Moon-orbiting satellite tracking techniques. Placing two spacecraft in the lunar orbit at a time will permit retransmission of signals from the satellite which becomes invisible from the Earth, whereby more reliable data on the far-side gravitational field can be obtained. Errors in determination of the lunar gravitational field arise because the exact position of the ALS relative to the Moon is not known and as a result of inexact subsequent reductions of the measured field parameters. Serious errors occur during mathematical processing of Doppler tracking data, which includes filtering operations and conversion of the Doppler frequency variations into the gravitational field characteristics. In addition, various instrumental errors are inevitable as well as errors due to refraction of radio waves in the Earth’s ionosphere and uncertainty as regards the position of the Moon-orbiting spacecraft with respect to the terrestrial tracking station. This uncertainty stems from inexact knowledge of the lunar ephemerides and the position of the tracking station on the Earth. Some of the above errors are systematic. They can be eliminated by closely studying their sources through observations of other celestial objects rather than the Moon-orbiting satellite. Particularly promising for studies into at least the high-frequency portion of the gravitational field are ALS-mounted gradiometers. Apart from high resolution, gradiometers offer a number of advantages for investigating the gravitational field on the far side of the Moon. Since they are more or less selfcontained and remain functional beyond the zone of visibility from the Earth, they can be used to measure the gradient of accelerations on the Moon’s far side. 1.6 Elements of ALS Orbits and the Concept of Perturbations in Motion as a Result of Irregularities of the Lunar Gravitational Field

Let point 0 stand for the Lunar centre of mass and S be an ALS (Fig. 1.4). The origins of the rectangular (X, Y, Z) and spherical (p, CD, A) coordinate

20

Lunar Gravimetry

Point of the vernal equinox Fig. 1.4. ALS orbit elements.

systems coincide with the lunar centre of mass, and Y is the vernal equinox. The equatorial plane of the Moon coincides with the plane XO Y. The orbital plane of the ALS intersects the equatorial plane along the nodal line K K 1 .It is known that the spatial position of a satellite at any instant is characterized by its radius vector (X,Y, 2 ) and velocity vector (X, Y, Z). Instead of these six parameters, one can use six elements of the Keplerian ellipse having one of its foci at the origin of coordinates. The elements representative of the ALS position in its orbital motion are (Fig. 1.4): R, longitude of the ascending node; i, inclination of the orbital plane of the ALS to the equatorial plane of the Moon; a and b, semimajor and -minor axes of the orbit; e = [(a’ - b’)/~’]’’~,its eccentricity; o,angular distance of the pericentre from the node; and M, mean anomaly M

271

2a T

= - ( t - to) = -t - Mo

T

where T is the satellite’s orbital period, 2 4 T is its mean motion (mean angular velocity), and to is the instant of transit of the ALS through the

21

The Gravitational Field of the Moon

pericentre. In addition, true ( u ) and eccentric (E) anomalies are used. The latter is related to the mean anomaly by the Kepler equation M = E - e sin E

while the true anomaly u is expressed in terms of the eccentric one as follows: tan

2

=

(g)"' ):( tan

The true (u) and eccentric (E) anomalies are necessary to determine the distance p between the satellite S and the origin of coordinates, coinciding with one of the foci of the elliptical orbit: a(l -- e2) p = ( X 2 + y2 + Z2)1/2 = = a(1 - ecos E) (1.6.1) 1 + ecosu Sometimes, use is made of the pericentre's longitude p=R+o

It comprises angles R and o measured in different planes. The differential equations of the satellite's motion in the irregular gravitational field of the Moon, in the rectangular system of coordinates ( X , Y, Z ) , are d2X G M , =+p3X=d2Y -+dt2

aT

ax

aT y=p3 ay

GM,

d2Z G M , =+p3Z=-

aT

az

where G M , is the selenocentric constant of gravitation, and T is the perturbing potential of the Moon, due to the difference of the true lunar potential from that of the point mass. The same equations written in the selenocentric spherical coordinate system take the following forms: GM, (p2:)

+ pz&y

aT

sin @ cos @ = aT

am

.(1.6.2)

22

Lunar Gravimetry

in this case, P = p(a, e, i, 0, Q, M )

CD = @(a,e, i, o,R, M)

(1.6.3)

A = A(a, e, i, o,R, M) It is convenient to represent the perturbing potential Tof the Moon expanded into a series of spherical functions:

2 [(ty

P n=2 T = G“,

(C,, cos mA

m=O

+ S,,

sin mA)P,,(sinCD)

1

(1.6.4)

Here, R is the mean radius of a spherical Moon. The expansion begins with n = 2 because the perturbing potential does not contain the zeroth harmonic corresponding to the point Moon potential; first-order harmonics are absent because the origin of the selected system of coordinates is assumed to coincide with the lunar centre of mass. If T = 0, which means that no perturbations affect the ALS motion, then the satellite is propelled by the gravitational field only of the point Moon. In that case, the motion of the ALS would have followed an elliptical orbit and obeyed Kepler’s laws. The presence of perturbations T complicates the spatial motion of the ALS. The elements of its orbit change in the course of time or, in other words, undergo perturbations. The magnitude and character of these perturbations depend on the coefficients C,, and S,, which represent the irregularity of the lunar gravitational field. Observational data are used to compute the perturbations affecting ALS orbit elements, from which the numerical values of C,, and S,, are determined. Let us provide a general scheme for derivation of the formulae correlating the perturbations of the elements with C., and S,,. To do this, we shall turn from the differential equations system (1.6.2) to a system of six first-order differential equations known as Lagrange equations (see, for instance, Caputo, 1967). They can be used to represent the perturbations for any element (daldt, deldt, di/dt, do/dt, dR/dt, dMu/dt) in terms of the derivatives of the perturbing function T with respect to the elements (dT/da, dT/de, dT/di, dT/do, dT/dR, dT/dMu) and the elements themselves (a, e, i, o,R, Mu). Consider in greater detail only the perturbations in the ascending node longitude R and angular pericentre distance o. Lagrange equations give dR -

dt

cosec i dT (CMu)’”[a(l - e2)]”2 di

(1.6.5)

23

The Gravitational Field of the Moon

and do

dt

=-

dT cot i (GMu)’/’[u(l - e2)]’/2 di

dT (GMu)’/2(ae)’/2de 1-e’

-+

(1.6.6)

Turn now from the time argument t to the true anomaly argument u, using the equation du 1 - = - (GMU)’/’[a(l - e2)]’/’ dt p’ Equations (1.6.5) and (1.6.6) can be rewritten as

dsz _ du

p’ cosec i

dT

GMaa(l - e’)

di

(1.6.7)

and do_ _ du

-

p’coti dT p2 dT -+-GMua(l - e’) di GM(ae de

(1.6.8)

Sequential transformations may yield the final expressions for dR/du and do/du. The result is cumbersome expressions whose arguments will be the desired harmonic coefficients, orbital elements, and the mean lunar radius R: dR - dR (C,,, S,,, R, orbital elements) du du d o -d o du du

(C,,, S,,, R, orbital elements)

Thus, in determining the zonal harmonics, assuming that rn = 0 for dR/du, we obtain

(:Yc2.

-

i + -32 (1 cot -

e2)3

x (1

):(

+

sin2 (o u)(l+ e cos u) 3

+

+

c30[5 sin3 (a u) sin’ i - sin (o u)]

+ 2e cos u + e’

+ -5 (1cosi - e2)4) :( ~ x (I + 3e cos u + 3e’ 4

cos’ u)

+

~ sin4~ (o[ u ) 7sin2 i - 3 sin2 (o+ u)]

cos’ u

+ e3 C O S ~u) + . . . (1.6.9)

24

Lunar Gravimetry

The perturbations in the orbital elements are secular and periodic. The periodic perturbations are, in turn, divided into long- and short-period perturbations. In particular, in the case of dR/dv, short-period perturbations are due to changes in v-that is, with a period close to the orbital period of the ALS. To eliminate them, it is sufficient to carry out averaging over v :

-=-I-& 2r

-

dR dv

dR dv

1 271

0

Then,

In order to eliminate the long-period perturbations as well, the averaging should also be performed over o: 15

('p40(

1

1 - sin2 i)

m=O

x(l

+iez)+E 6 (1

- e2)6 ('Y ac60

1+5e2+-e4 l5

8

) ] + . a .

(1.6.11)

In conclusion, here is a brief summary of secular variations in other elements, as a function of the harmonic coefficient CzO:

(1.6.12)

The odd zonal harmonics of the gravitational field,just as the tesseral ones,

25

The Gravitational Field of the Moon

do not give rise to secular perturbations. The odd zonal harmonics are responsible for long-period variations in orbital elements. For example, these variations are related to C30 in the following manner:

-

):(

3ecoso

("y ( y

cot i -sin2

= 2(1 - e2)3

1 + 4e2

3 sin o 3esino

(p

.

)

i -1

(1.6.1 3)

sin i - e cot i)(y sin2 i -

)

cot i -sin2 i - 1 (:)3

c 3 0

C30

Obviously, the best way to determine the harmonic coefficients of the lunar gravitational field would be from the secular perturbations in the orbital elements of a Moon-orbiting satellite. The maximum secular variations occur at the ascending node longitude Iz and at the distance w of the pericentre from the node. Long-period perturbations are predominant in the inclination i and eccentricity e. For instance, over 46 revolutions of Luna 10, the variations in Iz amounted to -7.7", and those in o,to + 11.8".

1.7 Methods for Determining the Harmonic Coefficients of the Gravitational Field from ALS Tracking Data

The problem of defining the lunar gravitational field is inverse to that of studying the ALS motion in this field. Tracking data are used to determine the perturbations in the ALS motion, due to anomalies in the gravitational field. It is assumed that the left-hand sides of (1.6.9)-(1.6.13) are known from tracking data, and the approximate values of the ALS orbital elements, used in the calculations of the factors of the coefficients C,, and S,, in the righthand sides of these equations, have been defined. Such a method for determining the coefficients C,, and S,, may be conventionally referred to as analytical because it is based on analytical solution of the differential equations of the ALS motion. The analytical method has provided several modifications of the solution, based on different initial tracking data, as well as several procedures for their processing. The processing procedures differ in

26

Lunar Gravimetry

the arrangement of the ALS tracking data obtained over the entire period when the ALS is under observation. Distinction is made between the method of long orbital arcs covering tens and hundreds of ALS revolutions and that of short arcs covering just a few revolutions. Experience has shown that, due to insufficient diversity of ALS orbit parameters, in the early determinations the harmonic coefficients C,, and S,, were cross-correlated. For example, as was shown by Lorell and Sjogren (1968), derivations of harmonic coefficients up to the fourth order yielded coefficients of correlation between Cz0 and c40, Goand c60, CZland c61,c 3 0 and GO, s 3 2 and C42r c 3 2 and s42, S33 and C43, C40 and c60, Cs0 and CT0 exceeding 0.8. Therefore, in later investigations, use was made of tracking data processing procedures that ensured a smaller cross-correlation between the harmonic coefficients to be determined. Let us dwell on the tracking data processing procedure whereby the harmonic coefficients are determined through analytical solution of equations of ALS motion, with reference to Ferrari (1973, 1977) and Ferrari and Ananda (1977). The Doppler tracking data are used first to derive the Keplerian elements (a, e, i, o,Q) for each orbit. Each element is approximated, with the aid of splines, by a time function. Differentiation of the latter permits calculating the long-period variations in the orbital elements (6, i, h, 0).The semimajor axis a is not perturbed by the gravitational field, therefore a = 0. Furthermore, Lagrange equations are used to calculate the harmonic coefficients C,, and S,, from long-period perturbations. The equations of orbital element perturbations can be written in the following vector form which is more compact: E = F(E)P, (1.7.1) where E is the vector of mean elements (a, i, e, Q, o,M ) , and F ( E ) is a matrix composed of partial derivatives of element rates, with respect to harmonic coefficients:

[; __

.. .

ac20

ah ai

ai

acnm

as21

...

]

ai

as,,

(1.7.2)

F(E) =

ah G

o

...

-

ac,,

ah ~

... an as,, ~

The matrix dimensions are [(n + 1)2 - 41 x 4. Let P be the vector of the desired harmonic coefficients C,, and S,, of expansion. of the lunar gravitational field into a series of spherical functions. Its matrix has the 1)' - 41 x 1 and takes the form dimensions [(n

+

P = C C Z O , C Z I , C Z ZCnrn,S21,S~~,..*,Snrnl ,...,

27

The Gravitational Field of the Moon

In the mean-square approximation, the solution algorithm for (1.7.1) is written as follows (Bryson, 1969): p =(FTw-~F

+ K - 1 ) - 1 ( ~ -1p* + ~ T w -E1)

(1.7.3)

where W is a weighting matrix for long-period element rates, having dimensions 4 x 4. It is assumed to be diagonal-that is, the covariance between the derivatives of different elements is neglected: 2

Oi 2

W=

Oi.

(1.7.4)

4 2 u6J

The variances ui can be calculated using their approximate relationship with the element errors 6 E . The above-mentioned quantity K is a covariance matrix for the desired harmonic coefficients. In the case of lower harmonics, its elements are determined from an a priori estimated vector P*, whereas in the case of higher harmonics, they are determined from an empirically established law of monotonically decreasing variances numbered n. The matrix K is also assumed to be diagonal

I 0

","

C

Using this technique, Ferrari (1977) derived harmonics up to order (16, 16) from Lunar Orbiter 5 and Apollo 15 and 16 subsatellite tracking data. Apart from determining the harmonic coefficients through analytical solution of equations of ALS motion in the irregular lunar gravitational field, a technique based on numerical integration of these equations has been used. The vector form of the differential equations of ALS motion can be written as

A=GM,V

(:-+- AJ

where A is the radius vector from the lunar centre of mass to the ALS, V is a Hamiltonian operator, and Tis a perturbation function which takes the form of (1.6.4) in the case of expansion in terms of spherical functions. However, it can also be expressed differently, particularly, in the form of the gravitational

28

Lunar Gravimetry

potential of a plurality of mass points: -

L

d

i=l

ri

where mi are mass anomalies within the Moon, and ri is the distances between these anomalies and the ALS. In practice, this boils down to differential determination of harmonic coefficients or mass points. To this end, a lunar gravitational field model characterized by coefficientsC,, and S,, (or masspoint parameters) in a zeroth-order approximation is derived. This field model is used to calculate the intermediate ALS orbit, then the observed orbit is compared with it. It is assumed that the difference between the two arises from the errors inherent in the adopted model of the lunar gravitational field. Differential corrections are introduced to handle the gravitational field coefficients of the initial model. Such an approach is more direct in certain respects than the analytical method described above. The latter calls for, first, determination of the orbital element rates from which the gravitational field coefficients are then derived. The tracking data are used to find the coordinates and velocity of the ALS, which is why the coefficients are related to them only through element rates. The discrepancy between the intermediate and observed ALS orbits stems from the disparity between the initial gravitational field model and the actual field. The model must be adequately representative of the gravitational field on the near and far sides of the Moon. Therefore, the lunar gravitational field coefficients determined by the above selection method must correspond, in principle, to both sides of the Moon. Errors may occur, of course, due to the fact that all gravitational field coefficients are determined from the tracking data covering only the near side-that is, the observations are not equally valid. The method of numerical selection of the coefficients C,, and S,, of the lunar gravitational field model has been used in some works, particularly those of Michael et al. (1970) and Michael and Blackshear (1972). It should be emphasized that the analytical methods are indispensable for qualitative analysis of the behaviour of solutions of the equations of ALS motion in the irregular gravitational field of the Moon. They are necessary for finding optimal solutions, optimizing ALS tracking programmes, and the like. It is quite obvious that reliable determination of the gravitational field coefficients requires sufficiently long tracking periods (weeks, months). Besides, a greater number of ALS with various orbital elements (inclination, eccentricity, semimajor axis) is desirable. Tables 1.4-1.6 list 'the ALS orbit elements used in determining the coefficients C,, and S,, of the lunar gravitational field. Theoretical analysis has shown that the optimal ALS orbits for studying the gravitational field are those with minimal semimajor

29

The Gravitational Field.of the Moon

TABLE 1.4 Basic parameters of some ALS used for determination of the lunar gravitational field: a, semimajor axis; e. eccentricity; i, inclination of the orbit to the lunar equator; T. period

ALS

a(km)

e

i

T(min)

Lunar Orbiter 1 Lunar Orbiter 2 Lunar Orbiter 3 Lunar Orbiter 4 Lunar Orbiter 3 Lunar Orbiter 4 Lunar Orbiter 5 Explorer 35 Explorer 49

2670 2702 2688 3751 1968 6150 2832 5980 2803

0.327 0.371 0.332 0.516 0.062 0.654 0.317 0.576 0.002

12 18 21 84 21 85 85 169 62

206 210 208 344 130 72 1 225 690 220

Lengths of arcs used for analysis (days)

Number of observations

9.7 8.5 15.9 8.6 17.1

2076 2662 2502 3739 3322

19.9 2138 831

5847 133 64

axes a and maximal inclinations i. In this respect, the orbital elements of Lunar Orbiters 1-3 were not optimal because of the small inclination of their orbits, whereas in the case of Lunar Orbiter 4 the value of a was too high. The latter becomes an advantage when only lower harmonics are to be derived. In this case, the perturbing effect of the high-frequency gravitational field components is minimized. For the gravitational field of the Moon to be defined, the ALS must not be subject to any accelerations other than the gravitational ones due to irregularities of the internal structure of the Moon and its figure. The satellite must not execute propulsive manoeuvres. However, since ALS are normally launched to accomplish a broad range of tasks in addition to gravity studies (photography, investigation of radiation, meteorites, and other phenomena in the circumlunar space), they have to perform manoeuvres, which is why the time alotted for gravity studies is limited. For example, with regard to all five Lunar Orbiters, throughout their active lifetime (45 satellite-months), only 24 individual orbits between 3 and 20 days long were available for gravity determinations. The most tangible perturbing effect on the ALS motion is exerted by sunlight pressure. Its magnitude can be calculated using the formula F=

CSo(1 C

+ K ) cos u

where So = 1.39 x erg cm-2 s-' is the solar constant representing the solar radiation intensity per unit area per unit time, C is the cross-sectional area of the satellite, c is the velocity of light, K is reflectance (0 < K < l), and a is the angle of incidence of sunlight on the satellite's surface. To minimize this effect use should be made of satellites with a small cross-sectional areato-mass ratio. The pressure of sunlight on, for example, Lunar Orbiter l led

30

Lunar Gravimetry

TABLE 1.5 ALS parameters used by Akim and Vlasova (1983) to derive harmonic coefficients of the Moon

T (min)

ASL Luna 10 Luna 11 Luna 12 Luna 14 Luna 15(1) Luna 15(2) Luna 15(3) Luna 16(1) Luna 16(2) Luna 16(3) Luna 17(1) Luna 17(2) Luna 18(1) Luna 18(2) Luna 18(3) Luna 19(1) Luna 19(2) Luna 19(3) Luna 19(4) Luna 19(5) Luna 20(1) Luna 20(2) Luna 21(1) Luna 21(2) Luna 21(3) Luna 22(1) Luna 22(2) Luna 22(3) Luna 22(4) Luna 22(5) Luna 23(1) Luna 23(2) Luna 23(3) Luna 24(1) Luna 24(2) Luna 24(3)

242 1 2417 2662 2250 1866 1896 1801 1848 1808 1798 1824 1796 1837 1875 1805 1869 1969 1969 1969 1970 1835 1800 1838 1806 1812 1978 1978 2543 2543 2543 1837 1197 1799 1853 1805 1804

178.1 177.7 205.4 159.6 120.6 123.5 114.3 118.9 115.0 114.0 116.5 113.8 117.8 121.4 114.7 120.9’ 130.7 130.7 130.7 130.7 117.6 114.2 117.8 114.8 115.3 131.5 131.6 191.8 191.8 191.8 117.7 113.9 114.1 119.3 114.7 114.6

e 349 129 101 163 44 97 16 99 21 15 83 21 73 90 15 126 12 83 134 161 89 25 89 7 15 194 234 171 172 193 95 17 17 108 14 10

0.138 0.228 0.309 0.155 0.045 0.032 0.026 0.006 0.028 0.025 0.001 0.020 0.014 0.025 0.029 0.003 0.08 1 0.075 0.050 0.036 0.005 0.021 0.006 0.034 0.033 0.023 0.003 0.249 0.249 0.240 0.002 0.023 0.024 0.004

0.029 0.03 1

71.9 9.7 17.8 41.8 125.8 125.9 126.7 70.2 70.5 70.8 141.5 141.1 35.2 35.7 35.4 40.6 40.6 40.6 40.2 40.9 64.6 64.1 62.1 62.8 62.7 19.7 19.6 19.3 19.3 19.9 137.7 137.4 137.5 119.4 120.5 120.3

132.3 80.3 78.7 311.6 65.2 40.9 159.8 222.0 320.8 343.7 137.3 98.2 14.4 89.4 4.7 103.9 261.6 294.9 329.5 202.7 223.8 347.8 82.3 131.7 137.3 144.6 267.9 77.7 1 19.4 220.4 52.7 159.4 157.1 140.8 159.5 155.4

’

113.9 225.1 237.9 293.7 285.6 286.9 287.7 61.6 60.9 60.0 40.8 43.2 61.3 57.7 57.1 25.5 339.7 288.2 163.5 73.2 54.6 53.8 226.0 225.1 224.6 7.0 302.4 93.5 354.6 298.3 226.4 228.6 229.0 235.3 237.1 237.1

to its acceleration equal to 1.2 x 10-5cms-2. Accurate account of the perturbations due to light pressure is rendered difficult by the fact that its perturbing effect on the orbiting circumlunar satellite changes abruptly each time the latter disappears in the Moon’s shadow and re-emerges from it. Moreover additional perturbations are caused by the tidal action of the Earth, the Sun and planets of the solar system. The perturbing effect due to the Earth can be expressed, to a sufficient degree of approximation, as T e = G M , ,P2 P,,(cos $)

A

31

The Gravitational Field of the Moon

TABLE 1.6 Some ASL parameters used by Ferrari (1977) to derive the harmonic coefficients of the Moon

ASL Apollo 15 subsatellite Apollo 16 subsatellite Lunar Orbiter 5

Length of arcs used Inclination (days) (deg)

222 34 8

150 170 85

Coordinate variations Period, T(min)

120 120 191

Eccentricity e cP(deg)

0.02 0.02 0.28

+27 +10

+40

I(deg) 0 - 360 0+3m +50

where G M Q is the geocentric constant of gravitation, A is the Earth-Moon distance, $ is the angle between the radius vectors A and p from the lunar centre to the satellite. The perturbing potential TQ comprises invariable and time-dependent portions. A similar formula is applicable to the perturbing effect produced 'by the Sun, which is much less pronounced. The perturbations due to the Earth, the Sun and planets are of secular and long-period nature. For example, the perturbing effect exerted by the Earth at an altitude of 2000 km is roughly equal to that due to lunar oblateness. The effect of the Sun amounts to 0.5 x low2of that of the Earth, while the effect of the planets is even less. Relativistic effects are responsible for secular motion of the perigee: d o - 3(GMg)3/2 dt c2a5/2(l- eZ) Other possible perturbations are of electromagnetic origin, if it is borne in mind that the ALS travels through an ionized medium and in a field of solar radiation. All these effects contribute to the satellite's drag. Also, the attractive forces of the Earth and the Sun bring about the redistribution of mass within the Moon, which must also affect the satellite's motion. However, with the accuracies currently achieved in ALS, the effects of electromagnetic origin and due to mass redistribution inside the Moon are ignored. As regards the perturbations due to sunlight as well as the tidal action of the Earth and the Sun, one must find the derivatives, along the respective coordinates, of the perturbing potential of these effects and introduce appropriate terms into the right-hand sides of equations (1.6.2) 1.8 Determination of Line-of-Sight Accelerations due t o the Earth's Rotation and Orbital Motions of the Moon and ALS

A new satellite method started being used in the late nineteen sixties for studying the lunar gravitational field (Muller and Sjogren, 1968), which had

32

Lunar Gravimetry

never been applied before to the gravitational field of the Earth. According to this new method, the ALS is regarded as a test body which changes its direction and velocity, during its orbital motion, under the effect of the irregular gravitational field of the Moon. The satellite velocity changes are measured from terrestrial tracking stations, using the Doppler technique. This permits the measurement of only one component of the velocity, namely, the projection on the ALS-terrestrial tracking station direction, known as the line-of-sight velocity. To determine the line-of-sight velocity of the ALS with respect to a terrestrial tracking station by the Doppler method, a highly stable generator of standard frequency radio signals is mounted on board the satellite. These signals are received by some tracking stations on the Earth. It is known that, if the distance between the source of radio signals and tracking station changes, the Doppler effect will manifest itself in an observed frequency different from that of the radio signals transmitted from the satellite. The difference between the transmitted signal (fo) and observed (f) frequencies is

where V, is the component of the relative velocity along the line from the ALS to the terrestrial tracking station, and V, is the radio signal propagation velocity. The "+" sign corresponds to shortening of the distance between the satellite and station, and " -"indicates that this distance increases. Doppler measurements yield what amounts to line-of-sight velocity averaged over time T:

0

0

where K = V,J0 is the factor of conversion of the averaged frequency difference Af into velocity. It can be easily seen that the line-of-sight velocity V, may vary for a number of reasons, namely: the tracking station moves about the axis of the Earth's diurnal rotation; the lunar centre of mass revolves, over a complex orbit, around that of the Earth-Moon system; the satellite moves along a Keplerian orbit around the point Moon; and the irregularity of the lunar gravitational field distorts the Keplerian orbit. The satellite isalso affected by perturbations due to the Sun, planets, light pressure, and other factors. All these factors render the time variations in the observed line-of-sight velocity highly complex. Let us now examine the line-of-sight velocity components one by one and assess the contribution of each component to the observed

The Gravitational Field of the Moon

33

line-of-sight velocity V,. Determination of the anomalous gravitational field of the Moon calls for elimination of the velocity components which are due to all of the above factors, except for the component arising from the irregularity of the lunar gravitational field. We shall attempt to describe analytically variations in the line-of-sight velocity and line-of-sight acceleration, due to (1) diurnal rotation of the Earth, (2) orbital motion of the Moon, and (3) orbital motion of the ALS in the central gravitational field of the Moon. First, let us introduce a geocentric ecliptic rectangular system of coordinates (X, Y, 2) (Fig. 1.5) and take this system as the main one as opposed to some other coordinate systems which will be auxiliary. Thus, we shall denote the coordinates of the terrestrial Doppler tracking station P, in the main system, by X p , Y p ,and Z p , while the coordinates of the lunar centre of mass (0,)will be ( X ( , Y,, Z,) and those of the satellite’s centre (0,) will be (Xs,Y,, 2,). The geocentric equatorial coordinate system will have the following notation: ( x , y , z ) . Its origin coincides with that of the main coordinate system (X, Y, 2).

Fig. 1.5. Systems of coordinates of the terrestrial Doppler tracking station ( P ) , the Moon’s centre of mass (00, the ALS centre (OJ, and the distances between them.

34

Lunar Gravimetry

1

The coordinates X p , Y p ,and Z p vary with time t in the following manner:

[ [A =

0

[

0 COSE

si:&]

-sin&

COSE

pp

cos ( p p cos A p

ppcos(ppsinAp p ~ s i n( p p

(1.8.1)

where E is the inclination of the Earth's equatorial plane to the ecliptic; p p , ( p p , and AP are the geocentric coordinates of the station P , A p = (27c/n/T)(t- to), T being the diurnal period of the Earth's rotation and t o being the initial instant at which the point P passes through the plane Oxz. Now, let us find the expression for time changes in the position of the lunar centre of mass. We have already mentioned the complexity of the true orbital motion of the Moon even in the context of the main lunar motion problemthat is, with due account for the perturbing effect of the Sun as the only factor affecting this motion and assuming that all three bodies (Moon, Earth, Sun) are points and that the centre of mass of the Earth-Moon system revolves around the Sun along a Keplerian orbit. We shall resort to the formulae of lunar motion theory developed by Brown (1 919), which represent the ecliptic geocentric spherical coordinates of the lunar centre of mass (A, &, L,) as the following sums:

B,

=

A=

1i bj sin

$j

(1 3.2)

a@

c cj cos

*j

j

Here, Xis the mean longitude of the Moon, u @is the equatorial radius of the Earth ll/i = k l l kzl' k3D k4F are combinations of angles 1, l', D and F. They are dependent on the lunar orbital elements and time t. The coefficients k l , k 2 , k 3 and k4 assume, in various combinations, the values 0, f 1, f 2 , . . .. The subscript j stands for one of the combinations of coefficients k l , k 2 , k3 and k4. The coefficients a j , bj and c j of the trigonometric functions in (1.8.2) are complex series in terms of the manner in which the mean motions of the Sun and the Moon, the semimajor axes of their orbits, and the inclination of the Moon's orbit plane to the ecliptic are correlated. The coefficients uj, bj and c j are expressed in angular units. If we restrict ourselves only to the linear relation of I, I', D and F with time t, we have

+

+

+

1 = 296O6'25.31"

+ 17,179,167.085,94t"

35

The Gravitational Field of the Moon

L‘ = 358’28’33.60” + 1,295,977.415,16tr’

+ 16,029,616.645,69t” F = 11’15’11.92“ + 17,395,266.093,19t”

D

=

350’4423.67”

where t is expressed in Julian years reckoned from the epoch 1900. The harmonics in (1.8.2) representing periodic and secular variations in L,, B, and A arise from perturbing motions and are referred to as inequalities. Using the equations (1.8.2), we can express the rectangular coordinates of the lunar centre of mass as a function of time t in the following manner:

Xu = A cos B, cos L,

=

Y, = A cos B, sin L,

=

Z, = A sin B,

=

a& cos

[cjbj sin $j(t) cos I + Cj

sin $j(t)I

Qj

Cj cj cos $j(O

+ Cj aj sin +hj(t)]

cos [Cj bj sin t,bj(t) sin 1 Cj cj cos $j(t)

acBsin [Ij bj sin IClj(t>] Cj cj cos $j(t)

(1.8.3)

Finally, let us examine the changes, in time t , in the coordinates of the satellite executing an unperturbed motion. The rectangular coordinates of the ALS in the main coordinate system (X, Y, Z ) will be represented as the sum of the corresponding coordinates of the lunar centre of mass (Xu, Y,, Z , ) and the relative coordinates of the ALS (X,,j,,Z,). The system of relative coordinates X,, j , and 2, has its origin 0,at the lunar centre of mass, which is to say that it is selenocentric,its axes being parallel to the corresponding axes of the main system (X, Y, Z). The satellite’s coordinates in the main system are therefore

x,= x,+ x,,

Y, = Y, + y,,

z,

=

z,

+ 5,

(1.8.4)

The coordinates X,, j , and 5, will be expressed in terms of the ecliptic selenocentric spherical coordinates p,, b, and I,:

X, = p , cos b, cos I, J,

=p,

cos b, sin 1,

(1.8.5)

Z, = p , sin b,

Let R be the longitude of the satellite’s ascending node, reckoned from the vernal equinox to the line of intersection of the ALS orbit plane with the ecliptical plane, w be the angular distance from the node to the pericentre, i be the inclination of the ALS orbit plane to that of the ecliptic, a be the semimajor axis, e be the eccentricity of the ALS orbit, and u be the true anomaly. The ecliptic latitude b, and longitude I, are related to R, i, w and u

36

Lunar Gravimetry

by the formulae 1, = R + tan-' [cos i tan (o+ u)] 6, = sin - [sin i sin (w + u)]

'

(1.8.6)

It is known that the radius vector p, is expressed in terms of orbital elements as a(1 - e') ps = 1

(1.8.7)

+ e cos u

Substitution of (1.8.6) and (1.8.7) into equations (1.8.5) gives

1 + 1 +

a(1 - eZ) a 2, = 1 +ecOSu[2sin i sin (o u)

+

x cos [Q

+ tan-'

- - sin i sin (o

Ys =

x -

z, =

cos i tan (o+ u)]

sin [Q

+ tan-'

a(1 - e')

1 + e cos u

u)

cos i tan (o u)]

(1.8.8)

sin i sin (o+ u )

Thus, the selenocentric rectangular coordinates (X,, Y,,27), are expressed in terms of ALS orbital elements. If it is assumed that the satellite moves in the central gravitational field without any perturbations, then it will follow a Keplerian orbit. Only the true anomaly u will vary in time t , the rest of the orbital elements remaining constant. The distance between the Doppler tracking station on the Earth and the monitored ALS will be denoted by

6

=

(l' + q'

+

[')1'2

(1.8.9)

where O

C

j=o

Qn+l,m+l+~j

(1.9.10)

(2n

+ 3)(n + rn)! ( n - rn)!

where bmO= 1 at m = 0 and bmO= 0 at rn # 0. The practical realization of the above two methods for transforming the observed field of line-of-sight accelerations r into the gravitational potential V encounters difficulties due to the absence of r values over the entire closed surface S , to say nothing of the fact that the surface S is far from spherical because of the ellipticity of the ALS orbits. The impossibility of monitoring the line-of-sight accelerations when the satellite is above the far side of the Moon adversely affects the derivation of its general gravitational field. With this in view, Brovar and Ganifaeva (1974, 1975) considered the following problem. A closed surface S enveloping the entire Moon and satisfying Liapunov’s condition is given. Also given are line-of-sight accelerations T(x, y, z ) at points on that part of the surface S which faces the Earth and will be denoted by Sl-that is, the following boundary

48

Lunar Gravimetry

condition is set:

w,Y , 4 Is,

=f1k

(1.9.11)

Y , 2)

It is assumed that the rest of the surface ( S , ) is characterized by some other given parameter of the gravitational field. These may be potential increments along the surface Sz or gravitational potential derivatives tangential to it. In view of the future ALS projects involving auxiliary relay satellites placed in high orbits in addition to the main satellite, it may be expected that lunar gravitational potential derivatives along two orthogonal directions p and q tangential to the surface S2 will be determined. The boundary conditions associated with them can be written as

Proceeding from these boundary conditions, one can determine the potential V(x, y, z) at points on the surface Sz to within constant C

V(x, y, 4 Is, =f4(x, y, z)

+c

The above-described exterior mixed boundary-value problem has a unique solution at boundary conditions (1.9.11) and (1.9.12), provided the Moon's mass is known, the direction of x nowhere coincides with that of the tangent to S , , and the boundary conditions at the line joining the surfaces S1 and Sz coincide. The line-of-sight accelerations T(x, y, z) at points of .a selected sphere So are determined, from given line-of-sight accelerations Ti(x, y, z) at N points (xi,yi,zi)of the external space above the surface S1 and from the tangent derivatives d V/dp and d V/dq on the surface S , , by minimizing the functional

FCW, Y , 41 = [[[fl Sl

I'

Y , z)K(x1 - x,Yl - Y , 21 - z)dSo

SO

where K(xl - x, yl - y, z1 - z) is the kernel of Poisson's integral (1.9.1), while K 1 and K 2 are the kernels in the formulae of Moritz and Brovar, which

The Gravitational Field of the Moon

49

take the form of (1.9.6). Minimization (1.9.13) of the functional is performed on a set where the desired function of r belongs to Hilbert’s space L2 and is bounded in the norm. In addition, r satisfies the bounding condition of the type of orthogonality with respect to the known sets of spherical harmonic of Y, on S o . Mathematically, these conditions for the desired function can be written as

The constant C depends on known lunar gravitational potential expansion coefficients. Narrowing the class of functions to be found enhances the stability of the sought-after solution. The mixed boundary-value problem under consideration belongs to ill-posed ones. Minimization (1.9.13) of the functional F(T) is performed using the penalty function method in combination with the method of conditional gradients.

1.10 Gravity Measurement Concepts and Requirements of Lunar Gravimeters

Lunar gravity, just like terrestrial gravity, manifests itself in many phenomena. The free fall acceleration, pendular oscillation frequency, deformation of springs and torsion wires with test bodies attached to them, and the like depend on the magnitude of gravity. Quantitative measurements of these manifestations of gravity may yield its actual value. The instruments for gravity measurements can be divided into two major groups: (1) static and (2) dynamic (Veselov and Sagitov, 1959). The instruments based on the dynamic principle are used to measure time-a quantity in one or another way associated with gravity (time of free fall of a test body from a particular height, period of oscillations, etc.). Let us begin our description with the static methods of gravity measurement.

1.1 0.1 Static Methods

These methods are used to determine the magnitude of a balancing force equivalent to gravity. The function of such a reference force may be performed by the elastic force of a deformed spring, a torsion wire, or a compressed gas, the force of a magnetic or electrostatic field, the centrifugal force, and so on. This reference force must be invariant with time and unaffected by temperature, pressure, and other factors. In studying the

50

Lunar Gravirnetry

gravitational field of the Earth, the most commonly used instruments are gravimeters in which the reference force is produced by elastic elements in the form of various springs and torsion wires. In terms of design, such gravimeters can be divided into two groups differing in the motion of the test body under the effect of gravity, which may be (a) linear or (b) rotary. They are shown schematically in Figs 1.7a and b, respectively. The balance equations for these gravimeters can be written as mg

+ W(x) = 0,

mglk(cp)

+ %R(cp)

=0

(1.10.1)

Here, g is gravity, 1 is the distance between the centre of gravity of the test body of mass m and the axis of rotation, x is the displacement, cp is the angle of rotation of the test body, W(x) is the reference force, W(cp) is the reference force moment, and mglk(cp) is the gravity moment. k(cp), W(cp), and %R(x) are nonlinear functions of cp and x, whose concrete analytical form depends on the gravimeter design. Equations (1.10.1) can be used to derive expressions for the sensitivity of gravimeters; that is, a relationship between the linear or angular motion and gravity acceleration:

Let us now carry out some estimations of the gravimeter systems in which the balancing force W(x) and force moment W(cp) are linear with respect to x and cp,

w-4= t1(x - xo),

W(cp) = %(cp

- cpo)

( 1.10.3)

where t1 and z2 are constants of the elastic elements (“rigidity”), xo and cpo are the initial deviations corresponding to the deformation of the elastic elements at g = 0. In the case of the design shown in Fig. 1.7b, k(q) = cos cp. Substitution of (1.10.3) for (1.10.2) gives the following simple expressions for the gravimeter sensitivity: dx_- -_ m _ dg

tl’

dcpml cos cp dg mglsincp - z2

--

(1.10.4)

The first equation indicates that the sensitivity depends on rigidity t and mass m. As can be inferred from the second equation, the sensitivity depends not merely on the mass m but on its product by arm 1. The sensitivity of rotary gravimeters (see Fig. 7b) can be enhanced, in accordance with (1.10.4), by ensuring that the gravity moment mgl is balanced out by the elastic force of the torsion wire twisted once or several times so that cp x 2pn. To estimate the sensitivity of gravimeters, it is useful to resort to relationships between sensitivity and the period of the oscillation of the test

51

The Gravitational Field of the Moon

(a)

(b)

Fig.l.7. Gravimeterswith (a) linearand (b) angularmotionofthetest bodyundertheeffectofgravity.S. spring; 0, axis of the horizontal torsion wire.

body in the gravimeter:

T 2 rnlk(0) -

1 T2

14n2 In other words, the sensitivity of static gravimeters is proportional to the square of the period T of free oscillation of the test body.

1.1 0.2 Dynamic Method

The classical dynamic instrument for measuring gravity g is the pendulum instrument. Its basic component is the pendulum undergoing free oscillation. It is known that g=-

471’1

MlT

where I is the moment of inertia of the pendulum with respect to the axis of oscillation, while M , I and T stand for the mass, reduced length, and free oscillation period, respectively, of the pendulum. Practical uses of pendulum instruments include relative measurements of gravity-that is, determination

52

Lunar Gravimetry

of the gravity difference between the measurement points and the datum point at which gravity go is assumed to be known. Then, the problem of measuring gravity at a point is reduced to determining only one quantity, namely, the change in the period of the pendulum’s free oscillation with respect to the datum point. It is assumed that the mass M , moment of inertia I , and reduced length 1 of the pendulum remain invariant. If the free oscillation periods at the measurement and datum points are T and To, respectively, then gravity g at the measurement point is 2

9 = go(;)

Pendulum measurements of gravity involve various correction factors introduced into the measured periods and taking into account the damping of free oscillation, differences in oscillation amplitudes, time variations of the pendulum temperature and its gradient, air density, sympathetic oscillation of the tripod, and so on. Should the need arise to measure lunar gravity to within f 1 mGal, the period of a quarter-second pendulum would have to be determined to within + 8 x l o w ’ s (Korzhev, 1979). In recent years, so-called vibrating-string gravimeters began to be used. Their operating principle is very simple. The upper end of an elastic string is secured to a frame, while suspended from the lower end is a test body. The gravity-dependent quantity is the frequencyf of natural transverse vibrations of the string, by virtue of its being related to tension under the effect of gravity by the relation ( 1.10.5)

where M and m are the masses of the test body and string, respectively, and L is the length of the string. The idea of using a vibrating string loaded with a test body for gravity measurements was proposed as far back as 1935 (Melikyan, 1938), yet the first gravimeter built on this principle was described only in 1949 (Gilbert, 1949). Subsequently, various designs of such gravimeters were proposed in the Soviet Union and abroad. They have been used to measure gravity from sea-going vessels, including submarines, on the bottom of shallow water basins, and in drilling wells. Vibrating-string gravimeters then began to be employed as accelerometers to measure acceleration in inertial navigation systems. Dynamic gravimeters also include instruments based on a balanced gyro. The amount of precession of such a gyro, which depends on the moment of gravity, is also a measure of gravity. Measurement of the absolute gravity on the Earth is one of the most difficult tasks of modern science: its determination on the Moon is even more

The Gravitational Field of the Moon

53

complicated. The absolute gravity determination provides a vivid example of the difficulties involved in absolute measurements of even a relatively simple physical quantity having length and time as its only dimensions. The most accurate and promising technique of determining absolute gravity is currently considered to be the free fall method. Gravity g is determined from measurement of the time it takes a test body to fall freely within an exactly known range of heights. On the Earth, an accuracy of several pGal has been achieved. No measurements of this kind have been carried out on the Moon, but lunar conditions are highly favourable for absolute gravity determinations, primarily because of the natural high vacuum. Also, the lower magnitude of lunar gravity, as compared to terrestrial gravity, results in a slower fall of a test body, making measurements easier. Of course, metrological measurements in outer space without direct participation of man are an extremely difficult task. The gravimeters intended for lunar gravity measurements must meet a number of special requirements: (1) First, the wide difference between the terrestrial and lunar gravitiesas great as 820Gal-must be compensated. The compensation may, in principle, be attained in a number of ways, depending on the gravimeter design (by tensioning a special compensating spring, by varying the voltage applied to a compensating capacitor, by removing some of the mass from the test body, by changing the tilt of the instrument, etc.). (2) The measuring system of the automatic lunar gravimeter must provide for digital data output as the most convenient and noise-immune form for transmission to Earth. (3) The lunar gravimeter must have an adequate thermostatic temperature control to maintain the temperature inside the instrument stable within 0.0014.01"C.Under lunar conditions, such temperature control is extremely difficult in view of variations in the external temperature from -100°C to + 120°C. (4) The instruments designed for unattended operation must have a sufficiently sophisticated automatic remote control system. Provision must be made for automatic levelling of the gravimeter, de-arresting of its sensitive system, activation of the measuring system, remote measurement of gravity, processing of the measurement results in a code suitable for transmission to the Earth, and transmission of the code. Then, the instrument must be arrested and ready for transportation to a new site. The accuracy with which gravimeters must be levelled is confined to a few minutes of arc. This can be done by positive shifting of the instrument into the necessary position with the aid of the sensor of the vertical designed on the free pendulum or level principle.

54

Lunar Gravimetry

The data recording and transmission procedures can be simplified if the measurement is carried out by an astronaut. Lunar gravimeters must withstand impact loads and considerable accelerations due to vibration of the running booster engine. ( 5 ) Another difficult task is calibration of lunar gravimeters or, in other words, matching the gravity values exactly with the gravimeter readings. Under the natural conditions of the Earth, gravimeters can be calibrated only within 5.3 Gal (difference between the gravity on the pole and the equator) near the mean value of 980Gal. Use of a gravimeter calibrated in this fashion on the Moon calls for reliable extrapolation into the range of g x 162 Gal. Some gravimeters can be calibrated additionally by the reversal method which will be considered in what follows in the context of a particular vibrating-string gravimeter design. (6) Lunar gravimeters are limited in mass, size, and power consumption. Overall analysis of various gravimeter designs does not permit one to select the optimal design for lunar gravity measurements. The selection of a gravimeter depends on, among other things, the tasks to be accomplished by gravimetric surveying-that is, on whether absolute or relative gravity values are the target, whether the scope of measurement is global or confined to detailing of a gravitational field within a limited area, and whether we are interested in the spatial distribution of the gravitational field or in its time variations at a certain point.

1.ll Direct Gravity Measurements on the Moon’s Surface

No matter how accurate, satellite methods cannot replace direct gravity measurements on the Moon with the aid of pendulum instruments and gravimeters. Satellite observations permit defining the anomalous portion of the gravitational field, averaged for certain areas. The linear dimensions of these areas amount to hundreds or, in the best case, tens of kilometres. It should be remembered that an ALS flying over the Moon’s surface covers about two kilometres per second. The acceleration acting upon a test body (in this case, the ALS) can be measured more accurately if it persists for a sufficiently long period of time, which is why the slower movements of the ALS and, of course, at low altitudes are preferable. The finer details of the gravitational field, associated with the density heterogeneities of the upper layers of the Moon and its surface features, can be detected using gravimetric instrumentation positioned directly on the lunar surface. So far, only four direct gravity measurements have been performed on the Moon’s surface, involving the Apollo 11, 12, 14 and 17 spacecraft (Nance, 1969, 1971; Talwani et al., 1973; Talwani and Kahle, 1976). According to

55

The Gravitational Field of the Moon

Nance (1969), the accuracy of gravity determination from Apollo 11 was & 13 mGal, which is about the same as for Apollos 12 and 14 where similar instruments were used. The instruments were in fact updated threecomponent accelerometers known as PIPA-pulsed integrating pendulous accelerometers, previously employed for gravity determinations at sea and in air from moving craft (Bowin et al., 1969). The accelerometers were mounted on board the lunar module which performed soft landing on the Moon's surface. If x, j ; and i' are the accelerometer readings in three orthogonal directions, then gravity g = (9+ j j z + zz)l'z. The absolute gravity measurements during landing of Apollo 17 were carried out by the astronauts using the traverse gravimeter TG developed especially for lunar gravity measurements at the Massachusetts Institute of Technology. Talwani and Kahle (1976) estimated the accuracy of absolute gravity determination to be k 5 mGal. This undertaking may be considered as the first sufficiently accurate measurement of the difference between the terrestrial and lunar gravities. Table 1.8 summarizes the results of direct lunar gravity measurements with the aid of the above instruments. In addition to the observed gravity values gobs, the table also gives the corresponding normal gravity values y. The latter were calculated assuming that the Moon is a uniform sphere. The Moon's rotation was ignored, and the selenocentric constant of gravitation was assumed equal to GM,

= 4902.71

km3 sec-2

The first gravimetric profile about 20 km long was obtained on the Moon with the aid of the above-mentioned TG gravimeter mounted on the lunar roving vehicle. Designed similarly to the TG is the vibrating-string gravimeter in which gravity is determined by measuring the changes in natural frequency of transverse vibrations of the string under the effect of gravity. The basic sensing element of the gravimeter is the vibrating-string accelerometer (VSA). It comprises (Fig. 1.8) two test bodies suspended from elastic strings TABLE 1.8 The results of direct gravity measurements on the Moon

Spacecraft

cp

I

Elevations of landing sites under sphere with R = 1736 km (km)

11

0'40 N 3"12'S 3"W s 20"13' N

23"29' E 23"24 W 17"28 W 30"42' E

-0.53 0 0.39 1.19

Landing site coordinates

Apollo Apollo Apollo Apollo

12 14 17

Observed gravity value, G (mGal) 162852 13 162674 162653 162695 & 5

56

Lunar Gravimetry

Fig. 1.8. Sensitive element of a vibrating-string accelerometer. 1, test bodies; 2, strings; 3, spring; 4 solenoids.

made of beryllium bronze. The test bodies are interconnected by a soft spring and linked with the housing via braces eliminating transverse displacements of the test bodies. The strings are arranged between the poles of permanent magnets, and, when an a.c. voltage is applied to the strings, they start vibrating transversely. As has already been mentioned, frequency is dependent on the gravity acceleration g acting upon the test bodies with mass M and on the string tension F o due to the intermediate spring. In the case of a vibrating-string accelerometer with a tensioning spring, the frequency f of transverse vibration is given by the formulaf= $[(Fo + M g ) / L m ] ” 2 , rather than (1.10.5).Because of technical difficulties, it is impossible to make L, M and rn exactly equal in the upper and lower accelerometers, which is why the difference between their frequencies (fi -fi) is also dependent on the even power exponents of g : F

=fi

-fz

= ko

+ k l g + kzg2 +

1 . .

(1.11.1)

In general, F is a nonlinear function of g . If L, M and m had been equal, all the terms with even power exponents of g would have been equal too. The feequency F was measured at each measurement point, and the gravity difference at these points was determined from the difference between the corresponding values of F. It is assumed that the coefficients ko, k l and k z in

57

The Gravitational Field of the Moon

(1.1 1.1) are known. The terms nonlinear with respect to g are necessary only in determining wide gravity differences, such as in the case of terrestrial and lunar gravities. The gravimeters were designed to allow for periodic checking of the coefficient ko on the Moon, To this end, the sensitive system of the gravimeter may be turned through 180". Equation (1.11.1) holds for the reversed system with substitution of -g for g.

F,

=

ko - klg

+ kzg2

(1.11.2)

Measurement of F and F , at the same point permits the coefficient ko to be determined using (1.1 1.1) and (1.1 1.2). Such measurements were performed at every point along the path of the lunar roving vehicle. This served as an additional check of the zero point drift. The frequency was measured in the gravimeter automatically by way of comparison with the reference frequency of 100 kHz of the quartz oscillator mounted inside the gravimeter. The gravity measurement results were digitized and transmitted to the astronauts in the form of a pulsed code. The vibrating-string accelerometer was placed in a gimbal mount which allowed the accelerometer to rotate within 30". The rotation was imparted by a step motor associated with the sensor of the vertical. The sensor was essentially a pendulum with two degrees of freedom. It produced a continuous signal indicative of position with respect to the vertical. The faster levelling was performed up to gravimeter tilts equal to 32', whereas the slower levelling was carried out within k 32'. A step of the motor corresponded to a + 1 ' tilt of the instrument. The overall levelling accuracy is estimated to be f7'. The instrument was provided with a twostep thermostat rated at 50°C. The outer thermostat module operated on an on/off principle, while the inner one worked continuously. The temperature stability inside the instrument was maintained to within +0.005"C. Additional thermal insulation was provided (multilayered coatings, gold-plated surface, etc.). The total capacity of the battery supplying power to the entire gravimeter was 300 Wh over a period of fifteen days. Special shock absorbers were used to minimize the effect of vibration, impacts and g-loads both during flight toward the Moon and at landing. The gravity measurement range was 170 Gal. The overall dimensions of the gravimeter were 48 cm x 26 cm x 23 cm. It weighed a total of about 15 kg. Figure 1.9 shows the profile of "free-air" gravity anomalies in the Taurus-Littrow Valley near the south-eastern edge of Mare Imbrium, obtained by with the aid of the T G gravimeter (Talwani and Kahle, 1976). The measurement accuracy is k 2 mGal. Of particular interest is the possibility of using self-contained capsules with gravimetric instrumentation, hopping (flying) from one lunar site to another. Relative gravity measurements will be taken at each new site with the horizontal and vertical coordinates of the capsule being determined with a

+

58

Lunar Gravimetry

LM

Lo ,~ I

-6

I

, I

-4

1. Y

, I

LM Y

0

-2 0 Distance (km)

2

Fig. 1.9. Profile of gravity anomalies across the Taurus-Littrow Valley, derived with the aid of the TG gravimeter. 1, gravities measured with respect to the landing site of the LRV; 2, free-air anomalies.

high degree of accuracy if accelerations are measured in transit with the aid of a three-component accelerometer. This approach to investigating the gravitational field seems to be energetically more efficient than the use of lunar roving vehicles. Let us now estimate the possible accuracy of determining the coordinates (x, y, z) of the observation point with respect to the initial point. If use is made of accelerometers whose acceleration , accuracy of determinmeasuring accuracy is E . ~= ci; = E! = 0.01 cm s - ~ the ing the increments Ax, Ay and Az of the initial coordinates is

Let the flight over a distance of 5 km take 10 s. This means that, while measuring acceleration with the above accuracy, the acelerometer will permit the coordinates of the new point to be determined, relative to.the initial ones, with an accuracy of E, = cy = E, = f0.5 cm. This chapter has been concerned with gravity measurements. Of greater interest are measurements of gravity gradients on the lunar surface; however, none have been carried out so far.

59

The Gravitational Field of the Moon

1.12 Studying Second Derivatives of the Lunar Gravity Potential

The main advantage of the method proposed by Muller and Sjogren is that it provides information about the gravitational field “free of charge” (as Forward put it) because this method does not require any special instrumentation using, instead, measurements of the carrier frequency of the telemetering transmitter. However, the method has a drawback-measurements are possible only on the near side of the Moon. Information from the far side can be transmitted only via a special relay satellite placed in a high-altitude circumlunar orbit. Yet, even when the experiment is thus made much more complicated and expensive, about 40% of the far-side trajectory will be beyond the tracing zone. Therefore, global mapping of line-of-sight accelerations will take several months with the terrestrial stations operating full time. Hence, it would be of particular interest to carry out independent measurements of the gravitational field with the aid of gravimetric instruments on board ALS. The results of such measurements, as opposed to Doppler tracking data, can be stored and transmitted to the Earth at any time convenient for communication. By virtue of the equivalence principle, a freely flying spacecraft cannot be used to measure the gravitational intensity-that is, the vector g = V V. The physically measurable quantity in this case may only be the difference between the accelerations of test bodies separated in space, which is determined, by the second and higher spatial derivatives of the gravitational potential V(r). This is why independent measurements of the gravitational fields of planets, their satellites, and asteroides from spacecraft are possible only using gradiometers. The second derivatives of the gravitational potential V make up a symmetrical tensor of rank 2, which takes the following form in the local rectangular basis:

Vik

=

Vxx

Vxy

Vxz

Vyx

Vyy

Vyz

Vzx

VZY

Vzz

The diagonal components are known as in-line gradients, and the nondiagonal ones are referred to as cross-gradients. The spur of the tensor Vik equals zero, according to the Laplace equation

therefore, this tensor is unambiguously determined by five independent

60

Lunar Gravimetry

tY

Fig. 1.10. Force moment acting upon a dumbbell in a nonuniform gravitational field. x, y, z, local rectangular base.

quantities, namely, three cross-gradients V,,, V,,, V,,, and any two in-line gradients. All existing gradiometers and those currently under development can be classed as being in one of two groups. The instruments belonging to the first group measure the force moment M acting upon an extended body in a nonuniform gravitational field. For example, if two point bodies with mass m are interlinked by a weightless rod whose length is 2R (Fig. l.lO), such a dumbbell is acted upon by,a force moment rotating it about the axis z : M,(cp)

= 2mR2( VAsin

2cp

+ V,, cos 2cp)

(1.12.1)

In Eotvos’ gradiometer, the dumbbell is suspended from a thin torsion wire, and the gravitational moment is measured by the angle of its twisting. Such a gradient measurement method may be called static. Another modification of the static method is used in the gradiometer developed at the Charles Stark Draper Laboratory (CSDL) (Fig. 1.1 1). In this gradiometer, the dumbbell is enclosed in a spherical envelope immersed into a special liquid where it has zero buoyancy. Such a suspension provides the dumbbell with all six degrees of freedom. Highly sensitive transducers of linear and angular displacements control electromagnetic servomotors maintaining the orientation of the dumbbell along the x axis. The outputs of the gradiometer is compensating force moments with respect to the y and z axes: M , = VxyA J ,

M y = - V,, AJ

where AJ = J,, - J,, N J,, is the difference between the principal moments of inertia of the sensor. Under laboratory conditions, the sensitivity of the gradiometer has been

61

The Gravitational Field of the Moon

',

I

a- , '

'\

m

.I

Fig. 1.11. CSDL gradiometer

brought to tenths of an eotvos at an averaging time of 10 s (Ames et al., 1977). Development of a low-noise liquid suspension has turned out to be a technically difficult task; in particular, thermostatic control to within 10-6"F was necessary. In the existing form, the three-axis version of the CSDL gradiometer is rather cumbersome (100 dm3, 100 kg) and consumes about 20 W. It should be pointed out, however, that the instrument is intended for terrestrial and marine measurements. There is every possibility that in the satellite modification some of the technical problems will be obviated. According to Heller (1979), the most promising application would be the use of the sensing element of the CSDL gradiometer as a test body in a drift-free satellite. Under conditions of complete weightlessness, the liquid suspension would no longer be needed and, theoretically, the intrinsic noise could be reduced by several orders of magnitude. There is no other information concerning the development of a satellite version of the CSDL gradiometer. The static method is not the only way to measure gravity gradients. Indeed, let us consider the equation describing the motion of a dumbbell suspended from a torsion wire: J$

+ Hqj + D(cp - cpo) = J ( V Asin 249 + V,, cos 249)

'

(1.12.2)

Here, cp is the angular coordinate of the dumbbell in the xy plane, J = 2mRZ is the moment of inertia, H is the coefficient of viscous friction, D is the torsional rigidity of the wire, cpo is the angular coordinate of the dumbbell at which the moment of elastic forces is nil. Since the second derivatives of the

62

Lunar Gravimetry

potential of real gravitational fields are small, the deflections of cp from the balanced cpo, caused by them, may also be considered small:

*=

cp - cpo > l), and the optimal algorithm for finding VxYand V,, will be sin 2pt cos 2pt 0

which corresponds to synchronous detection of the y(t) signal. Since noise p(t) is present in the y(t) signal, (1.12.10) yields a random value. It can easily be shown that its variance is D ( 4 = N1IEo Substitution of N 1 = ~ x T H J into - ~ this formula and calculation of Eo for each measurement method gives: (1) For the static method: S(t) = 1;

Eo

= 2;

4x TH

D(f1) = Nl/z =J 22

(1.12.11)

69

The Gravitational Field of the Moon

(2) For the dynamic method: S ( t ) = 211/(t) = 2t,b0 sin (mot + Do);

(1.12.12) 2xTH D(f,) = Eo = 211/;z; 21+@ ll/$Jzz (ll/o is the dumbbell oscillation amplitude determined by the initial conditions). (3) For the modulation method: 8x TH D(P,,) = D(P,) = (1.12.13) J 2z N1

~

~

Formulae (1.12.11)-(1.12.13) permit the different methods for gravity gradient measurement to be compared in terms of the limit of accuracy, determined by thermal fluctuations. As can be inferred from (1.12.12), the thermal limit of accuracy of a dynamic gradiometer depends on the initial dumbbell oscillations amplitude. Since the initial equation (1.12.5) holds only for lowamplitude oscillations when ll/o 0;that is, if Dn+ > d,+ 1. Thus, the optimal order of the expansion depends on the ratio between the degree variances of the errors involved in the determination of the coefficients and those of the coefficients themselves. The first expansion of the Moon’s relief in spherical functions up to the eighth order was carried out by Goudas (1968) who used absolute elevations of the near-side relief. The total lack of data on the Moon’s far side has given rise to certain assumptions. Goudas proceeded from symmetry of the relief on the near and far sides. This is why his expansion could adequately describe the near-side relief and have nothing to do with the far-side relief. Since the first expansion of the Moon’s relief, more data on the latter have become available. The early selenodetic catalogues of absolute elevations on the near side were revised and compiled catalogues have been prepared (Gavrilov et al., 1977; Lipsky et al., 1973; and others). The elevations were

136

Lunar Gravimetry

reduced to the Moon’s centre of mass. The basic methods for compiling unified catalogues have been outlined (Gavrilov, 1969; Lipsky et al., 1973). The principles of establishing base networks and their relative deformations, techniques of lunar polygonometry and triangulation have also been described (Gavrilov, 1969; Gurshtein and Slovokhotova, 1971; Habibullin et al., 1972; Light, 1972; Helmering, 1973). The most serious drawback of the available elevation data is the limited knowledge about absolute elevations on the far side. This drawback has begun to be remedied by satellite observations, which have been instrumental in the determination of the selenodetic longitudes, latitudes, and absolute elevations of several fundamental base points. In particular, Wollenhaupt et al. (1972) used the results of numerous optical measurements from Apollos 8, 9, 10, 11, 12, 14 and 15 to determine 31 base points along the equatorial belt of the Moon. Their latitudes range from - 1 1 ” to + 26”;21 out of these points are on the near side and 10 on the far side. The coordinate determination errors are 0.7 ( q ) and 0.6 (A) km, while those of absolute altitude determination do not exceed 0.4 km. The coordinate errors were determined by the accuracy of location of the Apollo spacecraft. Extremely valuable data on absolute elevations were obtained by laser altimetry from Apollos 15, 16 and 17 (Wollenhaupt and Sjogen, 1972; Kaula et al., 1973, 1974; Sjogren, 1977). Elevations of the physical surface were measured along profiles (Fig. 2.5) extending across the entire Moon in its equatorial zone. They are given on the profiles relative to a spherical Moon with a radius of 1738.0 km. The centre of this sphere coincides with the Moon’s centre of mass. Laser measurements were taken from the orbiting Apollo spacecraft at 20 s intervals, which corresponds to 30 km of covered distance. Although the altimeter was sensitive to within about 2 m and the spot of the laser beams on the lunar surface was only about 30 m in diameter, there was no need to probe at shorter intervals because the uncertainty in the Apollo’s coordinates remained considerable. The position of the spacecraft was determined from the Earth by Doppler tracking at 10 s intervals. For intermediate points in time, the position of the spacecraft on its orbit was computed using a model of the lunar gravitational field. The uncertainty in the Apollo’s position in the direction of the laser beams did not exceed an absolute value of 0.4 km, while the error in the relative position between measurements was 0.1 km. Laser altimetry has made it possible, in addition to studying the Moon’s relief, to solve a number of other problems. In particular, the position of the centre of the Moon’s figure with respect to its centre of mass was determined more accurately and the geometrical figure of the Moon was defined more exactly. Some common features of the Moon’s relief were established. Table 2.7 lists the mean elevations of some features on the Moon’s near and far sides

-E *

1

re

4

o

U

-4 -81

I

90"

I

I

120"

I

I

150"

I

I

180"

I

210"

I

1

240"

I

I

270"

I

1

300"

1

I

330"

I

I

360"

I

I

30"

I

I

60"

I

I

90"

I

8-

-

-E @ Y U

1

:i-flkg+q% --4

-

-

-----

------

#

------

Fig. 2.5. Intervals in elevations of the physical surface on the near and far sides of the Moon, relative to a spherical Moon with radius R = 1738 km, measured by laser altimetry from Apollo 15 (top) and Apollo 16 (bottom) (Kaula eta/., 1973, 1974; Siogren and Wollenhaupt. 1972, 1973).

138

Lunar Gravimetry

TABLE 2.7 Mean elevations of some formations on the Moon, relative to a sphere with radius R = 1738 km (Kaula ef a/., 1973)

Mean elevation according to (km) Formation type Far-side highlands Near-side highlands Circular maria Other maria

% of

Apollo

surface

15

57 23 6 14

+ 1.9

Apollo 16

Apollo

f2.1

+0.9 - 1.3 -3.7

- 1.7

- 1.2

-4.1 -2.0

-4.1 -2.5

17

-2.1

Weighted mean elevations (km)

+ 1.8

- 1.4 -4.0 -2.3

above a sphere with R = 1738 km, determined from these measurements (Kaula et al., 1973). The tabulated elevations belong to the equatorial zone and are not adequately representative of the Moon as a whole. The elevation profiles constructed from laser altimetry data (see Fig. 2.5) attest to the qualitative difference in relief between the near and far sides of the Moon. The maria have smooth floors inclined from west to east at about 1 : 500 to 1 :2000 (Kaula et al., 1973). An inverse correlation is observed between the depths of circular maria (Mare Serenitatis, Mare Crisium, Mare Smythii) and their ’diameters. The interferometric measurements with the aid of Earth-based radars, which preceded laser altimetry, allowed determining the elevations of only large features of the lunar surface, their accuracy being in the neighbourhood of 200 m (Zisk, 1972). The first data on the far-side relief elevations in the western hemisphere were provided by the photographs taken from the automatic probe Zond 6. They revealed a wide depression on the southern far side (Rodionov et al., 1971, 1976; Ziman et al., 1975). The level of the physical surface over a sizeable area (Fig. 2.6) was about 4.7 km below the mean level of the Moon’s surface. The upland region in the south, having a relative elevation of about 2.6 km, extends into Mare Australe on the near side. Satellite data on the geometrical figure of the Moon cannot substitute for the knowledge accumulated over more than fifty years of astronomical investigation of lunar surface elevations. Yet they provide the only insight into the Moon’s far side. It was extremely important to obtain information on the overall geometrical figure of the Moon with due account for both the near and far sides as well as its relationship with the centre of mass, which has been partially accomplished by way of laser altimetry. The results of the photogrammetric processing of the pictures taken from Apollos 15,16 and 17 are going to be used in establishing a highly accurate base network covering

139

Normal and Anomalous Gravitational Fields

0"

r

+

Fig. 2.6. The Moon's physical surface elevations, in a section from a plane near to 2. = 180", from the photographs taken from Zond 6 with a 20-fold magnification (Rodionov et al., 1971).

20% of the lunar surface (including its far side) and having a density of one base point per 900 km2 (Light, 1972; Helmering, 1973). The accumulated elevation data have made it possible to achieve more or less reliable expansions of the lunar relief in spherical functions. Bills and Ferrari (1977) handled in their work the results of 5800 laser altimetric measurements, 1400 photographs taken from Apollo spacecraft with an accuracy of f0.3 km, and 3300 elevation measurements based on photographs taken from the Earth. The latter were said to be accurate to within f 1.0 km. The terrestrial measurements were corrected for displacement of the Moon's centre of mass by 1.77(f0.16) km toward a point with coordinates 25"s and 191"E. The elevations were determined with respect to a sphere with R = 1737.46 km. The map of elevations of the smoothed Moon's relief, based on the work by Bills and Ferrari (1977), covers the zone confined within zk 45" latitude. The elevations vary from +5.5 to -2.5 km; that is, the total variation range is 8 km. Proceeding from the up-to-date lunar relief elevation data, Chuikova (1975a, 1975b, 1978) expanded the relief. She used a hyposometric map of the

140

Lunar Gravimetry

Moon’s near side confined within f70” latitudinally and logitudinally, based on Mills’ catalogue, the coordinates of the above-mentioned 3 1 base points (Wollenhaupt et al., 1972), the results of laser altimetry from Apollos 15 and 16 (Sjogren and Wollenhaupt, 1973), the absolute elevations of 68 points of the liberation zone on the far side in the Moon’s western hemisphere, determined from space probes Zond 6 and Zond 8 (Ziman et al., 1973)’ the elevations determined with the aid of Zond 6 along the far-side meridional profile (Rodionov et al., 1976), as well as the catalogue of elevations in the peripheral region, compiled at the Main Astronomical Observatory of the Ukrainian Academy of Sciences of particular interest were the data provided by Zond 6 and Zond 8; they have not been used in other works. This is precisely why on the averaged relief map drawn in this work (Fig. 2.7) the regions in the southern part of the western hemisphere, polar regions, and the peripheral region better represent the real Moon. Consider now the degree variances, not of the relief elevations themselves, but of their horizontal gradients along tangents to meridians and parallels. The expressions for these degree variances as applied to the relief will take the forms n (Dh,)n =

+ Br%)

m=O

(2.10.3) The coefficients A,,,, Brim, R,, and fin,,,can be computed according to the formulae (3.3.3) where the coefficients a,,, and 6,,,of expansion of the relief Figure 2.8 presents graphs elevations must be substituted for Cnmand Snm. showing the degree variations in r.m.s. values ahq = [(Dhq)n/(2n and a h l = [(Dh,),J(2n 1)]’/2 calculated using the relief expansion coefficients (Chuikova, 1978). Autocorrelation functions of the Moon’s and the Earth‘s reliefs are shown in Fig. 2.9. Let us examine the anomalous gravitational field due to the Moon’s relief. The masses constituting the visible relief will be represented condensed on a sphere’s surface as a simple layer with surface density

+

+

On(%

4= aowcp, 4= N

aoR

n

1C n=l

(anm

cos mA

m=O

+ bn, sin ml)P,,(sin

cp)

(2.10.4)

where a. is the mean density of the relief-forming (2.10.4) rock. The gravity potential of the masses in the simple layer is V(P, cp,

4=G

jj S

CP,1)dS

r

Fig. 2.7. Elevations of the lunar relief relative to a sphere with radius R = 1738 km (isolines taken at 0.5 km intervals)

r;

40'

c

I I

'E Y E 30 E

I

I

IIII

20

n Fig. 2.8. R.m.s. values of the spherical harmonics of the horizontal relief elevation gradients along meridians and parallels plotted against position in degrees. (1 ) (uhJn along meridians; (2) ( u , , ~along )~ parallels.

143

Normal and Anomalous Gravitational Fields

where I is the distance between a current point on the surface and the point (p, cp, A) at which the potential is considered. Substitution of the expansion for l/r (1.3.2) and that for 6, (2.10.4) into this equation gives V(p, cp,

A) = GaoR

ss

c N

n

1(anrncos rnl + 6,, sin rnA)Pnm(sincp) n = 1 m=O

S

In expansion (1.3.2),it was assumed that p1 = R. By resorting to the theorem of restoration of spherical functions (Idelson, 1936), we can simplify the righthand side of the latter equation:

x

(anrncos rnl + li,

sin rnA)Pnm(sincp)

(2.10.5)

Comparison of the coefficients of the resulting expansion with the corresponding ones of expansion (2.3.4)gives (2.10.6) Introducing the Moon’s mean density au instead of its mass Mu, we can of the gravitational potential expansion to relate the coefficients Cnmand Snm

Fig. 2.9. Normalized autocorrelation functions KOn($) of the (1) Moon’s and (2) Earths reliefs.

144

Lunar Gravimetry

the corresponding coefficients an, and 6,, of the relief expansion in the following manner: (2.10.7) Assuming that the Moon’s density varies with depth as a(p) = 0 0

+ upp

where u and p are constants governing the density variation pattern, instead of formula (2.10.7), we have (Goudas, 1968, 1973): (2.10.8)

Assuming that u = 0, we have (2.10.6) instead of (2.10.8), which corresponds to the relief density being constant and equal to the mean density of the Moon. Given in Table 2.8 are the values of normalized harmonic coefficients C,,, and S,,, of the gravitational potential, calculated using formula (2.10.6) based on the coefficients of the Moon’s relief expansion executed by Chuikova (1975a,b) as well as by Bills and Ferrari (1977). This table also lists for comparison the harmonic coefficients derived by Ferrari (1 977) and Akim and Vlasova (1977) from the perturbed motion of the ALS. The values of en, and S,,,, corresponding to the relief, are much higher. Also compare the degree variances of the gravity potential calculated with reference to the Moon’s relief and determined from tracking of the perturbed motion of the ALS (Fig. 2.10). The first degree variances are much greater than the corresponding variances determined from the perturbed ALS motion. The TABLE 2.8 Comparison of the normalizedharmonic coefficientsof the gravitationalpotential, calculatedfrom the observed Moon’s relief and determined from ALS tracking data ( x

Go

Gl

From relief (Chuikova, 1975)

-607.6

-603.8

From relief (Bills and Ferrari, 1977)

-212.3 f25.8

-605.8 f17.5

Determination

Go

Ctl

G 2

-21.9

-119.3

-78.98

87.23

-147.5 f13.6

-135.9 f22.1

-149.8 f26.5

11.5 f5.7

-91.52 51.7

2.04 f1.8

f 1.8

-90.03 f 1.0

0.7 f1.4

35.47 f0.5

&I

From ALS tracking data (Ferrari, 1977) From ALS tracking data (generalized model)

0.58 f 1.7

1.97 f9

1.3 k1.4

33.66

145

Normal and Anomalous Gravitational Fields

n Fig. 2.10. R.m.s. values of the spherical harmonics ( u T ) , for the lunar gravitational potential plotted against degree, n. (1 ) Calculated from the Moon's relief; (2) determined from ALS tracking data.

difference diminishes as the order n of the harmonics increases. This pattern can be explained by the different degree of isostatic compensation of masses differing in extent. The relief masses over small areas cannot overcome the strength of the lunar crust and remain isostatically uncompensated, which is to say that they are responsible for a stronger gravitational field. As regards regions characterized by low harmonics, the isotatic compensation is virtually complete; that is, corresponding to excess of relief masses is a

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146

Lunar Gravirnetry

deficiency of mass below and vice versa. As a result, the anomalous gravitational field approaches zero, as can be seen on the grounds of Fig. 2.10. It can be inferred from Table 2.8 that the geometrical figure of the Moon cannot be derived from the coefficients of expansion of the lunar gravitational field determined from perturbations in the ALS motion. This is precisely what Volkov and Shober (1969) did following Goudas (1968, 1973). Disregarding the internal mass distribution will not produce anything close to the actual relief of the Moon.

Chapter Three

Spatial Variations in the Lunar Gravitational Field and Their Use in Studying the Figure and Internal Structure of the Moon

3.1 Covariance Analysis of the Moon's Gravitational Field

The currently existing methods for measuring the Moon's gravitational field yield information only within a discrete set of points with inevitable random errors. Handling of the gravitational field in most selenodetic and selenophysical problems reduces to its integration with a particular kernel in a certain region which, as a rule, comprises the entire surface of the Moon. There arise problems of error filtration and interpolation (prediction) of gravity anomalies in areas inadequately covered by measurements, if at all. Kaula (1959, 1967) and Heiskanen and Moritz (1967) demonstrated that these problems can best be resolved by applying the Kolmogorov-Wiener linear filtration method well known in the theory of stochastic processes. The Kolmogorov-Wiener method is based on examination of the statistical relationship involving the parameters of a random process (field) at different points. Such a relationship is characterized by a covariance function determined as the expected value of the product of the values of a random

148

Lunar Gravirnetry

function f(x) at two different points: K = E(f(Xdf(X2))

(3.1.1)

The argument x is essentially time t , if f ( t ) is a random process, or the coordinates of points in space, if f(x) is a random field. The way (3.1.1) is written implies that the random function f(x) is centred; that is, its mean value is zero:

3=~ { f ( x )=l 0

(3.1.2)

Otherwise, deviations f ( x ) -7must be substituted for f ( x ) in expression (3.1.1). E is an operator averaging over all possible realizations of the random process (field f ( x ) . In reality, we are always dealing with a single realization of the gravitational field of any object which is invariable in time, including the Moon. Then, the question arises: can we regard the Moon's gravitational field as a realization of a random field, and if so, what is the exact definition of the averaging operator E? This question and its implications were covered at length by Moritz (1980). Random processes f(t) are known to exist whose covariance can be found unambiguously from any single realization. Substituting the following operator of averaging over the function domain for the expected value operator E , we obtain: E = {f(t)f(t

li

+ At)} = T

f(t)f(t

0

+ A t ) dt

(3.1.3)

The random processes exhibiting this property are referred to as ergodic. As can be inferred from (3.1.3), the covariance function of an ergodic process does not depend on the values of the variables t l and t2 themselves, but is determined by the interval between them or, in other words, it is a function of the variable At. In his work, Moritz (1980) demonstrated that it is possible to find random fields f(0,A) on a sphere which also exhibit the property of ergodicity. The covariance function of such fields K = M{f(P)f(Q))

(3.1.4)

must be dependent only on the angular distance Ic/ between the points P(0, A) and Q(W,A'). Therefore, the averaging operator M is derived a s follows. First, one of the points (say, P ) is fixed, then, the product f(P)f(Q) is averaged over all points at a given angular distance from P (Fig. 3.1). This is done through integration with respect to the azimuth a of the point Q between 0 and 2.n and provides for isotropism of the operator M. The resulting mean value with

+

149

Spatial Variations in the Lunar Field

Fig. 3.1. Averaging of the product over all points at a given angular distance

respect to a is then averaged over all possible positions of the point P on the sphere, which ensures uniformity of the operator M. To simplify the subsequent expressions, we shall denote the averaging over the sphere by operator M p : ’

.

2n r

n r

(3.1.5) .

.

e=o A = O

where 8 and l are the coordinates of the point P. In this case, the complete operator for finding the covariance (3.1.4)can be written as 2n

1

M{.} = 2A

Mp{-}dcr

(3.1.6)

a=O

Assume that the values of the anomalous lunar gravitational potential T (8, A) on a sphere with radius R can be regarded as a realization of an ergodic random field. The validity of such an assumption was treated at length by Moritz (1980). Therefore, we shall not dwell on this problem but proceed directly to computation of the covariance function of the anomalous potential. The term “anomalous lunar gravitational field will be used here and in what follows to denote the field of lunar mass attraction ( V )minus the central term

150

Lunar Gravimatry

For the anomalous potential T defined in this manner, the expansion in spherical functions begins with n = 1: P

n=l

(Cnmcos mA + S,, sin rnA)Fn,(cos e) (3.1.7)

To simplify the formulae, let us introduce the following designation of normalized spherical functions and harmonic coefficients: v

in

With such designations, the series (3.1.7) takes the following form: (3.1.9) where P is the point with coordinates p, 8, and A. The relations of orthonormality of the spherical functions (3.1.8) take the following simple form: M p { ynrn(P)xk(P)} = dnidmk

(3.1.10)

The mean values of the spherical functions are zero: MP{ Ynm(P)} = 0

therefore, the anomalous potential T is centred: MP{T(P)} = 0

and, according to (3.1.4) and (3.1.6), we have the following expression for the covariance function K: 2%

(3.1.11) Let us find K ( $ ) in the form of expansion in Legendre polynomials Pn (cos JI): m

K($) =

C D~P~(cos JI) = i=O 2n

M p { ( T ( P ) T ( Q ) }da 0

(3.1.12)

151

Spatial Variations in the Lunar Field

Multiply both sides of this equation by Pn(cos $) and integrate with respect to $ between 0 and 7c with weight sin$. In view of the orthogonality of Legendre polynomials, we have

11 n

2 2n+ 1D

n=g

Zn

yl=O

P,(cos $)Mp{T(P)T(Q)}sin $ d$ da (3.1.13)

a=O

The right-hand side of (3.1.13) can be rearranged in the following manner:

{

n

Dn = (2n + l)Mp T(P) x

d, j

2n

-

v=O a=O

The double integral in the right-hand side of (3.1.14) is nothing but an operator of averaging over the coordinates of the point Q on the surface of a sphere (see Fig. 3.1): n

Zn

jj

47c -

yl=O

Pn(cos $)T(Q) sin $ d$ dcr = MQ{P,(COS$)T(Q)} (3.1.15)

-

a=O

We shall use the known formula of addition of spherical functions for its computation. In the above-adopted notation, this formula takes the following simple form:

Multiply both sides of (3.1.16) by T(Q) and apply the operator MQto them: in= - n

With the orthonormality relation (3.1.10), at p = R, we have GM MQ{Ynm(Q)T(Q)} = 7 Ynm

(3.1.18)

Substitution of this expression into the right-hand side of (3.1.17) gives

(2n + 1)MQ {Pn(COS $)T(Q)} =

GM R 1

m= -n

Then, according to (3.1.14) and (3.1.15), we have

YnmYnm(P)

(3.1.19)

152

Lunar Gravimetry

Rearranging the right-hand side of this equation and using formula (3.1.18) again, we obtain the coefficients Dn of interest: (3.1.20)

or, if we turn back to the usual harmonic coefficients, (3.1.21)

Thus, the covariance function of the anomalous potential on a sphere with radius R is

A similar approach can be used to find the covariance for points located on different spheres with radii p1 and p2 when P = P (pl, 8, A), Q = Q ( p 2 , 8’,X), and the angular distance between P and Q is still In this case, the same formulae are used, with the difference that a general potential expansion of the (3.1.7) and (3.1.9) type is substituted for T As a result

+.

K(4h P1, P 2 ) =

c (Ky+’ DnPn(cos$) PlP2

(3.1.23)

n=l

where the coefficients Dn are determined using the same formula (3.1.21). Let us assume that P = Q in formula (3.1.4); that is, )I = 0 and p1 = p2 = p. Then, from (3.1.23) we derive

K(P, P ) = M ( T 2 ( P ) } that is, the mean value of the square of the potential anomalies over the entire sphere. If we use statistical terminology, this quantity should be called “anomalous potential variance”. Denoting it by D, we have m

D=K(O) =

1 Dn n=l

As regards D,, they have the meaning of variances of the nth harmonic of the anomalous potential and are known as degree variances: Dn

=

MP { T m }

The quantities Dn considered as functions of the harmonic number n form a spatial spectrum of the gravitational field potential, which makes studies into degree variances a matter of special interest.

153

Spatial Variations in the Lunar Field

3.2 Degree Variances and Covariance Functions of Various Characteristics of the Gravitational Field and Figure of the Moon

The above-derived formula (3.1.21)for degree variances applies not only to the anomalous potential, but also to any other characteristic of the gravitational field if the coefficients of its expansion in spherical functions (3.1.7)are known. Hence, we can immediately write, the expressions for the degree variances of the quantities for which transformation factors had been obtained (Tables 2.5 and 2.6),namely, for the level surface elevations [,mixed gravity anomalies Aqm, gravity anomalies T p , anomalies of the vertical gravity gradient T,, and anomalies of the second vertical gravity gradient Tppp. Here are the formulae for the degree variances of these quantities on a sphere with radius p > R:

(3.2.1)

The covariance functions of the above characteristics of the gravitational field on a sphere with radius p are determined from formula (3.1.12): K { .1 =

2 Dn{ .}Pn(COS $1

(3.2.2)

n=2

where Dn{ . } stands for the corresponding degree variances determined from formulae (3.2.1).For example, m

KT,,(+,P ) =

1 ~n(T,)Pn(COS+)

n=1

=

(Fy2

n=l

(n + 1)2('yn+2pn(cos

+) m = O

(ctm+ s:.) (3.2.3)

The general rule in the computation of covariance functions of various characteristics of the gravitational potential is as follows (Moritz, 1980). Let

154

Lunar Gravimetw

j and g be arbitrary characteristics of an anomalous gravitational field, which can be represented as linear functions of the anomalous potential T: f=FT

(3.2.4)

g=GT where F and G are linear transformation operators. Consider the quantities f ( P ) and g(Q) at two different points P = P ( p , cp, 1)and Q = Q(p', cp', A'), the angular distance between them being II/: cos $

= sin cp

sin cp'

+ cos cp cos cp' cos (1

-

A')

(3.2.5)

According to (3.2.4), these quantities can be written as (3.2.6) where the subscripts in FP and GQ indicate that F operates upon the coordinates of the point P, while G operates upon the coordinates of the point Q. Let us derive the covariance .off and g: K,g = M{FPT(P)GQVQ)) = M{FPGQT(P)T(Q)}= FpGpM(T(P)T(Q)} = FPGQK(P,Q)

where K ( P , Q) is the covariance function of the anomalous potential. Formula (3.2.7) provides the desired law of transformation of covariances. Its derivation involves the commutativity of the averaging operator M [see (3.1.6)] as well as the operators F and G. A proof of this generally obscure fact can be found in Morirz's work (1980). Let us illustrate the application of formula (3.2.7) by the following example. Suppose we want to find the covariance between the selenoid elevations i= T/y on a sphere with radius p = p1 and the mixed gravity anomalies bg, = -(aT/ap) - 2T/p on a sphere with radius p = p z . Then, the operator F takes the form l/y while the operator G = (a/ap) - 2/p. According to formula (3.2.7), we have 1 aK

KC, &,

hence

=

2~

-i [Q-k p']'

155

Spatial Variations in the Lunar Field

In particular, if l and Agmare given on the same sphere with radius R, 1

m

Ks,A~,(+> = - C (n - 1)DnPn(cos yR n = l Here is another example. We want to find the covariance between the horizontal components of the plumb-line deflection:

1 aT ( ( P ) = ~- and YP1 acp

1 v(Q) = yp2 cos cp'

aT

. anl

According to (3.2.7), K