Physics of the Sun and Its Atmosphere: Proceedings of the National Workshop (India) on ''Recent Advances in Solar Physics'' Meerut College, Meerut, India 7 - 10 November 2006

Physics of the Sun and its Atmosphere Proceedings of the National Workshop (India) on “Recent Advances in Solar Physics”

This page intentionally left blank

Editors

B. N. Dwivedi Banaras Hindu University, India

U. Narain Meerut College, India

Phy SICS of

the Sun and its

A tmospher

Proceedings of the National Workshop (India) on "Recent Advances in Solar Physics" Meerut College, Meerut, India 7-10 November 2006

World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI

A-PDF Merger DEMO : Purchase from www.A-PDF.com to remove the watermark

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PHYSICS OF THE SUN AND ITS ATMOSPHERE Proceedings of the National Workshop (India) on “Recent Advances in Solar Physics” Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-283-271-9 ISBN-10 981-283-271-8

Printed in Singapore.

ChianYang - Phys of the Sun.pmd

1

9/18/2008, 1:31 PM

PREFACE

This book emerged from a successful workshop on “Recent Advances in Solar Physics”, held at Meerut College, Meerut in November 2006. We thought a book covering the modern view of the Sun from its interior to its exterior from a pedagogical viewpoint will be a valuable input to the beginners pursuing solar physics. With this objective, we decided to publish this volume by World Scientific, Singapore. In this process, we invited most of the leading experts who lectured at the workshop apart from inviting some internationally reputed scientists to make this volume more valuable. It is satisfying to see this book in print at a time when one of us (Dr. Udit Narain) superannuates after pursuing an active solar physics research for over three decades while teaching physics at Meerut College. All this could be possible with the kind and generous support of our esteemed colleagues from all over and above all, the authors of this book for which we cannot thank them enough.

B.N. Dwivedi & Udit Narain

v

This page intentionally left blank

ACKNOWLEDGEMENTS

It has been the voice of my soul to organize a National Solar Workshop at Physics Dept, Meerut College before my superannuation on 31 July 2007. Prof. B.N. Dwivedi (IT-BHU) was the first person to support me and the national workshop on “RECENT ADVANCES IN SOLAR PHYSICS” was held at Meerut College premises entirely through his efforts. I carried on his advice and programmes mechanically since the preparation for the workshop started. Prof. N.K. Dadhich, Director, IUCAA wanted to hold the solar workshop at BHU under the leadership of Prof. Dwivedi who convinced him to let it take place at Meerut College in view of my superannuation. Dr H.P. Mittal, Head of Physics Dept and Dr S.K. Agarwal, Principal, Meerut College kindly allowed and supported it without any financial support. I gratefully acknowledge the SOC members: S.M. Chitre, B.N. Dwivedi (chair), R. Jain, P.K. Manoharan, J. Singh, W. Uddin and P. Venkatakrishnan. Prof. Dwivedi started looking after the academic part, namely Scientific Organising Committee, speakers, schedule of lectures, publication of proceedings etc. I started looking after local organization part, namely LOC, financial aspects, accommodation, transport etc. The application for finacial assistance was submitted to IUCAA, DST, UGC, CSIR, ISRO and INSA. The request for financial assistance was also made to IIA, PRL, and ARIES. IUCAA provided 25000/=; IIA 30,000/=; CSIR 20,000/=; INSA 10,000/= and UGC 50,000/=. PRL allowed four speakers, ARIES and RAC/TIFR one each with their travel expenses. I am very grateful to them as it was not possible to organize the workshop without their help. Individuals, namely Late Prof. Rajkumar, Dr Mukul Kumar, Dr Rakesh Kumar Sharma, Dr Sushil Kumar, Mr Nishant Mittal and Mr Joginder Sharma provided crucial financial assistance for which I am very grateful to them. I am very much grateful to Prof. S.S. Hasan, Director, IIA, Bangalore for inaugurating and delivering the keynote address at the workshop which was highly appreciated.

vii

viii

Acknowledgements

Prof. S.P. Khare (Former Head, Pro-VC, CCS Univ., Meerut) has been very kind in providing guidance and help from beginning to the end and I wish to express my gratitude to him. Help and support from Dr H.P. Mital and other faculty members, non-teaching staff, research scholars and PG students of Physics Dept, are greatly appreciated. Special thanks are due to Dr Ajay Chauhan for his help in all computer related problems and presentations. The help, cooperation and encouragement by Dr S.K. Agarwal is highly appreciated. I also thank Prof. V.K. Rastogi (CCS Univ, Meerut) for help, support and his personal involvement. Last but not the least, I wish to express my gratitude to the retired and active faculty members and individuals of Meerut who enlivened the proceedings of the workshop by their participation.

Meerut

Udit Narain

CONTENTS

Preface

v

Acknowledgements

vii

Chapter 1: Recent Advances in Solar Physics B.N. Dwivedi

1

Chapter 2: Overview of the Sun S.S. Hasan

9

Chapter 3: Seismic View of the Sun S.M. Chitre and B.N. Dwivedi

21

Chapter 4: Solar Magnetism P. Venkatakrishnan and S. Gosain

39

Chapter 5: Waves and Oscillations in the Solar Atmosphere R. Erdélyi

61

Chapter 6: VUV Spectroscopy of Solar Plasma A. Mohan

109

Chapter 7: Active Region Diagnostics H.E. Mason and D. Tripathi

127

Chapter 8: Hall Effect and Ambipolar Diffusion in the Lower Solar Atmosphere V. Krishan

ix

151

x

Contents

Chapter 9: On Solar Coronal Heating Mechanisms K. Pandey and U. Narain

173

Chapter 10: Coronal Mass Ejections (CMEs) and Associated Phenomena N. Srivastava

193

Chapter 11: The Radio Sun P.K. Manoharan

215

Chapter 12: The Solar Wind P.K. Manoharan

235

Chapter 13: The Sun-Earth System: Our Home in Space J.L. Lean

267

CHAPTER 1 RECENT ADVANCES IN SOLAR PHYSICS

B.N. DWIVEDI Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

1.1. Introduction For millennia, the Sun (and the universe) has been viewed in the visual light. As the bestower of light and life, the ancients made God out of the Sun. With the Babylonians, or with the multiple origins with the Chinese, Egyptians and Indians, quoting the Rig Veda: “All that exists was born from Sūrya, the God of gods.”, we have come a long way to understanding the Sun. In the early seventeenth century, however, Galileo showed that the Sun was not an immaculate object. Thus began our scientific interests in our nearest stellar neighbour, the Sun (cf., Figure 1.1.), with its sunspots and the related solar activity. The observations of the Sun and their interpretations are of universal importance for at least two reasons: First, the Sun is the source of energy for the entire planetary system and all aspects of our life have direct impact on what happens on the Sun; and second, the Sun’s proximity makes it unique among the billions of stars in the sky of which we can resolve its surface features and study physical processes at work. Observations of the solar atmosphere led to the development of the theory of radiative transfer in stellar atmospheres and the discovery of the element helium. Moreover, the Sun is the principal magnetohydrodynamic (MHD) laboratory for large magnetic Reynolds numbers, exhibiting the totally unexpected phenomena of magnetic fibrils, sunspots, prominences, flares, coronal loops, coronal mass ejections (CMEs), the solar wind, the X-ray corona, and irradiance variations etc. It is the physics of these exotic phenomena, collectively making up variations of solar activity, with which we are confronted today. The activity affects the terrestrial environment, from occasionally knocking out power grids to space weather and most probably general climate. 1

2

B.N. Dwivedi

Beginning with the first solar ultraviolet light from space in 1946, X-rays in 1948, hard X-rays and γ-rays in 1958; many experiments have been conducted or being conducted using balloons, rockets and satellites (e.g., OSOs, Skylab, SMM, Yohkoh, SOHO, TRACE, RHESSI, Hinode and STEREO etc.). Artificial satellites have provided the unique opportunity to have uninterrupted observations of the Sun from the vantage points, such as the Sun-Earth Lagrangian point L1 (e.g., SOHO), or from outside the ecliptic plane (e.g., Ulysses), or in stereoscopic modes using different orbits (e.g., STEREO). All these have provided a rich source of data, unlocking the secrets of the Sun and addressing some of its outstanding riddles (e.g., coronal heating, solar wind acceleration etc.)

Figure 1.1. Brief view of the Sun: The Sun’s energy derives from nuclear reactions that occur in its core which is at temperatures of 15 million degrees Kelvin. This energy moves outward, first in the form of electromagnetic radiation (e.g., X-rays and γ-rays) in the radiative zone. Energy then moves upward in photon-heated solar gas through convection in the convective zone (outer 200 000 km). Because of tremendous pressure, this energy is continually absorbed and re-absorbed and may take millions of years to reach the surface of the Sun. Convection motions in the Sun’s interior generate magnetic fields, emerging at the Sun’s surface as sunspots and loops of hot gas called prominences. Most solar energy finally escapes from a thin layer of the Sun’s atmosphere called the photosphere, which is the part of the Sun observable to the naked eye. Image credit: NASA.

Recent Advances in Solar Physics

3

Ground-based observations suffer from the effects of the Earth’s atmosphere such as atmospheric extinction resulting in the limited radiative spectrum of the Sun, and turbulence resulting in image distortions. None the less, making use of adaptive optics system, solar images with resolution of about 0.13" (90 km on the Sun), or even smaller structures down to 60 km, have been obtained by the Swedish 1-meter Solar Telescope (SST) on La Palma. Neutrino detectors have provided a unique tool for probing the Sun’s interior by comparing the emitted flux with the predictions of the standard solar models. Helioseismology from space and from ground (e.g., GONG) have revolutionised our understanding of the workings of the Sun. Against this brief background, it is intended to bring some of these developments in a limited but pedagogical and updated way to help beginners pursue solar physics research. 1.2. Main Contents This book contains 13 chapters beginning with a glance of main contents of each chapter as follows: Chapter 2: Overview of the Sun (Hasan): This chapter begins with how solar physics is going through an exciting period, particularly due to new insights obtained from space and ground observations. These have contributed significantly to improving our understanding of fundamental processes occurring in the solar atmosphere, from the interior to the heliosphere. By combining the information obtained through observations with theoretical developments, a holistic view is slowly beginning to emerge of the physical mechanisms taking place on the Sun. Chapter 3: Seismic view of the Sun (Chitre and Dwivedi): This chapter presents the solar seismology which probes the internal structure and dynamics of the Sun using hundreds of thousands of accurately measured frequencies of solar oscillations. With the accumulation of the helioseismic data obtained with Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) instruments over the past solar cycle, it is possible to study temporal variations that occur within the solar interior with the progress of the cycle. Chapter 4: Solar Magnetism (Venkatakrishnan and Gosain): This chapter is basically divided into two parts. In the first part, the important properties of the solar magnetic field are summarized. The discussion begins with a simple

4

B.N. Dwivedi

introduction to solar magnetohydrodynamics, describing the current status of the solar dynamo theory. Some very curious and interesting results on magnetic helicity and force-free fields are then presented in very basic terms. Finally, the application of this theoretical frame-work to the problems of coronal heating, solar flares and coronal mass ejections are developed in a simple unified scheme, based on a hierarchy of physical conditions. The second part of this chapter consists of a tutorial on magnetographs. It begins with a description of polarization of light from very fundamental notions of coherence of light. This is followed by simple but comprehensive explanations of the Zeeman and Hanle effects along with the necessary basic ideas of quantum physics of scattering of light. Then the working of a few important magnetographs is outlined, with special emphasis on a solar vector magnetograph developed for USO, to provide a “hands on” perspective. The chapter concludes with a few brief remarks on the possible future directions for research in the domain of solar magnetism. Chapter 5: Waves and Oscillations in the Solar Atmosphere (Erdélyi): This chapter introduces waves and oscillations in the solar atmosphere in a lucid manner. Recent satellite and ground-based imaging and spectral instruments have observed a wide range of wave and oscillatory phenomena in the visible, EUV, X-ray and radio wavelengths in the solar atmosphere. Because in most cases these waves and oscillations are tied to the complex magnetic structure of the solar atmosphere, these oscillatory and wave phenomena are interpreted in terms of magnetohydrodynamic (MHD) waves. Waves and oscillations are crucial in the understanding of the diagnostics and dynamics of the magnetised solar atmosphere, as these periodic motions contain information about the medium they occur in. Using undergraduate tools of applied mathematics, the basic properties of MHD waves and oscillations are described. The theoretical description is then strongly linked to the latest observational findings with applications to the wealth of MHD wave phenomena present in the solar atmosphere. Observed MHD waves propagating from the lower solar atmosphere into the higher, often very dynamic regions of the magnetized corona, have the potential to provide an excellent insight into the physical processes at work at the coupling point between these apparently different regions of the Sun. High-resolution wave observations combined with advanced forward MHD modelling can give an unprecedented insight into the connectivity of the magnetized solar atmosphere, which further provides us with a realistic chance to reconstruct the

Recent Advances in Solar Physics

5

structure of the magnetic field in the solar atmosphere. This type of solar exploration is termed as atmospheric magneto-seismology. Some new trends in the observational study of waves and oscillations, discussing their origin, and their propagation through the atmosphere are summarised. Finally, the role of the leakage of photospheric motions, whether coherent (e.g., p-modes), random (e.g., granular buffetting), or casual (e.g., footpoint nano-scale energy release) on the dynamics present in the solar atmosphere is addressed. Chapter 6: VUV Spectroscopy of Solar Plasma (Mohan): Electron densities, temperatures, elemental abundances and emission measures of the space plasma are the basic parameters to give the informations regarding the generation and transport of mass, momentum and energy. The fundamental property of hot solar plasmas is their inhomogeneity. Using the spectroscopic diagnostic techniques for the temperature and density structures of hot optically thin plasmas, the solar atmosphere and its composition have been thoroughly examined. As an illustration, the potential for plasma diagnostics of forbidden transitions from ground levels in the nitrogen-like ions has been presented. Some of the lines considered in the present chapter have been measured by SUMER for the first time. Also using the SUMER spectra, electron density, temperature and abundance anomalies in the off-limb solar corona are discussed. In particular, the behaviour of the solar FIP (first ionization potential) effect with height above an active region observed at the solar limb is presented. Chapter 7: Active Region Diagnostics (Mason and Tripathi): Recent observations from SOHO, Yohkoh and TRACE clearly demonstrate the complex and dynamic nature of the solar atmosphere. In order to explore the nature of solar active regions, it is important to determine the local plasma parameters (electron density, temperature, emission measure distribution, element abundances, flows, non-thermal line broadening etc.). This can only be reliably achieved using simultaneous imaging and spectroscopic observations. The Hinode and STEREO spacecrafts, launched in autumn 2006, are providing some spectacular new observations and insights. This chapter focuses on what has been learnt about active regions in particular from recent observations using spectroscopic diagnostics in the UV and X-ray wavelength ranges. Chapter 8: Hall Effect and Ambipolar Diffusion in the Lower Solar Atmosphere (Krishan): This chapter highlights the realistic importance of incorporating multi-fluid system in the Sun’s atmosphere to understand the

6

B.N. Dwivedi

physical processes in operation. The lower solar atmosphere is a partially ionized plasma consisting of electrons, protons and predominantly hydrogen atoms. The discrete structures such as the sunspots, the prominences and the spicules also consist of the three main species of particles. This essentially forms a three fluid system and therefore, it is mandatory to go beyond the single fluid magnetohydrodynamic studies. One must include the Hall effect which arises from the treatment of the electrons and the protons as two separate fluids and the ambipolar diffusion arising due to the inclusion of neutrals as the third fluid. The Hall effect and the ambipolar diffusion have been shown to be operational in a region beginning from below the photosphere up to the chromosphere. In this three fluid system, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect, in addition to the usual resistive dissipation caused by the electron-proton and electron-neutral collisions. These effects produce novel modifications in the equilibrium configurations of the flows and the fields, the wave phenomena and the magnetic field transport processes. A first principle derivation of these effects in a three fluid system along with an account of their role in the characterization of the lower solar atmosphere is given in this chapter. Chapter 9: On Solar Coronal Heating Mechanisms (Pandey and Narain): The million degree temperature of the solar corona has been an outstanding astrophysical problem since 1943. A number of mechanisms, such as accretion of intergalactic matter, acoustic waves, magnetoacoustic waves, Alfvén waves, currents/magnetic fields, spicules, magnetic flux emergence, velocity filtration etc have been offered as possible explanation. Alfvén waves may heat coronal holes as well as coronal loops in the solar corona. These structures can also be heated by currents/magnetic fields (as nano- and micro-flares) generated by slow photospheric foot point motions. The expected behaviour of X-ray and EUV intensities from Alfvén waves and nano- and micro-flares are quite similar. Only suitable experimental techniques can discriminate between the two main mechanisms. Velocity filtration does not require any source of energy but it requires the existence of highly energetic particles (mechanism still not known) at the base of corona. Other mechanisms do contribute to the energy budget of the solar corona but they cannot resolve the coronal heating problem individually. Chapter 10: Coronal Mass Ejections (CMEs) and Associated Phenomena (Srivastava): Coronal mass ejections are spectacular expulsions of mass from the Sun that display a three-part structure comprising a leading edge, a dark

Recent Advances in Solar Physics

7

cavity and a bright knot. Since the discovery of coronal mass ejections (CMEs) in 1970s, our knowledge about CMEs has improved considerably. This is largely due to multi-wavelength observations with several instruments dedicated to the observations of the Sun both from ground and space. Our understanding of CMEs took a giant leap with the unprecedented observations obtained by several instruments aboard SOHO designed to probe different layers of the Sun. This chapter expounds physical and kinematic properties of CMEs and associated phenomena, such as flares, eruptive prominences, sigmoids and EIT waves. It also examines the links between these phenomena and the physical processes that lead to eruption of CMEs. Finally, the observations of CMEs from the recently launched STEREO and Hinode are highlighted and the problems that these missions might be able to address and resolve. Chapter 11: The Radio Sun (Manoharan): Solar radio observations from ground-based and space-based instruments have contributed a unique perspective on the physical phenomena occurring on the Sun. In particular, radio observations have played a key role in probing the different altitudes of the corona and provided the possibility to trace the three-dimensional structure of the coronal magnetic field. Moreover, the comparison of radio observations with other multi-wavelength data (e.g., X-ray, EUV, and optical) has clearly shown specific advantages and allowed for a deeper understanding of solar flares and coronal mass ejections and the physics behind the fundamental processes of the solar radio emission mechanism. This chapter gives the overview of radio observations of the quiet and active Sun and physics of the explosive energy release. Chapter 12: The Solar Wind (Manoharan): This chapter reviews the evolution of the solar wind, with the particular emphasis on the properties of the solar wind within about 1 AU of the Sun. To start with, a brief discussion of coronal heating is given followed by the energy balance in the solar atmosphere and the formation of the solar wind. The solar wind measurement using the interplanetary scintillation technique is explained in detail. The results on the large-scale properties and long-term changes of quasi-stationary structures of the solar wind are presented. The solar wind disturbances resulting from the solar phenomena and their heliospheric evolution in space and time are reviewed based on radio scintillation technique. The solar cycle changes of the solar wind in the three-dimensional heliosphere are reviewed in the aspect of space weather effects of the solar wind. The final part also includes the turbulence characteristics of the quasi-stationary and transient solar wind.

8

B.N. Dwivedi

Chapter 13: The Sun-Earth System: Our Home in Space (Lean): The concluding chapter of this book addresses the Sun-Earth system and our home in space. Energy flowing from the Sun heats the Earth, structures its atmosphere and organizes the surrounding space environment. Changes in this energy occur continually, with myriad terrestrial impacts, some of which have societal consequences involving climate change, the ozone layer and space-based enterprise. 1.3. Concluding Remarks A pedagogical updated modern view of the Sun from its interior to its exterior as well as the Sun-Earth system in this book by eminent solar physicists present a rich menu to motivate graduate students who wish to pursue solar physics research career.

CHAPTER 2 OVERVIEW OF THE SUN S.S. HASAN Indian Institute of Astrophysics Bangalore-560034, India

2.1. Introduction The Sun plays a central role in two important respects: firstly, it provides a cosmic laboratory for investigating processes that cannot be simulated in the terrestrial environment and secondly, because of its relative closeness it serves as a proxy for understanding conditions in other stars. Formed about 4.6 billion years from a cloud of gas dust and frozen ice, the Sun at the current epoch in its life is a normal main-sequence star of spectral classification G2 with an average surface temperature of around 5700 K. It displays an astounding range of phenomena on myriad spatial and temporal scales that have traditionally defied comprehension. The Sun’s magnetic field, that varies on a 22-year cycle, triggers activity and powerful eruptions that affect regions extending from the Earth’s atmosphere to the distant edges of the solar system. Despite the inherent complexities of these processes, some progress has been achieved in understanding them through recent spectacular advances in observational techniques coupled with theoretical modelling. The aim of the present review is to provide a broad overview of some modern developments in solar physics that have had a significant impact on the subject. We begin by discussing in Sect. 2.2 the internal structure of the Sun and the processes taking place in the interior. Section 2.3 deals with magnetic fields and their influence on the dynamics, heating and activity in the solar atmosphere. In Sect. 2.4 we discuss processes in the outer atmosphere that includes eruptive phenomena such as CMEs and flares. The final Sect. 2.5 looks at future perspectives and directions.

9

10

S.S. Hasan

2.2. Internal Processes The source of the Sun’s energy derives from thermonuclear reactions, which occur in the core, a region at a temperature of about 107 K that extends from the centre to about 0.28 R
, where R
is the solar radius. These reactions convert hydrogen into helium, principally through the proton-proton chain. This energy is transported through radiation up to about 0.7 R
beyond which this process is no longer effective. From this radius up to the surface, the dominant mechanism for energy transport is through convection, similar to the fluid motions that arise when water is heated from below. The solar interior is opaque to optical radiation in view of its high opacity: in fact, visible light from the Sun emanates from a thin layer of around 100 kilometres thickness in the surface layers known as the photosphere. However, neutrinos generated in the core by nuclear reactions can travel unimpeded into space since they interact very weakly with matter. The mismatch between the neutrino flux determined from theory and its experimental verification constituted what has been commonly referred to as the solar neutrino problem. Recently, a resolution of this conflict has been found by invoking a mechanism that permits neutrinos emitted from the Sun to change “flavours” during their passage to the Earth. A novel experiment at the Sudbury Neutrino Observatory in Ontario Canada1 that can measure all flavours of neutrinos was able to achieve an impressive agreement with the theoretical prediction (cf., Chitre & Dwivedi, Chapter 3, this volume for more details). 2.2.1. Helioseismology and rotation The thermodynamic state of the solar interior is now known to a high level of precision mainly due to helioseismology, which in simple terms uses the properties of acoustic oscillations with periods of around 5 min. to probe the deeper layers that are inaccessible to direct observations. The accuracy with which the sound speed can be inferred using this technique is higher than a fraction of a percent. Due to the thermal stratification, the sound speed increases with depth from the surface. As a consequence, an acoustic wave, propagating downward will be refracted and eventually reach a level where it will turn around and travel upwards. On reaching the surface, the wave encounters a sharp decrease in density and is reflected back into the interior. In this way acoustic modes are trapped in a cavity with an extension that depends on the horizontal wavelength of the mode. Longer wavelength modes penetrate deeper into the Sun and can be used as a diagnostic for probing conditions close to the core, whereas modes of short wavelengths provide information about physical

Overview of the Sun

11

conditions just below the surface layers. This subject is dealt in greater detail in Chapter 3 of this volume. In addition to revealing the sound speed, helioseismology can also be used to deduce the rotation profile of the plasma in the solar interior because rotation produces a splitting of the acoustic modes. A careful analysis of this splitting makes it possible to determine the rotation rate at each depth and latitude. From observations it is known that surface layers of the Sun rotate differentially with a period that increases from 25 days at the equator to about 35 days at the poles. From helioseismology investigations it is established that apart from a narrow region below the photosphere, the differential rotation preserves approximately the same form as the surface in the interior layers. This behaviour continues to the base of the convection zone, below which the rotation character changes abruptly from differential to rigid body in a thin shear layer, known as the tachocline. Below the tachocline, material in the radiative interior rotates homogenously with an intermediate period of around 27 days. 2.2.2. Magnetic field generation As mentioned earlier, the solar magnetic field has a profound influence on practically all observed phenomena. What is the mechanism for generating and maintaining this field? The solar plasma in the interior is in the form of a fully ionized conducting gas that is “frozen” into the magnetic field. It is now generally accepted that the magnetic field is generated through a dynamo action at the base of the convection zone in the tachocline, which has a high velocity shear. The mechanism involves the interplay of differential rotation and convective motions. In the first step a weak poloidal (i.e. oriented principally in the north-south direction) field is twisted by the differential rotation to generate an azimuthal (toroidal) component (ω -effect), which is gradually amplified due to the rotation. The next step in the process is the regeneration of the poloidal component from the toroidal field. Parker2 proposed a workable model for this process that incorporates the effects of helical motions arising from Coriolis forces. This is the so-called α-effect, which can be estimated more formally using mean-field magnetohydrodynamics3,4. By suitably fine tuning adjustable parameters, such models could reproduce many aspects of the “butterfly diagram” that depicts the sunspot distribution with time over a solar cycle. However, in order to accomplish this, the angular velocity in the interior needs to increase with depth by up to 40%, which is inconsistent with the results of helioseismology. Other difficulties with this model include the problem of storing strong toroidal magnetic fields for a significant fraction of a sunspot

12

S.S. Hasan

cycle to overcome their premature escape to the surface due to magnetic buoyancy. Furthermore, detailed numerical simulations yielded magnetic patterns on the surface that were inconsistent with observations. A phenomenological model of the solar dynamo developed by Babcock5 and Leighton6,7 that was remarkably successful in explaining several observational features such as Hale’s polarity reversal, migration of sunspots and the solar cycle has recently been revived8-13. The key elements in the model involve advection by meridional circulation of poloidal magnetic fields from the surface downwards in to the convection zone where they are both sheared by the differential rotation as well as transported towards the equator. The newly created toroidal flux rises due to magnetic buoyancy and produces sunspots at the surface. The decay of these spots generates poloidal flux and the cycle repeats again. A general difficulty with this class of models is that the surface fields at the poles tend to be too strong13. Incidentally, the majority of dynamo models to date are kinematic, i.e. they involve solely the amplification of field through fluid motions but ignore the back reaction of the field on the fluid motions. Some efforts to produce full dynamic models of the solar dynamo have recently started with varying degrees of success14 but they still have a long way to go before they can match observations. Despite the impressive progress that has been made in dynamo models, particularly in recent years through detailed numerical simulations, there are still many open questions such as: (a) Is the dynamo action global or local (i.e. does it occur in a narrow region at the interface between the convection and radiative zones)? (b) What limits the amplitude of the solar magnetic field; (c) Can models reliably predict the strengths of future cycles? Some aspects of these problems are under active examination and hopefully will be resolved in the near future. 2.3. Solar Magnetism Above the photosphere, the temperature decreases gradually reaching a minimum value of around 4200 K at a height of 500 km above the surface. This is the base of a region known as the chromosphere. The temperature now increases with height, slowly at first to a value of about 104 K, and thereafter extremely rapidly from a height of 2000 km to a value of 105 K in a thin layer (known as the transition region with an extension of some 100 km). Above this region is the corona, in which the temperature increases gradually to few million degrees. The focus of this section is on processes in the photosphere and chromosphere, while the following section will discuss phenomena in the corona.

Overview of the Sun

13

White light observations of the solar surface have revealed the presence of cellular motions that at the largest scales of convection defines a pattern known as supergranulation with a typical horizontal size of about 30,000 km. The boundaries of the supergranules15,16 are the sites of intense magnetic fields and define what is commonly referred to as the magnetic network. The regions between the supergranules with strong magnetic fields are also associated with jets of gas known as spicules that protrude high in to the chromosphere. Spicules play an important role in the dynamics and energy balance of the chromosphere, though not contributing to mass loss from the Sun. The mechanism responsible for producing spicules is unfortunately not fully understood, although there is strong evidence that it involves the magnetic field in a fundamental way. High resolution observations also reveal a cellular pattern on a smaller scale of about a 1000 km known as granulation. Ground-based telescopes equipped with adaptive optics and space telescopes with an angular resolution better than 1 arc sec like the Solar Optical Telescope on Hinode (a Japanese spacecraft launched in 2006) can easily resolve this pattern. Recent numerical simulations17 have been fairly successful in reproducing the broad observational properties associated with granulation. 2.3.1. Quiet Sun magnetism In the “quiet Sun” i.e. the regions outside sunspots and other centres of activity, the magnetic field broadly falls into two broad categories: network and internetwork fields. Various studies from the 1970s onwards confirmed that more than 90% of the quiet Sun flux is in the form of discrete bundles or flux tubes with field strengths in the kilogauss range and with diameters of the order of 100 km or less18,19. Observational techniques for magnetic field measurements are discussed by Venkatakrishnan (Chapter 4, this volume). Magnetic network elements can be identified with bright points in G-band (430.05 nm) images20. High-resolution observations indicate that these network bright points, (NBPs) located in “lanes” at the boundaries of granules, are in a highly dynamical state due to continuous buffeting by random convective motions in the subphotosphere. With the availability of new telescopes at excellent sites and sophisticated image reconstruction techniques it is now possible to examine NBPs with a resolution better than 70 km and investigate their structure in unprecedented detail21. This picture of the network is corroborated by recent observations22 images in G band of the molecular bandhead of CH (430.5 nm) and the line of Ca II H (396.8 nm). The G band and Ca II H lines are formed in the photosphere and chromosphere respectively. These observations reveal a

14

S.S. Hasan

picture of a network region consisting of bright multiple flux elements that are hotter than their surroundings. The physical processes that heat the network are still not fully identified. Is the network heated by wave dissipation and if so, what is the nature of these waves? These and other aspects related to the nature of waves in magnetic fields are discussed by Fay-Siebenburgen (Chapter 5, this volume). Let us now turn to the region in the interior of the supergranule cells that is often referred to as the internetwork (hereafter IN). Till recently it was believed that the IN magnetic field was weak with a mean field strength of a few Gauss. However, recent observations using high polarimetric sensitivity have provided new information on the nature of IN magnetism and revealed a field distribution with strengths ranging from a few Gauss to kilogauss23. Stronger fields have been identified with bright points in the cell interior23. In addition, observations have shown that a large fraction of magnetic fields in the cell interiors reside in small-scale weaker fields with a typical range of 20-150 G (ref. 24), a finding that is supported by the recent high resolution observations from the Hinode spacecraft25. Despite the impressive advances in observational techniques, there are many aspects about the internetwork magnetic field that need to be elaborated. For instance, the origin of this field is still unknown. Is the field a remnant produced from the decay of active regions or is it created by a turbulent dynamo26 ? 2.3.2. Sunspots 2.3.2.1. Structure Sunspots, in contrast to the thin flux tubes associated with NBPs, are much larger structures (with typical thickness in the range of 10000 to 20000 km) that are strongly magnetized (around 3000 G in the central regions). They are visible on the solar surface as dark features with a dark core called the umbra with a much lower temperature (around 4000 K) than the ambient atmosphere and surrounded by a lighter region called the penumbra. The darkness of sunspots has traditionally been attributed to suppression of convective energy transport (relative to the surrounding photosphere) by the strong magnetic field. The orientation of the magnetic field is mainly vertical in the centre of the umbra and becomes increasingly inclined with radial distance to about 70° (with respect to the vertical) at the edge of the penumbra, where the field strength drops to about 1000 G. The penumbra displays radial filaments along which fluid motions with speeds of several kilometres per second occur. This is the well-known Evershed

Overview of the Sun

15

effect, discovered in Kodaikanal, India in 1909, the origin of which is still being debated. Sunspot umbrae also reveal fine structure in the form of bright points or umbral dots with a typical diameter of about 150 km and brightness comparable to the photosphere. It was earlier believed that the magnetic field in umbral dots is reduced compared to the background umbra but recent observations do not indicate a decrease in field strength27. The physical mechanism responsible for their formation is most likely related to convection in a vertical magnetic field. Recent high-resolution observations have shown that the penumbral magnetic field exhibits an “interlocking comb structure”, consisting of two distinct groups of field lines associated with: (a) inclined bright filaments, and (b) almost horizontal dark filaments28,29. From a theoretical viewpoint this dual topology is also not well understood. It has been suggested that buoyancy and downward pumping of magnetic may contribute to creating and maintaining such structures30. 2.3.2.2. Solar cycle The number of sunspots and sunspot groups (sunspot number) present on the solar surface changes with time and exhibit a cyclic behaviour with an approximately 11 year period. The amplitude of the cycle (often called the strength of the cycle) varies from one cycle to another. Sunspots occur typically in the latitude range ± 35° and drift in latitude towards the equator as the cycle progresses (Spörer’s law). In recent years evidence has accumulated that the solar cycle has a long-term modulation consisting of epochs of hyperactivity (most recent being the Medieval maximum in the 12th century) as well as spells without sunspots (Maunder minimum during 1645-1715). These periods of abnormal activity are without explanation. Incidentally, the total solar irradiance (the energy from the Sun observed at Earth per unit area per unit time and unit wavelength interval) also exhibits a 11 year cycle which is in phase with the sunspot cycle and has implications for the terrestrial climate (see Lean, Chapter 13, this volume). 2.3.3. Active regions Active regions are areas on the Sun with enhanced magnetic fields and where magnetic flux has erupted through the photosphere into the chromosphere and corona. They are visible in white light, enhanced line emission (such as in Calcium), X-rays and radio wavelengths. Sunspots are ubiquitous features of active regions and are associated with eruptive phenomena such as solar flares,

16

S.S. Hasan

filaments, radio bursts, enhanced coronal heating, and coronal mass ejections. In addition ephemeral active regions that are bipolar in nature represent another group of regions on the solar surface. Considerable efforts have gone in to understand the birth, evolution, and decay of active regions, as well as the underlying mechanisms responsible for the above processes. Recently a new method for imaging active regions on the far side of the Sun has been developed31 using local helioseismology techniques. This will enable detection in advance of activity before these regions are accessible to direct observations. Detailed information on their properties has revealed by space and high resolution ground observations, which are discussed by Tripathi and Mason (Chapter 7, this volume). Possible scenarios for enhanced heating in active regions, particularly in the corona, are dealt by Mohan (Chapter 6, this volume). 2.4. Processes in the Corona 2.4.1. Eruptive phenomena Magnetic fields in active regions are responsible for a large number of dynamic processes that produce copious radiation over the full electromagnetic spectrum as well as acceleration of the solar plasma. Many of these occur on a short time scale of minutes to hours and generate effects that produce disturbances in the terrestrial environment and beyond. 2.4.1.1. Flares The most dramatic eruptive phenomenon in the atmosphere of the Sun is the solar flare producing radiation often over a wide range of frequencies (radio, optical, X-rays and even gamma rays) as well as energetic mass motions. The energy released in a large event is typically in the range of 1028 – 1033 erg. Smaller events called microflares (1026 erg) and nanoflares (1022 erg) have also been recently included in the category of flares. There have been suggestions that the latter are largely responsible for heating the corona32. Flares have traditionally been observed in the H-alpha (656.3 nm) line as bright kernels located on opposite sides of dark filaments that outline a region where the magnetic polarity changes sign (the so called neutral line). This suggests that magnetic field lines of opposite polarity must be involved in the physical mechanism that produces flares. Many flares begin with an eruption of the filament (filament activation). However, there are also events, particularly associated with small flares, which often involve an interaction of two loops and

Overview of the Sun

17

the release of energy through magnetic reconnection (i.e. the “annihilation” of magnetic field lines of opposite polarity). There is yet another class of flares in which pre-existing structures in the form of loops erupt – such events often produce the most powerful flares. It is generally accepted that the source of energy for flares comes from the magnetic field. However, what is still not firmly established are the precise mechanisms responsible for these phenomena. Considerable efforts have gone in to develop elaborate models to identify the processes that result in the sudden release of enormous energy. Most theoretical models are based on magnetic reconnection but the precise details are still under intense investigation. 2.4.1.2. Coronal Mass Ejections (CMEs) Coronal mass ejections (CMEs) are amongst the most energetic events in active regions involving the ejection of solar plasma into interplanetary space with speeds ranging from a few hundred to thousands of kilometers per second. They are often associated with flares, but there are events closely related to filament eruption without flares. CMEs have an important bearing on space weather and the Sun-Earth connection. A wealth of space and radio observations in recent years has provided considerable information on this phenomenon. Theoretical models for CMEs are under active study: in addition to specifying the mechanism that triggers CMES, models need to identify observable signatures, the necessary conditions for their onset and their relation to flares. It is generally agreed that they originate in regions of closed magnetic field lines. More details on CMEs and the associated radio bursts can be found in Srivastava (Chapter 10, this volume) and Manoharan (Chapter 11, this volume) respectively. 2.4.2. Other phenomena The outer solar atmosphere consisting of the corona is at a temperature between 1-2 × 106 K, making it visible in EUV and soft X-rays. Despite persistent efforts, there is no broad agreement on how the corona is heated (see Pandey & Narain, Chapter 9, this volume, for further details). Embedded in this hot tenuous atmosphere are the relatively cool dense prominences. New generation of telescopes from the ground as well as from space have revealed fine structure in prominences and provided valuable diagnostic information. A critical input for theoretical models is a precise knowledge of the vector magnetic field as well as of the thermodynamic quantities in prominences. Above active regions, loops are often observed, which are believed to be magnetically confined closed structures

18

S.S. Hasan

with an enhanced temperature (around 2.5 × 106) with a typical size between 104 km and 105 km. Smaller loops show up as bright points in X-rays. In addition the corona exhibits open magnetic structures known as coronal holes with a reduced density compared to the surrounding atmosphere. Dark regions in coronal holes are associated with the high-speed (600 – 800 km s-1) solar wind, which originates from magnetic funnels between supergranules (see Manoharan, Chapter 12, this volume). 2.5. Future Perspectives Solar physics has made impressive progress in recent years, particularly due to a good synergy between observations and theory. In the 1990s space missions such as YOKOH, SOHO and TRACE were launched which provided observations of solar features in unprecedented detail and contributed significantly to our knowledge of processes on the Sun. New missions launched recently such as STEREO and HINODE are supporting these efforts by providing multiwavelength information with high spatial and temporal resolution. The Solar Dynamics Observatory (SDO), the first major initiative of NASA as part of the Living With a Star (LWS) programme that is scheduled for launch during 2008, will focus on the causes of solar variability and its influence on the Earth’s environment. Long terms plans in the next decade include Solar Orbiter to study the Sun at a distance of 45 solar radii in an orbit co-rotating with the Sun. It will enable a study for the first time of the Sun’s polar regions. Another mission that is under study is Solar Probe that will come as close as 3 solar radii to the Sun’s surface in order to understand coronal heating and solar wind acceleration. A follow-up programme to SDO under LWS called the Solar Sentinels is also being considered. It consists of several spacecrafts at various distances from the Sun to study the origin of the solar winds and carry in situ measurements of energetic particles and interplanetary disturbances. In addition to space missions, several ground-based facilities have come up and many more are underway. The new developments in adaptive optics have provided a major fillip to the quality of ground-based observations. In fact, current ground-based telescopes at good sites and equipped with adaptive optics can provide angular resolution in optical wavelengths that is higher than is presently available from space. The Swedish Solar Telescope at La Palma and the German Vacuum Telescope in Tenerife are good examples of such facilities where a spatial resolution better than 0.2” (the limit on the Solar Optical Telescope on Hinode) can be achieved using adaptive optics. Two new facilities viz. the German 1.5-m GREGOR telescope at Tenerife and the 1.6-m NST (New

Overview of the Sun

19

Solar Telescope) of the Big Bear Observatory will shortly be commissioned. The Indian Institute of Astrophysics, Bangalore is planning a 2-m class National Large Solar Telescope (NLST) at a high altitude site, possibly in the Leh-Ladakh region. The main aims of this telescope is to resolve the fundamental scale in the solar atmosphere such as the pressure scale height, carry out spectroscopic observations with high spectral and spatial resolution of features close to the diffraction limit of the telescope and spectro-polarimetry in the visible and infrared to an accuracy of at least 0.1% to accurately derive vector magnetic fields. Proposals for larger facilities such as the Advanced Technology Solar Telescope (ATST) in the U.S.A. and the European Solar Telescope (EST) involving 4-m class instruments are under active consideration. A Frequency-Agile Solar Radio Telescope (FASR), a multifrequency (0.03-0 GHz) imaging radio interferometer consisting of 100 antennas, is also likely to come up in coming decade. Solar physics is going through an exciting phase with a large number of new programmes currently under way and several more on the anvil. These will provide multiwavelength information on a range of spatial and temporal scales that was unattainable earlier. Developments in observations and theory are beginning to yield a holistic picture of solar phenomena. References 1. Ahmad, Q. R., Allen, R. C., Andersen, T. C. et al., Phys. Rev. Lett. 87, 71301 (2001). 2. Parker, E. N., Astrophys. J. 122, 293 (1955). 3. Steenback, M., Krause, F. and Rädler, K. H., Z. Naturforsch. 21a, 369 (1966). 4. Stix, M., The Sun, Springer Verlag, Berlin (1989). 5. Babcock, H. W., Astrophys. J. 133, 572 (1961). 6. Leighton, R. B., Astrophys. J. 140, 1547 (1964). 7. Leighton, R. B., Astrophys. J. 156, 1 (1969). 8. Wang, Y.-M., Sheeley Jr, N. R. and Nash, A. G., Astrophys. J. 383, 431 (1991). 9. Durney, B. R., Solar Phys. 160, 213 (1995). 10. Dikpati, M. and Charbonneau, P., Astrophys. J. 518, 508 (1999). 11. Nandy, D. and Choudhuri, A. R., Astrophys. J. 551, 576 (2001). 12. Nandy, D. and Choudhuri, A. R., Science 296, 1671 (2002). 13. Charbonneau, P., Living Rev. Solar Phys., http://www.livingreviews.org/lrsp2005-2 (2005).

20

14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24. 25. 26. 27. 28. 29. 30. 31. 32.

S.S. Hasan

Brun, A. S., Miesch, M. S. and Toomre, J., Astrophys. J. 614, 1073 (2004). Simon, G. W. and Leighton, R. B., Astrophys. J. 140, 1120 (1964). Wang, H., Solar Phys. 117, 343 (1988). Stein, R. F. and Nordlund, Å., Solar Phys. 192, 91 (2000). Zwaan (1987). Solanki, S. K., Space Sci. Rev. 63, 1 (1993). Berger, T. E., Rouppe van der Voort, L. H. M., Löfdahl, M. G. et al., Astron. Astrophys. 428, 613 (2004). Rouppe van der Voort, L. H. M., Hansteen, V. H., Carlsson, M. et al. (2005). Rutten, R., Bettonivil, J. F. C. M., Hammerschlag, R. H. et al., in “Multiwavelength investigations of solar activity”, Eds. A. V. Stepanov, E. E. Benevolenskaya and A. G. Kosovichev, IAU Symp. (Cambridge) 223, 597 (2005). Sánchez Almeida, J., in “The Solar-B mission and the forefront of solar physics”, Eds. T. Sakurai and T. Sekii, ASP Conf. Ser. (San Francisco) 325, 115 (2004). Trujillo Bueno, J., Shchukina, N. and Asensio Ramos, A., Nature 430, 326 (2004). Orozco Suárez, D., Bellot Rubio, L. R., del Toro Iniesta, J. C. et al., Astrophys. J. 670L, 610 (2007). Petrovay, K. and Szakaly, G., Astron. Astrophys. 274, 543 (1993). Lites, B. W., Bida, T. A., Johannesson, A. et al., Astrophys. J. 373, 683 (1991). Degenhardt, D. and Wiehr, E., Astron. Astrophys. 252, 821 (1991). Thomas, J. H. and Weiss, N. O., Annual Rev. Astron. Astrophys. 42, 517 (2004). Weiss, N. O., Space Sci. Rev. 124, 13 (2006). Lindsey, C. and Braun, D. C., Solar Phys. 192, 261 (2000). Parker, E. N., Astrophys. J. 330, 474 (1988).

CHAPTER 3 SEISMIC VIEW OF THE SUN

S.M. CHITRE UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai-400 098, India B.N. DWIVEDI Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

3.1. Introduction The Sun has played a major role in the development of mathematics and physics over the past centuries and has been widely described as the “Rosetta Stone” of Astronomy. This is undoubtebly an apt description of a celestial object whose internal and external layers provide an ideal laboratory for testing atomic and nuclear physics, high-temperature plasma physics and magnetohydrodynamics, neutrino physics and general theory of relativity. The proximity of our star to Earth has enabled us to make a close scrutiny of its surface regions and the overlying atmosphere. It has provided a wealth of information of high spatial resolution about its surface features which is evidently not possible for other stars. Indeed, from very ancient times the Chinese and Greek astronomers had not failed to notice the dark spots on the otherwise immaculate surface of the Sun. Solar astronomers have, in fact, maintained systematic records of the appearance of these vivid regions on the visible disk of the Sun, hoping to understand the processes that drive the solar cycle and possibly shed some light on its role in influencing the terrestrial climate. The interior of the Sun is clearly not directly accessible to observations as the internal layers are shielded by the solar material beneath the photosphere. But nevertheless, it is possible to infer the physical conditions prevailing inside the Sun with the help of equations governing its structure together with boundary conditions provided by observations.

21

22

S.M. Chitre and B.N. Dwivedi

For a spherical star the governing equations for mechanical equilibrium are the following:

d P (r ) GM (r ) = − ρ (r ) , dr r2 d M (r ) = 4π r 2 ρ (r ) . dr Here P(r) is the pressure, ρ (r) the density and M(r) mass interior to the radius r for a spherically symmetric Sun. For maintaining thermal equilibrium, the energy radiated by the Sun, as measured by its luminosity, L(r), must be balanced by the nuclear energy generated throughout the solar interior,

d L( r ) = 4π r 2 ρ (r ) ε , dr

ε is the energy generation rate per unit mass and L(r ) = 4π r 2 ( Frad + Fconv ), Frad and Fconv being respectively the radiative

where

and convective flux of energy. The energy generation takes place in the central regions of the Sun by the thermonuclear reactions converting hydrogen into helium mainly by the protonproton chain which contributes over 98% to the energy production with the rest less than 2% from the CNO cycle reaction. The energy generated by these reaction networks is transported from the centre to the solar surface by radiative processes within the inner two-thirds of the Sun and in the outer third by convective processes. The radiative flux is given by Frad = − K rad

dT , dr

( K rad =

4acT 3 is the radiative conductivity) , 3κρ

where a is the Stefan-Boltzmann constant, c the speed of light and κ the opacity of solar material, and the convective flux modeled in the framework of a local mixing-length formalism can be expressed as

Fconv = − K turb ρT

dS (r ) , dr

where K turb is the turbulent diffusivity and S the entropy.

Seismic View of the Sun

23

The outstanding question is how to check the correctness of the theoretically constructed models of the Sun. It turns out that the solar interior is, indeed, transparent to neutrinos resulting from thermonuclear processes in the solar core and also to seismic waves generated through bulk of the body of the Sun. These serve as complementary probes which provide valuable information about the structure and dynamics of the solar interior. The standard solar model (SSM) used for this purpose1 is based on the simplest set of assumptions: The Sun is assumed to be a spherically symmetric body with negligible effects of rotation, magnetic field, tidal forces and mass loss on its global properties. It is supposed to be in quasi-mechanical and thermal equilibrium with the energy generation taking place in the central regions by nuclear reactions which convert hydrogen into helium mainly by the proton-proton chain. The energy is transported outwards principally by radiative processes, but in the top third by radius the energy flux is carried largely by convection modeled in the framework of a mixing-length prescription, and there is no other process for transporting energy such as any wave motion. There is supposed to be no mixing of nuclear reaction production beyond the convection zone, except for the slow gravitational settling of helium and heavy elements by a slow diffusion beneath this zone into the radiative interior. It is presumed that the standard nuclear and neutrino physics is applicable for constructing theoretical solar models satisfying the observed constraints, namely mass, radius, luminosity and the ratio of chemical abundances Z/X, where X and Z refer respectively to the fractional abundance by mass of hydrogen and elements heavier than helium.

3.2. Solar Neutrinos The early investigations in Solar Physics were largely concerned with an extensive collection of spectroscopic data for inferring the temperature, density and chemical composition in the surface layers of the Sun. Since the mid-1960s there have been experiments set up to measure the flux of neutrinos released by the nuclear reaction network operating in the solar core2. The neutrino count rate is highly sensitive to the temperature and composition profile in the central regions of the Sun. It was hoped that the steep temperature dependence of some of the nuclear reaction rates involved would enable a determination of the Sun’s central temperature to an accuracy of better than a few percent. The motivation for setting up the neutrino experiment was “to see into the interior of a star and thus verify directly the hypothesis of nuclear energy generation in stars” and it was thought that “The use of a radically different observational probe may reveal wholly unexpected phenomena; perhaps, there is some great surprise in store for

24

S.M. Chitre and B.N. Dwivedi

us when the first experiment in neutrino astronomy is completed”3. Indeed, there have been valiant efforts since the mid-1960s to set up experiments that are designed to undertake the exceedingly difficult measurement of neutrinos from the Sun.

PROTON-PROTON CHAIN

p + p → d + e + + νe −

p + e + p → d + νe 3

p+d → pp-I:

pp-II:

pp-III:

He + 3 He →

3

He + p →

3

He + 4 He → −

Be + e

7

Li + p →

4

4

7 7

→ 8

He + 2 p

He + e + + νe

(≤ 18.8 MeV)

Be + γ

Li + νe

(0.38, 0.86 MeV)

Be + γ

8

Be → 2 He

3

He + 4 He →

7

Be + p →

8

8

B →

8

(1.44 MeV)

He + γ

3

7

(≤ 0.42 MeV)

4

8

7

Be + γ

B + γ

Be + e + + νe

(≤ 14.6 MeV)

4

Be → 2 He

CNO CYCLE 12

C+ p →

13

N →

13

13

C + e + + νe

13

C+ p →

14

14

N+ p →

15

15

15

15

O →

N+ p →

N+ γ

N+ γ

O+ γ

N + e + + νe 12

(≤ 1.2 MeV)

4

C + He

(≤ 1.7 MeV)

Seismic View of the Sun

25

or 15 16

O+ p →

16

O+ γ

O+ p →

17

F+ γ

17

O + e + + νe

17

F →

17

O +p →

14

(≤ 1.7 MeV)

N + 4 He

Davis’s Chlorine experiment was located some 1480 m underground in the Homestake gold mine in South Dakota. It has a tank containing 615 tons of liquid perchloroethylene (C2Cl4) which is sensitive to intermediate and high energy neutrinos. In this experiment the Chlorine nuclei serve as solar neutrino absorbers according to the reaction 37

Cl + ν
→ 37Ar + e– (threshold = 0.814 MeV).

The count rate is dominated by the high-energy 8B neutrinos contributing 5.9 SNU, with 7Be neutrinos making a contribution of 1.1 SNU (1 SNU = 10–36 captures per target atom per second). The theoretically predicted capture rate for SSM for the Chlorine experiment is 7.6 ± 1.2 SNU4. Davis, however, reports measurement of the solar neutrino count rate of 2.56 ± 0.023 SNU which clearly shows a puzzling deficit by nearly a factor of 3 over the SSM prediction. Throughout the experimental runs in the Homestake mine, Davis has been consistently reporting a count rate which is significantly lower than that predicted by the standard solar model. This is the celebrated solar neutrino problem which has been haunting the community of solar and neutrino physicists for nearly four decades or so. There have been a number of ingenious suggestions5 which have been proposed to lower the central temperature of the Sun. These have included proposals invoking partial mixing in the solar core which can bring additional fuel of hydrogen and helium to the centre, thus maintaining the nuclear energy production at a slightly lower temperature; the presence of a small admixture of Weakly Interacting Massive Particles (WIMPs) in the central regions which would effectively diminish the temperature gradient as a result, contributing an increase in the thermal conductivity; the rapidly rotating solar core; the centrally concentrated magnetic field; lower heavy element abundance in the core. All these proposals lead to a slight reduction in the central temperature resulting in a lowering of the flux of high energy neutrinos. A quarter of a century after the Chlorine experiment, the Japanese experiment consisting of a 680 ton of ordinary water tank was located about a

26

S.M. Chitre and B.N. Dwivedi

kilometer underground in the Kamiokande mine. This experiment was designed to detect charged particles by measuring Cérenkov light through the elastic scattering reaction

νx + e– → ν′x + e– (threshold = 5 MeV). The Kamiokande and the later upgraded Super-Kamiokande experiment are sensitive to the count of the high-energy 8B neutrinos released in the reaction network. The measured flux from the Super-Kamiokande experiment again shows deficiency by a factor of 2 over the total flux predicted by SSM. It is clear that the Chlorine and Super-Kamiokande experimental measurements are inconsistent with the proposition of resolving the solar neutrino problem by lowering the central temperature. Such a reduction of the central temperature will cause even a larger suppression of the high-energy 8B neutrino flux to which the Super-Kamiokande is exclusively sensitive, while the Chlorine experiment which detects the intermediate as well as the high energy neutrinos shows even a larger deficit in the neutrino counting rate! This is a paradoxical situation which leads to the conclusion that a cooler solar core is not a viable solution of the solar neutrino puzzle. Besides these two experiments, there are three other radiochemical experiments, GALLEX, SAGE and GNO that use gallium detector for capturing the lower energy neutrinos via the reaction: 71

Ga + νe → 71Ge + e–

(threshold = 0.233 MeV).

The counting rate for the gallium experiments is on an average 74.7 ± 5.0 SNU, while the SSM prediction of the neutrino counting rate is 128 ± 8 SNU, again showing a deficit in the measured neutrino flux. A possible resolution of this conundrum is to endow neutrinos with a tiny mass enabling them to transform their flavour during propagation. Thus, the electron neutrinos released in the reaction network could get converted into neutrinos of a different flavour while transiting through the interior of the Sun and of the Earth and along their flight path through the interplanetary space. The first compelling evidence for such oscillations of neutrino flavours came from the Super-Kamiokande’s analysis of data on high energy cosmic ray produced neutrinos in the Earth’s atmosphere. The asymmetry in the measured up and down fluxes of neutrinos produced by cosmic ray interactions with the terrestrial atmosphere would be the result of passage of the upward moving neutrinos through the solid mantle of Earth, while the downward moving neutrinos coming

27

Seismic View of the Sun

from overhead are generated afresh in the Earth’s atmosphere and are less likely to undergo any flavour oscillations. The recent measurements by the Sudbury Neutrino Observatory (SNO) appears to provide convincing evidence for solar neutrino oscillations. The SNO experiment located at a depth of over 6000 meters of water equivalent in Sudbury (Canada) uses 1000 ton of heavy water containing the isotopes of deuterium. In both heavy water tank in Sudbury and the ordinary water detector at Super-Kamiokande, neutrinos can elastically scatter electrons to produce Cérenkov radiation, but such an electron scattering may be caused by any of the three neutrino flavours (x = e -, µ - and τ - neutrinos):

νe + d → p + p + e − (charged current) ν x + e − → ν′x + e −

(elastic scattering)

ν x + d → ν′x + p + n

(neutral current)

SNO’s heavy water detector is capable of isolating electron neutrinos via the charged current (CC) reaction. The neutral current (NC) reaction is equally sensitive to all the neutrino flavours, while the elastic scattering (ES) has significantly lower sensitivity to µ – and τ – neutrinos. SNO has reported the elastic scattering count rate which equals Super-Kamiokande’s event rate to within experimental errors. It is noteworthy that SNO’s count of the charged current reaction which is sensitive exclusively to the electron – neutrinos is lower than the count rate of Super-Kamiokande and SNO. Interestingly, the total 8 B neutrino flux as measured by the NC reaction is (5.09 ± 0.62) × 106 cm-2 s-1, in agreement with the prediction of SSM. Table 1. Solar neutrino experimental results. Expt.

Chlorine

Gallium

Superkamiokande

Sudbury

Threshold (MeV)

0.834

0.233

5

5

R

0.33 ± 0.03

0.55 ± 0.03

0.465 ± 0.015

0.36 ± 0.015

(0.36 ± 0.015)*

(1.0 ± 0.1)**

R = * **

Measured neutrino flux

.

Predicted model neutrino flux = Neutral current corrected neutrino flux. = Total neutrino flux measured by neutral current reaction.

28

S.M. Chitre and B.N. Dwivedi

These experimental measurements are a reassurance to Solar Physicists that the resolution of solar neutrino problem should be sought in the realm of Particle Physics and that non-standard neutrino physics is responsible for the deficit in the measured neutrino fluxes, thus validating the bold proposal made by Gribov and Pontecorvo6 that the discrepancy between theoretically predicted neutrino count rate and Davis’s experiment could be due to our inadequate understanding of neutrino physics!

3.3. Seismic Sun The surface of the Sun undergoes a series of mechanical vibrations which manifest as Doppler shifts in a spectral line. These solar oscillations were discovered by Leighton et al.7 when they measured the velocity at some point on the solar disk to find an oscillatory pattern centred around a period of 5 minutes. The nature of these oscillations was clarified8,9 as acoustic modes of pulsation of the whole solar body that are trapped below the surface. Subsequently Deubner10 established that the power in oscillations is concentrated along a series of ridges in the frequency – wavenumber diagram in exact accordance with the prediction of theoretical models for acoustic modes. The frequencies of oscillations representing a superposition of millions of independent modes with amplitude of the order of a few cm/s have been determined to an accuracy of better than 1 part in 105 largely because of continuous observations extending over very long periods of time achieved with the help of ground-based networks (GONG, BiSON, TON) and satellite – borne instruments (MDI) on board SOHO. The accurately measured oscillation frequencies have provided very stringent constraints on the admissible solar models. The oscillatory modes are characterized by the spherical harmonic degree, ℓ, the azimuthal order m and the radial order n. Were the Sun to be spherically symmetric, the frequencies would be independent of m, but on account of asphericities due to effects of rotation, magnetic field, thermal perturbation, velocities, indeed, the frequencies depend on m. Since the departures from spherical symmetry are ≤ 10-5, the oscillation frequencies may be conveniently expressed as

ν nℓm = ν nℓ + ∑ a knℓ p kℓ (m) . k

Here ν nℓ is the mean frequency determined from the spherically symmetric structure of the solar interior; a knℓ the splitting coefficients and pkℓ (m) are orthogonal polynomials of degree k in m.

Seismic View of the Sun

29

There are two main classes of waves generated inside the Sun: highfrequency acoustic p-modes which are driven by pressure forces and lowfrequency gravity g-modes for which buoyancy is the main controlling force, and in between are the f-modes which are surface gravity modes which are essentially independent of stratification. The helioseismic data of oscillation frequencies may be analyzed in two ways: i) Forward method, ii) Inverse method. In the forward method, an equilibrium standard solar model is perturbed in a linearized theory to determine the eigen-frequencies of solar oscillations and these are compared with the precisely measured p-mode frequencies11. The fit is naturally seldom perfect, but a comparison of the frequencies indicates that the thickness of the convection zone is close to 200,000 km and the helium abundance by mass Y, in the solar envelope is about 0.25. The direct method has had only a limited success, although it led to an improvement of the input microphysics such as opacities and emphasized the role of diffusion of helium and heavy elements beneath the convection zone into the radiative interior12. The limitations of the forward technique prompted the use of inversion techniques13 which have proved quite effective in inferring the acoustic structure of the Sun using only equation of mechanical equilibrium. While adopting the inversion method, it is convenient to write the adiabatic equations of solar oscillations in the variational form14 which may be linearized to express the differences between the frequencies obtained from the reference model and those measured for the Sun by relating to the differences in the sound speed cs and density ρ, say, as R δν nℓ R nℓ δ c2 F (ν nℓ ) δρ = ∫ κ c 2 , ρ (r ) 2s (r ) dr + ∫ κ ρnℓ,c 2 (r ) (r ) dr + , s ν nℓ 0 s E nℓ ρ cs 0

where the kernels κ cn2ℓ, ρ (r ) and κ ρnℓ,c 2 (r ) are determined by the eigenfunctions s

s

given by the reference model and δν nℓ , δc s2 and δρ represent the difference between the Sun and a solar model. The mode inertia is Enℓ and F (ν nℓ ) is the surface term. A triumph of the inversion technique has been a very reliable inference about the internal acoustic structure of the Sun15,16. The profile of the sound  ∂ ℓn P   being the adiabatic index, can be speed, c s = Γ1 P / ρ , Γ1 ≡   ∂ ℓn ρ  s determined through bulk of the solar interior to an accuracy better than 0.1% and

30

S.M. Chitre and B.N. Dwivedi

the profiles of density and adiabatic index to a somewhat lower accuracy. There is a remarkable agreement between the profiles of the sound speed (and also density) deduced from seismic inversions and SSM, except for a pronounced discrepancy near the base of the convection zone and a noticeable difference in the energy generating core (Fig. 3.1).

Figure 3.1. Relative difference in sound speed profile between the Sun (as inferred by seismic inversions) and a standard solar model (SSM).

The hump near the base of the convection zone may be attributed to a sharp change in the gradient of helium abundance profile almost certainly resulting from diffusion, and a moderate amount of rotationally-induced mixing just beneath the convection zone can smooth out this feature17. The dip in the relative sound speed difference around 0.2 R
may be due to ill-determined chemical composition profiles in the SSM, possibly resulting from inadequate understanding of the diffusion process or from the use of inaccurate nuclear reaction rates. The adiabatic index Γ1, which is normally equal to 5/3, is decreased below this value in the hydrogen and helium ionization zones and the extent of reduction is determined by the equation of state and by chemical abundances. The dimensionless ratio of acoustic and gravitational acceleration, represented by,

W (r ) =

1 dcs2 g dr

Seismic View of the Sun

31

has a value of ≈ –2/3 through bulk of the convection zone (cf. Fig. 3.2) and shows features resulting from the dip in the adiabatic index Γ1 in the helium ionization zone. Thus, the peak in the plot of W(r) around r = 0.98 R
may be calibrated to determine the helium abundance in the convective envelope18,19 to obtain Y = 0.249 ± 0.003. The large peak in W(r) near the surface (r ≈ 0.99R
) arises from hydrogen and singly-ionized helium ionization zones, and near r = 0.7R
. The striking discontinuity in the gradient of W(r) marks the base of the convection zone which can be used to determine the depth of the convection zone. The base of the convection zone from the seismic data comes out to be (0.7315 ± 0.0005)R
20. It is also possible to estimate seismically the heavy element abundance in a manner similar to that adopted for determination of helium abundance, since different elements leave separate imprints on W(r) below the He II ionization zone. The signatures are necessarily small because of the low abundance of heavy elements, but it becomes possible to determine Z with this technique to obtain Z = 0.0172 ± 0.00221. The extent of overshoot of convective eddies beneath the base of the convection zone can also be surmised from the oscillatory signal in frequency differences. It is found to be consistent with no overshoot, with an upper limit of 0.05 Hp (Hp = local pressure scale height)22,23.

Figure 3.2. The function W(r) for a solar model is shown by the continuous line, while the dashed line represents the same for the Sun using the inverted sound speed profile.

32

S.M. Chitre and B.N. Dwivedi

The recent downward revision of solar abundances of oxygen and other heavy elements by Asplund et al.24,25 from Z = 0.017 to Z = 0.012 has led to a serious discordance between SSM constructed with new abundances of heavy elements and the seismically inferred solar models. There have been a number of attempts to resolve this vexatious problem (e.g. increasing the diffusion rate of helium and heavy elements beneath the convection zone; increasing the opacity near base of the convection zone by ~ 15-20%, increasing the neon abundance), but none of them have had a universal acceptance. The global helioseismic structure of the Sun discussed hitherto is based on the equations of mechanical equilibrium. However, in order to determine the temperature and chemical composition profiles in the solar interior, it is necessary to supplement the seismically inferred structure through primary inversions by the equations of thermal equilibrium, together with the auxiliary input physics such as the opacity, equation of state and nuclear energy generation rates26,27,28. It turns out that the inverted sound speed, density, temperature and composition profiles, and consequently the resulting neutrino fluxes come close to those predicted by SSM. In general, the total solar luminosity computed from the seismically inferred profiles would not necessarily match the observed total luminosity (L
≈ 3.8 × 1033 erg s-1). The discrepancy between the computed and observed solar luminosity can, in fact, be effectively used to provide a diagnostic test of input nuclear physics; in particular, it can be demonstrated that the proton-proton cross-section, S11 needs to be increased slightly to S11 = (4.06 ± 0.07) × 10-25 MeV barn29. The calculations also enable us to set limits on the heavy element abundance, Z in the solar core30. The seismic models lead to a determination of the central temperature of Sun ~ (15.6 ± 0.4) × 106 K, allowing for an uncertainty of up to 10% in the opacities31. It is also possible to infer the helium abundance profile Y, assuming the heavy element abundance, Z. The resulting helium abundance profile comes to a fairly close agreement with that obtained with SSM which includes diffusion, except in the regions just below the convection zone where the abundance profile is essentially flat27. This is again indicative of some sort of a mixing, possibly resulting from a rotationally- induced instability. It is noteworthy that the temperature at base of the convection zone is ≤ 2.2 × 106 K which is not adequate to burn lithium. However, if there is some amount of mixing that extends beyond the base to a radial distance of 0.68R
, temperatures will exceed 2.5 × 106 K at which lithium can be destroyed by nuclear burning to explain the low lithium abundance at the solar surface.

Seismic View of the Sun

33

Figure 3.3. The inferred rotation as a function of depth inside the Sun at different solar latitudes.

The surface rotation of the Sun has been measured through observations of sunspots and other features. It is well known that the Sun has differential latitudinal rotation with the equatorial layers rotating faster than the polar regions. The rotation rate in the Sun’s interior could be determined from the frequency splittings of various modes, since each mode of solar oscillation is trapped in a different region and it is possible to infer both the radial and latitudinal variation of the rotation rate inside the Sun using the accurately measured frequency splittings of acoustic modes. The Coriolis force gives the first-order contribution from rotation with the resulting splittings having only odd powers of the azimuthal order m. These odd order splitting coefficients, a1, a3, a5, … enable a determination of the rotation rate as a function of radius and latitude. The centrifugal force which is of second order in rotation contributes only to even order splitting coefficients, a2, a4, a6, …. The rotation rate obtained by averaging the GONG and MDI data sets, as a function of fractional solar radius and latitude is displayed in Fig. 3.3. It is evident that the observed surface differential rotation persists through the solar convection zone, with the radiative interior rotating almost uniformly32. The transition region near the base of the convection zone, called the tachocline, is centred at a radial distance of r = (0.6916 ± 0.0019)R
has a half-thickness of

34

S.M. Chitre and B.N. Dwivedi

(0.0065 ± 0.0013)R
33. There is evidence for a shear layer just beneath the solar surface extending to r ≈ 0.95R
with the rotation rate increasing with depth. The helioseismically inferred rotation rate in the solar interior is consistent with the measured solar oblateness of approximately 10-5 (ref. 34). The resulting quadrupole moment turns out to be (2.18 ± 0.06)10-7 (ref. 35) which leads to a precession of perihelion of the orbit of planet Mercury of ~ 0.03 arcsec/century. This is clearly consistent with the framework of general theory of relativity.

3.4. Inconstant Sun The accumulation of helioseismic data by GONG and MDI projects over this past solar cycle 23 has enabled a study of the temporal variations in the solar structure and dynamics with the activity cycle. Earlier works11,36 of Libbrecht & Woodard36, and Elsworth et al.11 (1990) had, in fact, established temporal variations of the solar oscillation frequencies with the shift in the frequencies of acoustic modes up to 0.4 µHz during the course of the cycle, with maximum frequency occurring at the peak of solar activity. It was also demonstrated by Bhatnagar et al.37 that the frequency variation is well correlated with various solar activity indices. Furthermore, the frequency shift scaled by mode inertia depends on frequency alone, thus indicating that the frequency variations probably occur near the surface layers of the Sun and that the temporal variations in the structure are confined to the outermost layers below the photosphere and there is hardly any variation discerned in the solar interior. Likewise, temporal variations of the even splitting coefficients appear to be well correlated with the corresponding component of the surface magnetic flux38 and the surface term is found to incorporate most of the observed time variations. The time variations in the frequencies of f-modes which are surface gravity modes, however, seem to have two components: an oscillatory component with a period of almost exactly 1 year and a secular component well correlated to solar activity. The former is very likely reflecting the orbit period of Earth in data analysis, while the latter is probably on account of the magnetic field located in the superficial sub-surface layers varying with the solar activity cycle. One of the most striking observations about solar rotation rate was made by Howard and LaBonte39 to demonstrate that there is a temporal variation of the surface rotation rate with the solar activity cycle. These observations established the existence of faster- and slower- than average bands migrating towards the equator. This so called “torsional oscillation” was found to be highly correlated to the migrating field pattern in the ‘butterfly diagram’40. The helioseismic data accumulated over the solar cycle has now established that this zonal band pattern

Seismic View of the Sun

35

persists through the solar convection zone41,42. It is found that at mid-latitudes the bands of fast and slow rotation migrate equatorwards like the sunspot butterfly plot, while at high latitudes, they migrate towards the poles resembling the pattern of magnetic features observed at the surface43,44 (Fig. 3.4). Interestingly, helioseismic observations over the solar cycle indicate that the polar regions speed up while the equatorial latitudes slow down with the buildup of the activity cycle45. Through bulk of convection zone it appears that the temporally varying kinetic energy of rotation shows an increasing trend in high latitudes and a decreasing trend in the low latitudes with progress of the solar cycle. The residual kinetic energy of rotation and magnetic energy are, in fact, out of phase in the equatorial band and the magnetic field tends to show an increase from minimum to the maximum phase of the solar cycle. It is, therefore, tempting to speculate that angular momentum is being transported from the equatorial to polar regions during the course of the solar cycle, perhaps by meridional circulations serving as a conveyor belt !

Figure 3.4. Migrating zonal bands.

Solar irradiance is known to vary on at least two time-scales – one related to the period of solar rotation and the other on the longer time-scale of activity cycle with the amplitudes of both these variations ≥ 0.1% (ref. 46). The short time-scale changes (~ 1 month) are probably a consequence of surface magnetic features (e.g. sunspots, faculae, magnetic network), while the longer time-scale

36

S.M. Chitre and B.N. Dwivedi

(~ 11 years) variations which are in phase with the solar activity cycle may be primarily due to changes in the solar luminosity. It is likely that the solar irradiance variation arises partly from the competing effects of darkening due to sunspots and brightening due to faculae and also the network and also partly from cyclic changes in the luminosity caused by fluctuations in the thermal energy content in the outer layers of the Sun. During the course of the activity cycle, there is a slight brightening and darkening of the Sun and its total energy which is modulated periodically and gets channelled into different reservoirs such as potential, rotational or magnetic. One of the outstanding problems in Solar Physics is to identify a plausible underlying mechanism that is responsible for simultaneous temporal variation with solar activity cycle of oscillation frequencies, solar rotation, magnetic field and total solar irradiance.

Acknowledgements We thank H.M. Antia for valuable comments and for supplying the Figures.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15.

Bahcall, J.N. and Pinsonneault, H.M., Rev. Mod. Phys. 67, 781 (1995). Davis, R., Phys. Rev. Lett. 12, 303 (1964). Bahcall, J.N., Astrophys. J. 149, L7 (1967). Bahcall, J.N., Basu, S. and Pinsonneault, H.M., Phys. Lett. B433, 1 (1998). Chitre, S.M., Bull. Astron. Soc. India 23, 379 (1995). Gribov, V. and Pontecorvo, B., Phys. Lett. B28, 493 (1969). Leighton, R.B., Noyes, R.W. and Simon, G.W., Astrophys. J. 135, 474 (1962). Ulrich, R.K., Astrophys. J. 162, 993 (1970). Leibacher, J.W. and Stein, R.F., Astrophys. J. 7, L191 (1971). Deubner, F.-L., Astron. Astrophys. 44, 371 (1975). Elsworth, Y., Howe, R., Isaak, G.R., McLeod, C.P. and New R., Nature 347, 536 (1990). Christensen-Dalsgaard, J., Proffitt, C.R. and Thompson, M.J., Astrophys. J. 403, L75 (1993). Gough, D.O. and Thompson, M.J., in Solar Interior and Atmosphere (eds. Cox, A.N., Livingston, W.C., Mathews, S.M.), Univ. Arizona Press, p. 519 (1991). Chandrasekhar, S., Astrophys. J. 139, 664 (1964). Gough, D.O. et al., Science 272, 1296 (1996).

Seismic View of the Sun

37

16. Kosovichev, A.G. et al., Solar Phys. 170, 43 (1997). 17. Brun, A.S., Turck-Chièze, S. and Zahn, J.-P., Astrophys. J. 525, 1032 (1999). 18. Gough, D.O., Mem. Soc. Astron. Ital. 55, 13 (1984). 19. Däppen, W., Gough, D.O., Kosovichev, A.G. and Thompson, M.J., Lecture Notes in Phys., Vol. 388, 111 (1991). 20. Basu, S., Monthly Notices Roy. Astron. Soc. 298, 719 (1998). 21. Antia, H.M. and Basu, S., Astrophys. J. 644, 1292 (2006). 22. Basu, S., Antia, H.M. and Narasimha, D., Monthly Notices Roy. Astron. Soc. 585, 553 (2003). 23. Monteiro, M. J. P. F. G., Christensen-Dalsgaard, J. and Thompson, M. J., Astron. Astrophys. 283, 247 (1994). 24. Asplund, M., Greversse, N., Sauval, A.J., Allende Prieto, C. and Kiselman, Astron. Astrophys. 417, 751 (2004). 25. Asplund, M., Grevesse, N. and Sauval, A.J., in Cosmic Abundances as Records of Stellar Evolution and Nucleosynthesis, ASP Conf. Ser. 336, p. 25 (2005). 26. Gough, D.O. and Kosovichev, A.G., in Inside the Sun, IAU Coll. 121, 327 (1990). 27. Antia, H.M. and Chitre, S.M., Astron. Astrophys. 339, 239 (1998). 28. Takata, M. and Shibahashi, H., Astrophys. J. 504, 1035 (1998). 29. Brun, A.S., Antia, H.M., Chitre, S.M. and Zahn, J.-P., Astron. Astrophys. 391, 725 (2002). 30. Antia, H.M. and Chitre, S.M., Astron. Astrophys. 393, L95 (2002). 31. Antia, H.M. and Chitre, S.M., Astrophys. J. 442, 434 (1995). 32. Thompson, M.J., Toomre, J.; Anderson, E., Antia, H.M., Berthomieu, G., Burtonclay, D., Chitre, S.M., Christensen-Dalsgaard, J., Corbard, T., Derosa, M. and 16 coauthors, Science 272, 1300 (1996). 33. Basu, S. and Antia, H.M., Astrophys. J. 585, 553 (2003). 34. Kuhn, J.R., Bush, R.I., Scherrer, P. and Scheick, X., Nature 392, 155 (1998). 35. Pijpers, F.P., Monthly Notices Roy. Astron. Soc. 297, L76 (1998). 36. Libbrecht, K.G. and Woodard, M.F., Nature 345, 779 (1990). 37. Bhatnagar, A., Jain, K. and Tripathy, S.C., Astrophys. J. 521, 885 (1999). 38. Antia, H.M., Basu, S., Hill, F., Howe, R.W. and Schou, J., Mon. Not. R. Astron. Soc. 327, 1029 (2001). 39. Howard, R. and LaBonte, B.J., Astrophys. J. 239, L33 (1980). 40. Snodgrass, H.B., Astrophys. J. 383, L85 (1991).

38

S.M. Chitre and B.N. Dwivedi

41. Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R.W., Larsen, R.M., Schou, J., Thompson, M.J. and Toonre, J., Astrophys. J. 533, L163 (2000). 42. Vorontsov, S.V., Christensen-Dalsgaard, J. Schou, V.N., Strakhov, V.N. and Thompson, M.J., Science 296, 101 (2002). 43. Leroy, J.-L. and Noens, J.-C., Astron. Astrophys. 120, L1 (1983). 44. Makarov, V.I. and Sivaraman, K.R., Solar Phys. 123, 367 (1989). 45. Antia, H.M., Chitre, S.M. and Gough, D.O., Astron. Astrophys. 477, 657 (2008). 46. Fröhlich, C. and Lean, J., Astron. Astrophys. Rev. 12, 273 (2004).

CHAPTER 4 SOLAR MAGNETISM P. VENKATAKRISHNAN and SANJAY GOSAIN Udaipur Solar Observatory, Physical Research Laboratory, P. Box 198, Dewali, Udaipur 313001, Rajasthan, India

INTRODUCTION This chapter is basically divided into 2 parts. In the first part, the important properties of the solar magnetic field are summarized. The discussion begins with a simple introduction to solar magneto hydrodynamics. This introduction will be sufficient to understand the current status of the solar dynamo theory that follows. Some very curious and interesting results on force free fields are then presented in very basic terms. Finally, the application of this theoretical framework to the problems of coronal heating, solar flares and coronal mass ejections are developed in a simple unified scheme, based on a hierarchy of physical conditions. The second part consists of a tutorial on magnetographs. It begins with a description of polarization of light from very fundamental notions of coherence of light. This is followed by simple but comprehensive explanations of the Zeeman and Hanle effects along with the necessary basic ideas of quantum physics of scattering of light. Then the working of a few important magnetographs is outlined, with special emphasis on a solar vector magnetograph developed for USO, to provide a “hands on” perspective. The article concludes with a few brief remarks on the possible future directions for research in the domain of solar magnetism. PART 1. FUNDAMENTALS There is a theorem in astrophysics called the Vogt-Russell theorem, which states very confidently that the structure of a star can be completely determined once we know its mass and chemical composition. Alas, this confidence is not fully justified. The reason is that the atmospheres of two stars with the same mass, 39

40

P. Venkatakrishnan and S. Gosain

luminosity, and chemical composition are sometimes seen to be very much different from each other. What causes such differences? The answer is not completely understood by astrophysicists. What we are somewhat sure about is the fact that the study of the Sun’s atmosphere and its relationship with the features seen on the Sun’s surface will eventually provide the key to solve this puzzle. 4.1. Sunspots and the Eleven Year Sunspot Cycle The first inkling of the surprises in store for astronomers came with the invention of the telescope. Galileo turned a telescope at the Sun, and found that the Sun’s face was not pure white, but had several dark spots on it. These blemishes on the face of a heavenly body caused a lot of confusion to the religious leaders of that time who always imagined that heavenly bodies were free of defects. The dark sunspots remained a curiosity for many years. Careful recording of the sunspots’ positions, day after day, for many years by many scientists revealed a curious waxing and waning of the number of sunspots with a rhythm of about eleven years. The names of Schwabe, Carrington, Wolf, Maunder and Sporer, are linked with the exciting story of the discovery of the sunspot cycle1, shown in figure 1. What made these spots increase and decrease in number? Why are the spots dark? These were some of the provocative questions that arose at that time. Even though some progress has been made towards answering these questions, we are far away from a complete understanding. The first step towards a physical understanding of sunspots became possible soon after Zeeman discovered, in 1896, that spectral lines formed in magnetic fields split into many components and was awarded the Nobel Prize for this discovery. George Ellery Hale, of Mount Wilson Observatories in the United States of America, had noticed that the picture of sunspots taken in a spectral line of hydrogen showed whirlpool-like structures. He was reminded of the distribution of iron filings around a magnet. The Zeeman effect now gave Hale a way to detect magnetic fields in sunspots by looking for splitting of spectral lines in the light coming from sunspots. Indeed, he did see such a splitting in 1908, which confirmed that the sunspots had strong magnetic fields present in them (Hale, 1908). John Evershed, from Kodaikanal observatory wanted to see whether the field was produced by a whirlpool motion of gases within the sunspot. He went about measuring the motions of the gases using a spectrograph, and was surprised to see no whirlpool motion, but a radial outward motion from sunspots (Evershed, 1909). This discovery, in 1909, is yet to be satisfactorily explained. Evershed’s intuitive ideas about the possible ways of producing

Solar Magnetism

41

magnetic fields in electrically conducting fluids were much ahead of his times. The science of conducting fluids became fully developed only in middle of the 20th century, culminating in the discovery of hydro-magnetic waves by Hannes Alfven (Alfven 1942), for which he received the Nobel Prize in 1970. This new branch of physics began to be vigorously applied to the problem of magnetic field production in the Sun. In the regions of strong magnetic fields, the movement of plasma across the fields sets up an inductive electric field, much in the same way as the flow of induced current along a coil of wire that is rapidly moved in a magnetic field. The induced electric current in the solar plasma produces a magnetic field opposing the original motion of the fluid, which set up the induced current in the first place.

Figure 1. Plot of monthly averaged sunspot number showing the 11 year cycle.

42

P. Venkatakrishnan and S. Gosain

4.2. Magnetic Cycle Judging by the very “broken up” nature of the magnetic field on the solar surface (figure 2), we might even wonder whether one can talk about the magnetic field of the whole sun. However, there are several indications that there is something global about the magnetic field. First, we must realize that magnetic dynamos depend on the pattern of fluid motion. Evershed failed to find local dynamos for the individual sunspots, thereby indicating that the origin of the spot magnetic fields is related to some other pattern of fluid motions, not related to the individual spots. One such global motion is the rotation of the Sun, which Carrington first noticed from the systematic movements of the sunspots across the face of the Sun. Hale and Nicholson (1938) noticed that all the pairs of sunspots having opposite magnetic polarity behaved in a systematic way during every eleven year sunspot cycle (figure 3). First, these bipolar spots invariably had their axis almost parallel to the solar equator (as defined by the solar rotation). Each bipolar pair of spots had their magnetic polarity pointed the same way during every eleven years. After completion of eleven years, the pairs appearing in the new cycle had their polarities swapped with respect to that of the previous cycle. Further, the pattern in the southern hemisphere was opposite to that in the northern hemisphere of the Sun. This type of systematic behavior over a long time clearly indicates a global origin for the spot magnetic fields as can be visualized in a “magnetic butterfly diagram” in figure 4.

Figure 2. A full-disk magnetogram showing line-of-sight magnetic flux distribution, observed by SoHO/MDI satellite.

Solar Magnetism

43

Figure 3. A Cartoon of the Hale’s polarity rule. At T=0, (sunspot minimum) the spots of a new cycle appear at high latitudes. The polar field has maximum strength during sunspot minimum. The polarity of the polar field is the same as that of the leading spot of the corresponding hemisphere. The size of the spot cartoon is a measure of the number of spots. As we proceed into the cycle (T=2.75 years), the spots increase in number and appear at lower latitudes. At the same time, the strength of the polar field decreases (depicted by decreasing area). At solar maximum (T=5.5 years), the polar field goes through a minima reversing its sign, while sunspots appear at mid latitudes. At the next sunspot minimum (T=11 years), the old cycle spots are close to the equator while new cycle spots appear at high latitudes with their signs reversed. This continues through the new cycle maximum (T=16.5 years), until T=22 years, when magnetic cycle is completed.

Figure 4. Magnetic butterfly diagram indicates the global solar surface magnetic field over one complete magnetic cycle. The magnetic field is shown as gray scale vs. latitude and time as constructed from a sequence of Carrington maps obtained at NSO KP. White areas represent positive polarity; darker areas represent negative polarity.

44

P. Venkatakrishnan and S. Gosain

A simple model to explain this behavior of magnetic field was proposed by Babcock (1961). He assumed that the Sun has a seed field that stretches from one pole to the other, like what one would encounter in a bar magnet, for example. If the Sun rotated about its axis with the same period at all the latitudes, then this seed field would merely move round and round like the wires of a wicker basket would move around if we rotated the basket around (see figure 5). However, the Sun is known to rotate faster at the equator than at the poles. If we continued our experiments with the wicker basket and made the middle portion of the basket rotate faster than its end portions, then we will end up twisting up the basket into a sorry shape. The uneven rotation of the Sun will do very much the same thing to the seed field, producing large twists in the field. The twist in the lines of force can go unchecked, but nature has a safety valve. The twisted field has a tendency to push material out of the knotted places, making these portions lighter than their surroundings. The knots in the field then rise up to the surface of the Sun and produce the sunspots.

Figure 5. Generation of toroidal field from poloidal field by the action of differential rotation: (a) Purely poloidal field (b) & (c) progressive winding of the field lines (d) Purely toroidal fields.

A part of the knotted field gets straightened out again because of untwisting cyclonic motion of the gases in the convection zone, and we get back some of the seed field.

Solar Magnetism

45

If the twisting motions were to be stronger than the untwisting motions, then eventually all the seed field would get twisted up leaving no more seed field and the process would have to stop. If the untwisting motions were more effective, then we will have only the seed field with no sunspots. Thus, there is a balance between the two kinds of motions giving a continuing process. The sunspot fields eventually are fragmented into weaker fields that move to the poles and are destroyed. The game begins all over again, except that the sunspot fields change their polarity. This explanation by Babcock is only a description of the observed changes in the sunspot field using the movements of the solar plasma as a basis. Surprisingly no one has been able to give a proper explanation despite a lot of study about the generation of magnetic fields. In what follows, will present the current theoretical understanding of the solar dynamo after reviewing the basic magnetohydrodynamic equations. 4.3. Basic MHD Magnetohydrodynamics is the combination of electrodynamics with fluid dynamics. The Maxwell’s equations (Eqs. 1a-1d) provide the inter-relationship between the electric and magnetic fields in any medium, including a vacuum. The first equation expresses the electric field E for a given distribution of the electric charge density ρ , where ε ο is the permittivity of free space. The second equation denotes the non-existence of magnetic monopoles. The third equation is Ampere’s law for the creation of magnetic field B due to the distribution of current density j, with Maxwell’s contribution of the displacement current appearing as the second term of the R.H.S. Here µ is the permeability in a vacuum. The last equation is the Faraday’s law of electromagnetic induction. For magnetohydrodynamics, a simplified form of these equations is used as described in the following section. These simplified equations inter-relate the fluid velocity with the magnetic field through the induction equation. Finally, inclusion of the Lorentz force in the equation of motion of the fluid provides the generator for fluid motion caused by the magnetic field. This, basically, is the essence of magnetohydrodynamics. ∇ ⋅ D = ρ∗

(4.1a)

∇⋅B = 0 ∇ × H = j + ∂D / ∂t

(4.1b) (4.1c)

∇ × E = −∂ B / ∂ t

(4.1d)

46

P. Venkatakrishnan and S. Gosain

Here, B = µH and D = εE , µ is the permeability and ε is the permittivity of the medium. 4.4. MHD Approximation If L is a typical length scale and T is a typical time scale of the process, then MHD deals with the situation where L/T > displacement current. Thus, the displacement current can be neglected in the MHD approximation and Ampere’s law simplifies to curl B = µj. The second assumption for the MHD approximation is charge neutrality. From Gauss’s law, we have E=Le(n+ - n-)/ε, or n+ - n- = εE/(Le) = εB/(Te). For charge neutrality, n >> n+ - n-, or n >> εB/(Te). Finally, we need to modify the Ohm’s law for a conducting fluid in motion. A fluid moving in a magnetic field is subject to an electric field v × B in addition to the electric field E at rest. Ohm’s law then applies to the total field as j = σ(E + v×B). Using the above three ingredients of the MHD approximation (viz., Ampere’s law, j =∇×B/µ, charge neutrality, and the modified Ohm’s law E = j/σ–(v×B), we can arrive at the induction equation in magneto-hydrodynamics as ∂B/∂t = ∇× (v × B) –∇× ((∇×B) /σ). This is the fundamental equation that is used in solar dynamo theory for generation of magnetic field. 4.5. Solar Dynamo The aim of any solar dynamo theory is to explain the sustained cycle of the global magnetic field with the period of 22 years. A further goal would be to understand the slow variation in the strength of a cycle over a time period, which is much larger than 22 years. It is well known that any solenoidal vector field (which has zero divergence) like the magnetic field can be expressed as the sum of a poloidal and a toroidal field. In the case of the sun, we can imagine the process to begin with a completely poloidal field (as manifested during a sunspot minimum) which gets transformed into a chiefly toroidal field (as manifested during a sunspot maximum) through the action of differential rotation. This toroidal field needs to be converted back into a poloidal field, but with reversed polarity, at the end of eleven years. A similar half cycle is repeated to complete the 22 year magnetic cycle. Parker proposed the first theory in terms of a mean field kinematic dynamo (Parker 1955). Essentially, Parker separated the fluid variables into a slowly varying component and a fluctuating component. After

Solar Magnetism

47

taking an ensemble average over the corresponding 2 components of the induction equation, Parker obtained an induction equation in which the rate of change of the mean magnetic field is proportional to the mean field with a coefficient containing the ensemble average of the fluctuation of kinetic helicity. Similar to the situation in terrestrial cyclones, we expect this ensemble average to have opposite signs in the two hemispheres. Parker finally obtained an oscillatory solution for mean field in terms of the correlations of the flow field. The chief parameters which determine the functioning of the dynamo are 1) Ω parameter which is proportional to the gradient of rotational profile and 2) the Parker’s α parameter which is proportional to the chirality of the vortex motions of the rising eddies in the convection zone. The Ω parameter is mainly responsible for converting the poloidal field into the sunspot-forming toroidal field, while the α parameter is chiefly responsible for converting the toroidal field back into a poloidal field. By assuming suitable values for these parameters, Parker was able to obtain a period for the oscillation that was in the region of tens of years. The product of α and Ω had to be negative for the existence of an oscillatory dynamo. Later, Radler, Krause and Stix, made extensive numerical calculations and were able to simulate the various properties of the real solar dynamo with suitable models of the flow field (Stix, 1976). Further, other variants of this α-Ω dynamo, e.g. the α2 and α2-Ω dynamos were also proposed . A “wake up call” occurred for solar dynamo theorists when Peter Gilman completed a dynamical model of the solar dynamo, which calculated the flow field generating the magnetic field from the numerical simulation of magnetoconvection in a rotating sun (Gilman 1986). Gilman’s flow field followed essentially the Taylor-Proudman theorem, which states that the rotation of a slowly rotating fluid sphere will be constant on cylinders co-axial with the axis of rotation. However, the resulting “butterfly diagram” had spots moving towards the poles, opposite to the equator-ward movement seen in the actual sun! The solution to this paradox became clear when helioseismology enabled solar physicists to map the depth and latitude dependence of the solar internal rotation. The solar rotation was found to decrease with depth until the base of the convection zone, at which there was a sharp gradient of the rotation followed by a regime of rigid rotation. This gradient was named the tachocline. The problem of why the sun does not follow the Taylor-Proudman theorem is a topic of contemporary study. The tachocline is an ideal site for generating a strong toroidal field from the poloidal field. However, there are two basic problems associated with the

48

P. Venkatakrishnan and S. Gosain

tachocline. The first is that the gradient of the rotation profile is strongly positive at the tachocline. This, combined with the sign of Parker’s alpha parameter violates Parker’s criterion for an oscillatory dynamo. Secondly, the vigour of overshooting convection at the base of the convection zone is not strong enough to distort the very strong toroidal fields, that are necessary to exhibit Joy’s law according to the mechanism proposed by D’Silva and Choudhuri (1993). Thus, Parker’s suggestion for generating the poloidal fields from toroidal fields by the action of cyclonic eddies in the convection zone, would not work in the tachocline. A combination of Joy’s law and diffusion of magnetic flux on the solar surface does seem to generate the poloidal fields as demonstrated by the so-called flux transport mechanisms (Sheeley et al 1989). But these models do not seem to predict the strength of the poloidal field at sunspot minimum from a given amount of toroidal field seen at sunspot maximum. Further, the amount of “open magnetic flux” predicted from these models is not compatible with the latitudinal and temporal variation of the observed interplanetary magnetic field. Thus, there is plenty of scope for serious theoretical studies in this branch of solar magnetism. 4.6. Force-free Fields The main forces acting on a fluid parcel in the solar atmosphere are the Lorentz force, the gradient of plasma pressure and the weight of the material. The Lorentz force has two parts, 1) the tension force given by (B. ∇ )B/4 π and the pressure force given by ∇ (B2/8 π ). The plasma pressure and density decrease exponentially with height, while the magnetic field decreases less sharply as a polynomial function (Spruit,1983). Thus, the magnetic forces completely dominate the dynamics in the solar chromosphere and corona. At these heights, the magnetic tension and magnetic pressure force can no longer be individually balanced by the plasma pressure gradient and gravity force. Hence, for equilibrium, the magnetic tension and pressure forces have to balance each other, making the net Lorentz force zero. This configuration of magnetic field with zero Lorentz force is called a force free field. In a force free field, the current is aligned with the field and is proportional to it. Thus, ∇ × B = α B. This implies that (B. ∇ ) α = 0, or α is constant along a field line. This condition leads to the following problem of compatibility at the two foot-points of a magnetic field. We know that the force-free parameter alpha is determined by the 3 components of the photospheric vector magnetic field at each foot-point. However, the vector field at each foot point is independently specified by the sub-photospheric flow beneath each point. Since there is no

Solar Magnetism

49

physical condition to prevent arbitrary flow parameters at the foot points, the value of alpha can be quite different at the foot points. This violates the constancy of alpha for a force free field. Parker sought a way out of this dilemma, by invoking the spontaneous creation of discontinuities in the corona (Parker, 1994). These discontinuities could well produce electric fields and currents on small scales. These small scale currents are easily dissipated, but the discontinuity in alpha will be maintained by the photospheric driver. In fact, this is indeed Parker’s suggestion for coronal heating by the dissipation of small scale coronal currents which are maintained by the photospheric motions of the foot-points. This mode of coronal heating can be considered as a stationary stochastic process, as long as the rate of photospheric driving is equal to the rate of dissipation of the coronal currents. 4.7. Solar Eruptions The twisting action of the sub-photospheric flow can produce magnetic energy build-up on large scales. When the rate of energy supply is faster than the rate of energy dissipated by field discontinuities in the corona, the free energy or nonpotential energy of the magnetic field gets stored in the azimuthal component of the magnetic field. This component of the magnetic field can be detected only by a vector magnetograph. Several studies have shown that flares are chiefly produced in active regions possessing a large amount of free energy. It is also fairly certain that the interaction of existing field with newly emerging field acts as a trigger to initiate the onset of non-equilibrium of magnetic field that results in a solar flare. In the canonical picture of a solar flare evolved from concepts of Carmichael (1964), Sturrock (1966), Hirayama (1974), Kopp and Pneuman (1976), called the CSHKP model, the action begins in the corona (figure 6). Reconnection of coronal magnetic fields at the cusped summit of a loop system accelerates electrons (and rarely, protons) which stream down to the base of the chromosphere and generate Bremhstrahlung radiation in the hard X-ray part of the electromagnetic spectrum. H-alpha is also emitted due to recombination of the ionized hydrogen atoms. The process takes place so quickly, that the plasma has no time to relax to thermodynamic equilibrium. Thus, the hard X-ray as well as the H-alpha emission induced by electron precipitation are both non-thermal processes. The H-alpha emission is initiated in small kernels, which quickly spread out in the form of flare ribbons on either side of the magnetic polarity inversion line. By the time plasma relaxes to thermodynamic equilibrium, the intense heating of the plasma leads to explosive evaporation. The evaporation is explosive because of a thermal instability of hydrogen plasma which sets in at

50

P. Venkatakrishnan and S. Gosain

~ 10000 degrees K and catastrophically sets the temperature at 1 million degree K. Hot plasma expands quickly and fills the overlying coronal loops. The thermal radiation in the hot plasma is in the form of soft X-rays. Thus, soft X-ray onset will coincide with H-alpha flare and the hard X-ray spike, but soft X-ray peak emission occurs a little later, when the entire loop is filled and starts cooling. The source of energy for a flare is the non-potential magnetic energy of the active region, while the trigger is most likely to be a local re-arrangement of field lines caused by magnetic reconnection when a newly emerging flux system impacts on an existing system. Thus study of magnetic field of active regions is crucial for understanding flares.

Figure 6. Cartoon of the standard flare model, also known as CSHKP model.

4.8. Coronal Mass Ejections According to the most recent understanding of the CME phenomenon, the physical processes responsible for a CME start right from the solar interior, where the solar magnetic field is generated. This magnetic field emerges into the photosphere and later into the corona in the form of active region fields.

Solar Magnetism

51

Although the magnetic flux changes sign during each sunspot cycle, the magnetic helicity does not and keeps accumulating in the corona. Some part of the nonpotential or free energy is released through magnetic reconnection in the form of flares. The rest remains associated with the helicity in the form of a magnetic flux rope, which is confined by the remainder of the original magnetic field that escaped annihilation by the magnetic reconnection. The flux rope is also anchored by the weight of the associated prominence material. When the accumulated helicity attains a critical value, the confinement fails, either by reconnection above the rope (so-called break-out model of Antiochos, De Vore and Klimchuk 1999) or by reconnection below the rope (tether cutting model of Moore and La Bonte 1980) The flux rope then escapes from the sun in the form of a CME, with the canonical three part structure of the bright front (disturbed coronal streamer), followed by the cavity (the flux rope) and the core (part of the associated prominence; Zhang & Low 2005). PART 2. MEASUREMENT TECHNIQUES 4.9. Polarization of Light Visible light is part of the electromagnetic spectrum with frequencies in the range 1015 to 3 × 1014 Hz. Thus, the electric field of the electromagnetic wave oscillates over one cycle in 10-15 s. The source of the wave is any atom of the solar plasma, which is subjected to collisional and radiative excitation and deexcitation at typically 108 times per second. Moreover, several atoms undergo these processes at different times with no correlation between each other. The result is a set of broken wave trains with random phases. A light detector will detect a superposition of these randomly generated wave trains. Hence there will be no preferential plane of vibration for the electric field and the light is said to be unpolarized. However, any process which breaks this symmetry can produce a difference between the intensities of light measured in any two orthogonal planes around the direction of propagation, as well as a net phase difference between the oscillations of the electric field in the two directions. This results in the polarization of light. Mathematically, Stokes (1852) formulated the four parameters which fully characterize the polarization state of the light. These are given by I = < Ex.Ex*> + < Ey.Ey*> Q = < Ex.Ex*> - < Ey.Ey*> U = 2 Re <Ex.Ey*> V = 2 Im <Ex.Ey*>,

52

P. Venkatakrishnan and S. Gosain

where Ex and Ey are the complex amplitudes of the electric field and < > denotes ensemble average. Experimentally, I parameter is obtained by adding the intensities transmitted through a Polaroid whose axis is placed along x- and ydirections respectively, Q from the difference of the two intensities, U from difference in intensities along 45 and 135 degrees respectively to the x-axis, while V is obtained by doing a Q measurement after inserting a quarter wave retarder before the Polaroid. 4.10. Zeeman Effect As mentioned earlier, Zeeman detected a splitting of spectral lines when the source of light is placed in an external magnetic field. The so-called normal Zeeman effect shows right and left circularly polarized line components on the 2 sides of the central line and no line at the original position, when the direction of magnetic field is parallel to the light propagation direction. When the field is transverse to the propagation direction, there is a central or pi component which is linearly polarized in a direction parallel to the field direction, while the sigma components on either side of the pi component are linearly polarized, with the direction of polarization perpendicular to the magnetic field direction. The Zeeman effect can be explained in the following way. If at least one level of the atomic transition (producing the spectral line) is degenerate, then the degeneracy is lifted in the presence of the magnetic field. This is because the interaction energy of the magnetic field with the spin induced magnetic moment of the electron depends on the azimuthal quantum number. Thus, we now have the possibility of transitions between the various m-states of each level. However, the selection rule for change in m quantum number selects only certain types of transitions. When ∆ m = ± 1, circularly polarized light is emitted, while ∆ m = 0 transitions yield linearly polarized light The separation between the split components of the spectral lines depends on the strength of the magnetic field and splitting cannot be observed if this separation is smaller than the thermal broadening of the line. For such weak fields, one could use the variation of the polarization along the line profile to infer the strength and direction of the magnetic field. Typically, with a thermal broadening of 25 mÅ in visible Fe I lines of solar photosphere, the splitting is detectable only for fields stronger than about 1000 G. But splitting can be observed for weaker fields (300 G) in the infra-red Fe I lines at 15600 Å.

Solar Magnetism

53

Figure 7. Schematic of the classical picture of the Zeeman effect.

4.11. Hanle Effect In the Zeeman effect, the individual transitions from different m-levels of the excited state to the lower energy state produce separate spectral lines with slightly different wavelengths. In the case of a scattering, there is a transition from a lower energy level to a higher energy level followed by transition back to the lower energy level. Since the transitions occur in the same atom, there is stable phase relation between the incident and scattered photon. Thus, the process of scattering produces polarized radiation. This is called resonance scattering polarization. However, when the atom is immersed in an isotropic radiation field, the net polarization is zero. On the other hand, if the incident radiation field is anistropic, as for example near the solar limb, then we do detect the polarization in the scattered spectral line. When the scattering involves a degenerate upper level of the atomic transition, then a photon can excite all the degenerate states together in a coherent superposition of states. When deexcitation takes place, the resulting photon carries information about all the degenerate upper levels through a process of quantum interference. The emitted (or scattered) photon will continue to bear coherency relationships with the absorbed (or incident) photon. Now, if we introduce a magnetic field, then the degeneracy is lifted and we expect the quantum interference to be completely destroyed. This is not the case, as long as the interaction energy of the electron with the external magnetic field is smaller than the uncertainty in energy produced by the finite life-time of the upper energy level. Thus, the presence of a magnetic field that is weak enough to satisfy the above condition will modify the

54

P. Venkatakrishnan and S. Gosain

quantum interference and produce detectable effects like a) reduction in degree of polarization and b) rotation of the plane of polarization. These effects, which were first detected by Hanle in 1924 in laboratory sources, are together called the Hanle effect (Hanle, 1924). Since the Hanle effect is independent of the velocity of the frame of reference, we can detect this effect even in highly broadened lines, even for coronal emission lines. For the typical conditions in the solar photosphere, this effect can be used to detect fields as small a few tens of gauss in the solar photosphere. 4.12. Solar Magnetographs A solar magnetograph is an instrument that measures the polarization of sunlight emanating from a specific portion of the solar disk and within a narrow band of the solar spectrum. Older magnetographs exploited the Zeeman effect, where the polarization is a definite function of the position on the line profile. By using the values of the Stokes parameters at different positions on the line profile, we can estimate the chief parameters of the magnetized atmosphere at that particular region of the sun, e.g. the strength, inclination and azimuth of the magnetic field, as well as the temperature, fill factor or stray light component and few other parameters. Obviously, the more information we have, the better will our estimates approximate the “real” situation. In practice, there are different kinds of difficulties for each type of measurement and the observer often has to compromise on a few parameters and concentrate on those parameters, which make up his or her primary science goal. For example, people interested in the pre-flare evolution of magnetic fields would prefer a 2-D record of the magnetic field with high cadence. In this case, it is difficult to measure the Stokes parameters at several spectral positions. Hence, a filtergraph is used which records the 2-D image of the sun through a narrow spectral band. Different devices can be used to isolate the spectral band and we will give 2 examples below, one of which uses a Michelson interferometer, while the other uses a Fabry-Perot tunable etalon. Generically, the magnetograph will consist of a light collector, and an imaging system. There would be a polarization modulator followed by an analyzer, which forms the polarimeter. There would also be a spectral isolator, as mentioned earlier. 4.13. GONG Magnetograph The Global Oscillations Network Group consists of 6 telescopes distributed at convenient longitudes around the globe, so that the oscillations of the solar

Solar Magnetism

55

surface can be recorded continuously without data breaks. This is essential to avoid windowing effects while determining the Fourier frequencies and enhances the accuracy of the frequency measurements. Along with velocity oscillation measurements, the GONG instrument measures the line-of-sight magnetic field distribution over the entire solar disk with a cadence of 1 minute. It thus provides an automatic record of magnetic field evolution prior and after every solar flare, and has provided an extremely useful data-base to solar physicists interested in the dynamics and energetics of solar active regions that lead to solar flares. The light collector of the GONG magnetograph (Harvey et al 1998) consists of 2 flat mirrors which can be rotated along axes which are perpendicular to each other, forming an alt-alt system. The 75 mm beam is sent horizontally towards the North direction and into a laboratory building which houses the rest of the instrumentation. The alt-alt mount is robotic and moves to the sun’s direction automatically every day according to an almanac stored within the computer system. The telescope at each GONG site can be controlled remotely from the central GONG-OPS station located at NOAO, Tucson, Arizona, USA. The heart of the GONG magnetograph is a Michelson Interferometer (MI). The idea of using an MI to measure the spectral line profile is attributed to Michelson himself. The first complete implementation of this idea for solar work was done by Title and Ramsey (1980). The earliest users of MIs for measuring solar Doppler shifts were Brown (1980) and Kozhevatov (1983). A Lyot filter with a fixed bandpass of 0.1 nm (FWHM transmission) acts as the pre-filter to isolate the Ni I line at 676.8 nm. A polarizing, temperature-compensated, wide-field MI acts as the tunable filter. It has a “channel” transmission function in polarized light with a period equal to the typical width of the spectral line. The principle of the measurement process is given schematically in figure 8. The MI produces a modulation of the light intensity that is a function of the path delay. In GONG, the MI is a fixed interferometer with both arms transmitting orthogonal linear polarization states. A quarter wave retarder, combines both the orthogonal polarizations and encodes wavelength into intensity by having a linear polarization vector whose azimuth varies linearly with wavelength. A rotating analyzer then transmits different wavelengths as a function of the angle of the transmission axis. In this way, the channel spectrum of the MI is swept across bandwidth of the pre-filter. The presence of the absorption line produces the modulation. This modulated signal is sampled 3 times per rotation period and integrates the light from 3 different parts of the line profile as shown in the bottom right corner of the figure. The mean signal value is proportional to the intensity of the light transmitted by the pre-filter (including the spectral line).

56

P. Venkatakrishnan and S. Gosain

The amplitude of the modulation is a measure of the strength of the spectral line, while the phase of the modulation is proportional to the center of intensity or effective wavelength position of the line profile. The difference of this effective wavelength measured in right and left circularly polarized light is proportional to the line-of-sight magnetic field.

Figure 8. Schematic of the GONG magnetograph.

4.14. USO Solar Vector Magnetograph The Fabry-Perot (FP) based tunable filters are very useful in solar astronomy as imaging spectrometers. These filters basically isolate a single interference order (transmission maxima) of the FP etalon by blocking other orders with a suitable interference filter. The bandwidth (FWHM) of the interference filter is chosen to be half the Free Spectral Range (FSR), i.e., wavelength separation between two adjacent interference orders of the FP. The advantage of these filters is that they have higher throughput for a given spectral resolution R, compared to other type of spectrometers like prism or grating spectrograph (Chabbal and Jacquinot, 1955). Modern FPs use highly reflective multi-layer dielectric coatings deposited over substrates with surface flatness of about λ/200. These substrates are kept parallel to each other within λ/1000 using servo control system (Hicks et al 1976). The tuning of the FP pass-band is achieved by applying 12 bit digital voltage to piezo-electric crystals, which hold the reflecting glass plates, thereby

Solar Magnetism

57

adjusting the cavity length. Such filters deliver good spectral resolution, rapid scanning and good repeatability in their performance. The magnetographs based on such filters have the advantage of high SNR (due to higher throughput), simultaneous spectrometry of large field-of-view and large spectral range (by using different interference filters).

Figure 9. Schematic of the Solar Vectoer Magnetograph at Udaipur Solar Observatory.

Last decade has seen development of many solar magnetographs or imaging polarimeters, as some would prefer to call them, designed around a Fabry-Perot filter. Some examples are (i) Imaging Vector Magnetograph (IVM) of Haleakala, Hawaii (Mickey et al 1996), (ii) Tenerif Infra-red Polarimeter (TIP) at the Canary Islands, Spain (Martinez Pillet et al 1999), and (iii) Digital Vector Magnetograph (DVM) at the Big Bear Solar Observatory (BBSO) California, USA (Denker et al 2002). Here we give a brief description of newly built Solar Vector Magnetograph (SVM) at Udaipur Solar Observatory (Gosain et al 2006). The schematic of the instrument is given in figure 9. These instrument records polarized spectra of the selected active region in Fe I 630.25 nm line. The schematic layout of the instrument is shown in figure above. The essential components of a spectropolarimeter are a spectrometer and a polarization modulator and analyzer. The

58

P. Venkatakrishnan and S. Gosain

SVM uses as a spectrometer, an air-gap, piezo-tuned and servo-controlled FabryPerot etalon, similar to one described above. For polarization analysis of the light it uses a modulator unit with two rotatable quarter wave-plates. The instrument is fed by a Schmidt-Cassegrain telescope of eight-inch aperture. To avoid heating of the reflective coatings of the telescope mirrors due to excessive solar flux, the telescope is quipped with an eight-inch aperture interference filter in front. This filter limits the bandpass of the incoming solar radiation to a narrow range of about 15 nm, centered at Fe I 630.25 nm line. This also prevents the heating of the subsequent optics in the optical train. The instrument has a symmetric-design and avoids any oblique reflection for minimal instrumental polarization. The entire optical train is mounted on a computerized GermanEquatorial mount. The field-of-view of 4 arc-min diameter can be selected at the prime focus of the telescope. The subsequent optics consists of a polarimeter package, a FP spectrometer, the analyzer assembly and the CCD camera system. A polarimeter calibration unit can be inserted just after the prime focus for polarimeter calibration. There are two ways one can modulate the polarized light, namely spatial and temporal modulation. As the name suggests the former is sequential and latter is parallel. The temporal modulation is useful if the modulation frequency is faster than the frequency at which the atmospheric seeing changes. Although, modulators with very fast, kHz range modulation frequency, (much above frequency of seeing variation ~100 Hz) are available, the detectors at corresponding frequencies are not available. Thus one is forced to use a spatial or combination of both modulation schemes. In the latter case all the available photons are utilized for building up the SNR of the measurement. The SVM uses a spatial modulation scheme using two crossed calcites as the analyzer. The two beams polarized in orthogonal directions are recorded simultaneously on the same CCD chip. The retrieval of magnetic field vector using these spectro-polarimetric observations is done by fitting the observations with theoretical Stokes profiles generated using analytical solutions of radiative transfer equation in a model atmosphere. These analytical solutions were first derived by Unno (1956) and Rachkovsky (1962a, 1962b). It can be seen that in order to retrieve magnetic field vector one need to go through several steps. In order to expedite these numerous steps, that is, observing, reducing and calibrating the raw data, and deriving the magnetic field vector by fitting observed Stokes profiles a considerable amount of automation is required. The SVM is supported by many automated softwares with graphical user interface (GUI) to expedite these processes. The SVM is geared to measure magnetic fields during entire solar cycle 24 and shall contribute to the study of evolution

Solar Magnetism

59

of solar magnetic fields and also complement observations from space based instruments like HINODE and SDO in different spectral lines. 4.15. Future Directions It is very difficult to predict future directions in the study of solar magnetism. Solar physicists will be basically pre-occupied by two kinds of studies. The first kind would be on the long term and solar cycle modulation of active region parameters as well as the polar field evolution. For the former, we require more accurate polarimetry for precise determination of helicity related parameters, and further development of the dynamo theories. For the latter, we need to employ the Hanle effect over different spatial scales near the polar regions. The second type of study will be high resolution studies of the magnetic field on the smallest possible scales, for identification of the exact mechanism of magnetic heating of the atmosphere as well as the detailed behaviour of magnetic fields during the onset of solar eruptions. Both type of studies require great progress in observational as well as theoretical methods which will continue to occupy the attention of future generations of scientists. References Alfven, H., 1942, Nature, 150, 405. Antiochos, S. K., DeVore, C. R. and Klimchuk, J. A., 1999, ApJ, 510, 485. Babcock, H. W., 1961, ApJ, 133, 572. Brown, T. M., 1984, Bull. Astron. Soc. Amer., 16, 978. Carmichael, H., 1964, Proceedings of the AAS-NASA Symposium, Edited by Wilmot N. Hess., p.451. Chabbal, R. and Jacquinot, P., 1955, Nuovo Cim. 2, 661. Denker, C., Ma, J., Wang, J., Didkovsky, L., Varsik, J., Wang, H., Goode, P. R., 2002, Proc. SPIE, 4853, 223. D’Silva, S. and Choudhari, A. R., 1993, A&A, 272, 621. Evershed, J., 1909, MNRAS, 69, 454. Gilman, P. A., 1986, in P. A. Sturrock (ed) Physics of the Sun, vol I, Reidel, 95. Gosain, S., Venkatakrishnan, P. and Venugopalan, K., 2006, JAA, 27, 285. Hale, G. E., 1908, ApJ, 28, 315. Hale, G. E., Nicholson, S. B., 1938, Publ. Carnegie Inst. 498, Washington. Hanle, W., 1924, Z. Physik., 30, 93. Harvey, J. and GONG Team, 1998, Bull. Astron. Soc. India, 26, 135. Hicks, T. R., Reay, N. K. and Stephens, C. L., 1976, A&A, 51, 367.

60

P. Venkatakrishnan and S. Gosain

Hirayama, T., 1974, Solar Phys., 34, 323. Kopp, R. A. and Pneuman, G. W., 1976, Solar Phys., 50, 85. Kozhevatov, I. E., 1983, ISSLEDOVANIE. GEO. AERO. FIZ. SOLNSTA NO. 64, 42. Mart´ınez Pillet, Collados, S´anchez Almeida et al., 1999, in ASP Conference Series 183, Rimelli et al. (eds), p264. Mickey, D. L., Canfield, R. C., LaBonte, B. J., Leka, K. D., Waterson, M. F. et al., 1996, Solar Phys., 168, 229. Moore, R. L. and LaBonte, B., 1980, in IAU Symp. 91, Solar and Interplanetary Dynamics, ed. M. Dryer & E. Tandberg-Hanssen (Dordrecht:Reidel), 207. Parker, E. N., 1955, ApJ, 121, 491. Parker, E. N., 1994, Spontaneous Discontinuities in Magnetic Fields, Oxford Univ. Press. Rachkovsky, D. N., 1962a, Izv. Krym. Astrofiz. Obs., 27, 148. Rachkovsky, D. N., 1962b, Izv. Krym. Astrofiz. Obs., 28, 259. Schwabe, S. H., Astron. Nachr. 21, 2 (1844). Sheeley, N. R., Wang, Y. M. and Harvey, J. W., 1989, Solar Physics, 119, 323. Spruit, H. C., 1983, IAU Symp., 102, 41. Stix, M., 1976, IAU Symp., 71, 367. Stokes, G. G., 1852, Trans. Cambridge Phil. Soc., 9, 399. Sturrock, P. A., 1966, Nature, 211, 695. Title, A. and Ramsey, H., 1980, Applied Optics, 19, 2046. Unno, W., 1956, Publ. Astron. Soc. Japan, 8, 108. Zhang, M. and Low, B. C., 2005, Annual Re. of Astron. & Astroph., vol. 43, Issue 1, pp.103-137.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

CHAPTER 5 WAVES AND OSCILLATIONS IN THE SOLAR ATMOSPHERE ´ ROBERT ERDELYI Solar Physics & Space Plasma Research Centre (SP2 RC), Department of Applied Mathematics, University of Sheffield, S3 7RH, Sheffield, U.K. E-mail: [email protected]

1. Introduction The actual operating heating process that generates and sustains the hot solar corona has so far defied a quantitative understanding despite efforts spanning over half a century. Particular attention is paid here towards the exploration of the coronal heating problem from the perspectives of MHD waves and oscillations. Do MHD waves play any role in the heating of the solar atmosphere? In order to attempt answering this question, first we need do embark on the key properties of the heating of the solar atmosphere. Space observations, from Skylab in the 70th through SMM, Yohkoh and in very present times SoHO, TRACE, RHESSI and Hinode have investigated the solar atmosphere with unprecedented spatial and temporal resolution covering wavelengths from (E)UV, through soft and hard X-ray to even gamma rays. These high-resolution imaging and spectroscopic observations contributed to many discoveries in the solar atmosphere. The solar atmospheric zoo, to the best of our knowledge today, consists of features from small-scale X-ray bright points to very large coronal loops (Figure 1a). For an excellent textbook on the corona see, e.g. Ref. 1. Soon after the discovery of the approximately few MK hot plasma of the solar corona theoreticians came up with various physical models trying to explain the apparently controversial behaviour of the temperature in the atmosphere. The key point is the observed distribution of temperature: the 61

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

62

Ch5

R. Erd´ elyi

Fig. 1. Left: The very inhomogeneous and dynamic solar atmosphere. The full disk image is taken by SOHO/EIT in He II at 304 ˚ A. Right: Solar atmospheric temperature and density distributions as a function of height. The formation of some popular lines for observations is indicated by dots on curve T. Note the logarithmic scales.

solar energy is produced by thermonuclear fusion in the very hot (approximately 14 MK) internal core of the Sun. This vast amount of energy then propagates outwards, initially in the form of radiation (radiation zone) up to about 0.72R⊙ and later by convection (convective zone) right to the solar surface (photosphere) continuously cooling the solar plasma. Surprisingly, after reaching its minimum at the top of the photosphere, the temperature starts to rise slowly throughout the entire chromosphere (up to around 20,000 K), followed by a very steep and sharp increase in the narrow transition region (few 100,000 K) up to around 2 MK in the corona (Figure 1b). Although going continuously away from the energy producing solar core, instead of a temperature decrease, the tendency of temperature increase was found (Figure 1b). Maintaining this high temperature requires some sort of input of energy because without it the corona would cool down by thermodynamic relaxation on a minute-scale. Surprisingly, this non-thermal energy excess to sustain the solar corona is just a reasonably small fraction of the total solar output (see Table 1). It is relatively straightforward to Table 1.

Average coronal energy losses (in erg cm−2 sec−1 ).

Loss mechanism Conductive flux Radiative flux Solar wind flux Total flux

Quiet Sun 105

2× 105 < 5 × 104 3 × 105

Active region 105

107

5 × 106 < ×105 107

Coronal hole 6 × 104 104 7 × 105 8 × 105

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

63

estimate the entire energy budget needed for the solar corona: approximately just a tiny 10−4 fraction of the Sun’s total energy output is needed giving, at least in theory, a fairly easy task for theoreticians to put forward various mechanisms that could divert 0.01% of the total solar output into heating the corona. The question is today not where does the coronal non-thermal energy come from, but rather how is this energy actually transferred into the corona and how does it dissipate efficiently there. 1.1. Importance of Atmospheric Magnetism With increasing spatial and time resolution large-scale structures like sunspots, complex active regions, prominences, coronal loops, coronal holes are observed in great details. The improved resolution allowed to reveal fine structures like the magnetic pores, dark mottles, spicules, supergranular cells, filaments, X-ray and EUV bright points, etc. Since the discoveries of the solar cycle, the Hale’s polarity law, the butterfly diagram for sunspots and the cyclic variations in sunspot numbers the role of solar magnetic fields became a central theme. Skylab observations made it clear for the first time that the X-ray emitting hot and bright coronal regions and the underlying surface magnetic field concentrations are strongly correlated suggesting that coronal heating and solar magnetism are intimately linked (Figure 2). Today

Fig. 2. Approximately concurrently taken magnetogram, UV, EUV and X-ray full disk images. Observe that the locci of magnetic field concentrations at photospheric level coincide with the locii of high emissions in (E)UV and X-ray. Image credit: SOHO and Yohkoh.

it is evident that the solar atmosphere is highly structured and is most likely that various heating mechanisms operate in different atmospheric magnetic structures.2,3 In closed structures, e.g. in active regions, temperatures may reach up to 8 − 20 × 106 K, while in open magnetic regions like coronal holes maximum temperatures may only be around 1 − 1.5 × 106 K. Observations also show that density and magnetic field are highly inhomogeneous. Fine structures (e.g. filaments in loops) may have 3-5 times higher densities than

September 11, 2008

64

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

in their environment. The fluctuating brightness and the associated velocities, as opposed to the quasi-static nature of the corona, are far-reaching observational constraints what are not yet modelled adequately. There is also little known about how the heating depends on magnetic field strength, structure size (length, radius, expansion) and age. 1.2. Atmospheric Heating Mechanisms In order to explain the solar (and stellar) atmospheric heating mechanism(s) models have to provide a mechanism or mechanisms that result(s) in a steady supply of energy not necessarily on a steady way. Random energy releases that produce a statistically averaged steady state are allowed for to balance the atmospheric (chromospheric and coronal) energy losses and these models became more viable.4 -7 Testing a specific heating mechanism observationally may be rather difficult because several mechanisms may operate at the same time. Ultimate dissipation occurs on very small spatial scales, sometimes of the order of a few hundred metres that even with current high spatial resolution satellite techniques cannot (and will not for a while!) be resolved. A distinguished signature of a specific heating mechanism could be obliterated during the thermalisation of the input energy.4 We should, instead, predict the macroscopic consequences of a specific favoured heating mechanism2 and confirm these signatures by observations.8 For example one could predict the generated flows9 or specific spectral line profiles or line broadenings.10,11 The heating process is usually split into three phases: (i) the generation of a carrier of energy; (ii) the transport of energy from the locii of generation into the solar atmospheric structures; and finally (iii) the actual dissipation of this energy in the various magnetic or non-magnetic structures of the atmosphere. Without contradicting observations it is usually not very hard to come up with a theory that generates and drives an energy carrier. The most obvious candidate is the magneto-convection right underneath the surface of the Sun. Neither seems the literature to be short of transport mechanisms. There is, however, real hardship and difficulty in how the transported energy is dissipated efficiently on a time-scale such that the corona is not relaxed thermally. A brief and schematic summary of the most commonly accepted heating mechanisms is given in Table 2, see also Refs. 12-14. The operating heating mechanisms in the solar atmosphere can be classified whether they involve magnetism or not. For magnetic-free regions (e.g. in the chromosphere of quiet Sun) one can suggest a heating mechanism

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere Table 2.

65

Summary of various popular heating mechanisms, see also Refs. 13-14.

Energy carrier Dissipation mechanism Hydrodynamic heating mechanisms Acoustic waves (P < Pacoustic cutoff ) Shock dissipation Pulsation waves (P > Pacoustic cutoff ) Shock dissipation Magnetic heating mechanisms 1. Alternating current (AC) or wave mechanisms Slow waves Shock damping, resonant abs. Longitudinal MHD tube waves Fast MHD waves

Landau damping

Alfv´ en waves (transverse, torsional)

Current sheets

Mode coupling, res. heating, phase mixing, compressional viscous heating, turb. heating, Landau damping, res. absorption 2. Direct current (AC) mechanisms Reconnection (e.g. turbulent or wave heating)

that yields within the framework of hydrodynamics. Such heating theories can be classified as hydrodynamic heating. Examples of hydrodynamic heating are, among others, e.g. acoustic waves and pulsations. However, if the plasma is embedded in magnetic fields as it is in most parts of the solar atmosphere, the framework of MHD may be the appropriate approach. These coronal heating theories are called MHD heating mechanisms; for reviews see, e.g. Refs. 14-21 and 22. The ultimate dissipation in MHD models invoke Joule heating or, in a somewhat less extent, viscosity. Examples of energy carrier of magnetic heating are the slow and fast MHD waves, Alfv´en waves, magnetoacoustic-gravity waves, current sheets, etc. There is an interesting concept put forward by, e.g. Ref. 23, where the direct energy coupling and transfer from the solar photosphere into the corona is demonstrated by simulations and TRACE observations, see also Ref. 24. For a recent review on MHD waves and oscillations see, e.g. Refs. 25-27. Finally, a popular alternative MHD heating mechanism is the selective decay of a turbulent cascade of magnetic field.28 -30 Based on the times-scales involved an alternative classification of the heating mechanism can be constructed. If the characteristic time-scale of the perturbations is less than the characteristic times of the back-reaction, in a non-magnetised plasma acoustic waves are good approximations describing the energy propagation; if, however, the plasma is magnetised and perturbation time-scales are small we talk about alternating current (AC- )

September 11, 2008

66

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

heating mechanisms, e.g. MHD waves.21,25,31 On the other hand, if perturbations have low frequencies hydrodynamic pulses may be appropriate in a non-magnetised plasma, while if the external driving forces (e.g. photospheric motions) operate on longer times-cales compared to dissipation and transit times very narrow current sheets are built up resulting in direct current (DC-) heating mechanisms in magnetised plasmas.20 After it was discovered that the coronal plasma is heavily embedded into magnetic fields the relevance of the hydrodynamic heating mechanisms for the corona part of the atmosphere was re-evaluated. It is believed today that hydrodynamic heating mechanisms could still contribute to atmospheric heating of the Sun but only at lower layers, i.e. possibly in the chromosphere and up to the magnetic canopy.32 -34 At least as a first approximation the plasma is considered frozen-in in the various magnetic structures in the hot solar atmosphere. The magnetic field plays a central and key role in the dynamics and energetics of the solar corona (see Figure 2). High-resolution satellite observations show the magnetic building blocks that seem to be in the form of magnetic flux tubes (Figure 3) in the solar atmosphere. These flux tubes expand rapidly

Fig. 3. TRACE images of the highly structured solar corona where the plasma is frozen in semi-circular shaped magnetic flux tubes. Left: The magnetic field in the solar atmosphere shapes the structures that we see, as the emitting gas can generally only move along the field. Courtesy Charles Kankelborg. Right: The image shows the evolution of loop system: an increasing number of loops appears in the 1 MK range. Courtesy TRACE (http://vestige.lmsal.com/TRACE/Public/Gallery/Images/TRACEpod.html).

in height because of the strong drop in density. Magnetic fields fill almost entirely the solar atmosphere at about 1,500 km above the photosphere. This environment is well described within the approximation of MHD.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

67

2. Equations of Ideal and Dissipative MHD The MHD equations are discussed in many excellent books, see, e.g. Refs. 35-40. Here we only give a short account of the MHD equations and their main properties necessary for grasping the key points of MHD waves and the heating processes. We consider a collision-dominated plasma with isotropic kinetic pressure. The state of the plasma is described by the density ρ, the pressure p, the temperature T , the velocity v, the magnetic induction B, the electrical field E, the electrical current density j, and the electrical charge density ρe . We use the SI system of units. In this system ρ is measured in kg m−3 , p in kg m−1 s−2 , T in K, v in m s−1 , B in tesla = kg s−2 A−1 = 104 G (A is amper), E in Vm−1 , j in Am−2 , and ρe in A s m−3 . The quantities B, E, j, and ρe are related by Maxwell equations ∇ × B = µ0 j,

(1)

∂B = −∇ × E, ∂t

(2)

∇ · E = ǫ−1 0 ρe ,

(3)

∇ · B = 0,

(4)

where we neglected the displacement current in Eq. (1). Here µ0 (= 4π × 10−7 kg m s−2 A−2 ) and ǫ0 (= 8.854×10−7 m−1 s AV−1 ) are the magnetic and electrical permeability of vacuum. Eq. (1) is called the Ampere’s equation. The quantities B, E, j, and v are, in addition, related by the Ohm’s law, which we take in its classical from j = σ(E + v × B),

(5) −1

where σ is the electrical conductivity measured in m−1 AV . The derivation of Eq. (5) and the discussion of the framework of its applicability can be found elsewhere, see, e.g. Ref. 38. Since Eq. (5) expresses E in terms of B, j, and v, Eq. (3) can be considered as the equation determining ρe . This equation is almost never used in magnetohydrodynamics. We substitute Eq. (5) into Eq. (2) and use Eq. (1) to arrive at ∂B = ∇ × (v × B) + η∇2 B, ∂t

(6)

where η = (σµ0 )−1 is the magnetic diffusivity measured in m2 s−1 . Equation (6) is called the induction equation.

September 11, 2008

68

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

Plasma motions are described by the continuity, momentum, and energy equations. The continuity equation takes the form ∂ρ + ∇ · (ρv) = 0. ∂t

(7)

This equation expresses mass conservation. The momentum equation is   ∂v ρ + (v · ∇)v = −∇p + j × B + ρg + Fvis , (8) ∂t where g is the gravity acceleration and Fvis is the viscous force. This equation represents the second Newton law since its left-hand side is the acceleration of a unit volume of the plasma multiplied by its mass, while its right-hand side represents the force acting on this volume due to the pressure gradient, the Lorentz force, the gravity force, and viscosity. With the aid of Eq. (1) this equation can be re-written in the alternative form   1 ∂v ρ + (v · ∇)v = −∇p + (∇ × B) × B + ρg + Fvis . (9) ∂t µ0 The energy equation can be written as   ∂e ρ + v · ∇e + p∇ · v = −Q, ∂t

(10)

where e = cv T is the internal energy, cv the specific heat at constant volume, and Q the energy loss function describing the net effect of all the sinks and sources of energy. The first term on the right-hand side of Eq. (10) is the time derivative of the energy of a moving unit volume of plasma and the second term is the rate of work done by pressure on this volume. The temperature, pressure and density are related by the Clapeiron law p=

e R ρT, µ ˜

(11)

e = kB /mp is the gas constant, µ where R ˜ the mean atomic weight (the average mass per particle in units of the proton mass mp ), and kB the Boltzmann constant. For fully ionized plasmas consisting of protons and electrons only µ ˜ = 0.5. In what follows we also use the specific heat at constant pressure cp and the ratio of specific heats γ = cp /cv . The quantities e µ R, ˜, cv , and cp are related by cp = cv +

e R . µ ˜

(12)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

69

With the use of Eqs. (7), (8), and (12) we can rewrite Eq. (10) in the form     ∂ p p + v · ∇ = −(γ − 1)ρ−γ Q. (13) ∂t ργ ργ Since the entropy per unit mass of plasma is cv log(p/ργ ) + const, this equation is called the entropy equation. The term η∇2 B on the right-hand side of Eq. (6), the term Fvis on the right-hand side of Eq. (8), and the term Q on the right-hand side of Eq. (10) are dissipative terms. Their presence results in irreversible transfer of mechanical and magnetic energy into plasma heating. If we drop all these terms we arrive at the ideal MHD equations. The ideal momentum, induction, and entropy equations take the form   1 ∂v + (v · ∇)v = ∇p + (∇ × B) × B + ρg, (14) ρ ∂t µ0 ∂B = ∇ × (v × B), ∂t ∂ ∂t



p ργ



+v·∇



p ργ



(15)

= 0.

(16)

Equation (16) expresses the constancy of entropy in an elemental moving volume of plasma and it is called isentropic or adiabatic equation. 2.1. Linear MHD Equations Let us write all dependent variables in the MHD equations in the form f = f0 + f ′ , where f represents any dependent variable, the subscript ‘0’ indicates an equilibrium quantity, and the prime indicates the perturbation of a quantity. Then we substitute variables in this form into the MHD equations and obtain equations that contain terms of three types: terms that contain only equilibrium quantities; terms that are linear with respect to perturbations; and, terms that are nonlinear with respect to perturbations. Since the equilibrium quantities satisfy the MHD equations the terms of the first type cancel out. Let us now assume that |f ′ | ≪ f0 for any dependent variable. Then the terms of the third type can be neglected in comparison with the terms of the second type. As a result we arrive at the linear MHD equations ∂ρ′ + ∇ · (ρ0 v) = 0, ∂t

(17)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

70

ρ0

Ch5

R. Erd´ elyi

∂v 1 lin = −∇p′ + [(∇ × B′ ) × B0 + (∇ × B0 ) × B′ ] + ρ′ g + Fvis , (18) ∂t µ0 ∂B′ = ∇ × (v × B0 ) + η∇2 B′ , ∂t

(19)

∇ · B′ = 0,

(20)

∂ ′ (p − c2S ρ′ ) + ργ0 v · ∇ ∂t



p0 ργ0



= −(γ − 1)Qlin ,

p′ T′ ρ′ = + . p0 T0 ρ0

(21)

(22)

Here c2S = γp0 /ρ0 is the square of the sound speed and the superscript ‘lin’ indicates a linearized quantity. When deriving these equations we have assumed that an equilibrium state is static, i.e. v0 = 0. Because of this assumption we have dropped the prime when writing the velocity perturbation. 2.2. MHD Waves in Ideal Uniform Plasmas Let us look for solutions to the ideal MHD equation in the form of normal modes and take the perturbations to be proportional to exp[i(k · r − ωt)] with r = (x, y, z) and k = (kx , ky , kz ) in the Cartesian coordinates x, y, z. If we assume that ω 6= 0, we arrive at an equation for the displacement ξ as 2 2 )k(k · ξ) − vA kk(b0 · ξ) cos ϕ ω 2 ξ = (c2S + vA 2 2 2 kb0 (k · ξ) cos ϕ. + vA k ξ cos2 ϕ − vA

(23)

Here vA is the Alfv´en speed and ϕ the angle between the equilibrium magnetic field and the wave vector k. They are determined by 2 vA =

B02 k · b0 , cos ϕ = . µ0 ρ0 k

(24)

Here b0 is the unit vector in the direction of the equilibrium magnetic field. It is straightforward to rewrite Eq. (23) as one scalar equation for the quantity ξ⊥ = ξ · (b0 × k)/k, which is the component of the vector ξ perpendicular to both k and b0 , and two coupled scalar equations for (k · ξ) and (b0 · ξ). These equations are 2 2 (ω 2 − vA k cos2 ϕ)ξ⊥ = 0,

(25)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

71

2 2 3 [ω 2 − (c2S + vA )k 2 ](k · ξ) + vA k (b0 · ξ) cos ϕ = 0,

(26)

c2S k(k · ξ) cos ϕ − ω 2 (b0 · ξ) = 0.

(27)

First we study the case where ϕ 6= 0. Equation (25) has non-trivial solution when 2 2 ω 2 = vA k cos2 ϕ.

(28)

The waves with the frequency determined by Eq. (28) are called Alfv´en waves. We see that only the component of the plasma displacement perpendicular to both k and b0 is non-zero in an Alfv´en wave. The set of two linear Eqs. (26) and (27) has non-trivial solution when its determinant is zero. This condition results in the dispersion equation 2 4 2 k cos2 ϕ = 0. )ω 2 k 2 + c2S vA ω 4 − (c2S + vA

The solution to this equation is n o 1 2 2 2 2 ω 2 = a2± k 2 ≡ 2 k 2 c2S + vA ± [(c2S + vA ) − 4c2S vA cos2 ϕ]1/2 .

(29)

(30)

The signs ‘+’ and ‘−’ in this expression correspond to fast and slow magnetoacoustic waves. When waves propagate along the equilibrium magnetic field (ϕ = 0) it is straightforward to check that ω+ = vA k when cS < vA and ω− = vA k when cS > vA . Hence ω = vA k is the double root of the dispersion equation. It still corresponds to Alfv´en waves. The plasma displacement in Alfv´en waves can now be in any direction perpendicular to B0 . The other root of the dispersion equation is ω = cS k. It corresponds to a sound wave propagating along the equilibrium magnetic field. MHD waves play a very important role in coronal physics. Flux tubes are shaken and twisted by photospheric motions (i.e. by both granular motion and global acoustic oscillations, the latter being called p-modes). Magnetic flux tubes are excellent waveguides (see e.g. the coupling from photosphere to the transition region as observed by Ref. 24). If the characteristic time of these photospheric footpoint motions is much less than the local Alfv´enic transient time the photospheric perturbations propagate in the form of various MHD tube waves (e.g. slow and fast MHD waves; Alfv´en waves). The dissipation of MHD waves is manifold: these waves couple with each other, interact non-linearly, resonantly interact with the closed waveguide (i.e. coronal loops) or develop non-linearly (e.g. solitons or shock waves can form), etc. For an extensive review on the observations of MHD waves, see, e.g. Ref. 8, while on theory see, e.g. Ref. 26.

September 11, 2008

72

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

In an inhomogeneous and magnetised plasma there are two particular dissipation mechanisms of MHD waves that received extensive attention in the past decades: resonant absorption and phase mixing. Although there are major theoretical advances on these two particular dissipation mechanisms unfortunately we still have only indirect evidences that they may actually operate under solar circumstances. Thanks to the fantastic imaging capabilities of TRACE, plenty of observations of MHD wave damping in coronal loops are available8 and some of these cases may be an excellent candidate of resonant absorption. Further, it is less likely that phase mixing operates in closed magnetic structures, like solar coronal loops.

2.3. MHD Waves in Ideal Inhomogeneous Plasmas The analysis so far has been assumed a uniform medium. The mathematical advantage of this assumption is that the PDEs readily reduced to algebraic equations for the dispersion relation ω = ω(kx , ky , kz ). However this is only valid provided the wavelength is much smaller than the length-scale of the inhomogeneity. When the wavelength λ ≥ l0 , the length-scale of inhomogeneity, then the inhomogeneous nature of the medium determines the behaviour of the disturbances. In the solar atmosphere the principal causes of inhomogeneity are gravity and the structured magnetic field (magnetic pores, sunspots, chromospheric canopy, arcades or prominences, coronal loops, plumes, etc). Gravity creates a vertical stratification in plasma density and pressure, and the magnetic field can cause the plasma pressure to increase in a direction normal to the field. These stratifications introduce a few significant effects that affect MHD wave propagation: (i) amplification of the wave amplitude (e.g. chromospheric shocks); evanescence - regions in which waves, otherwise oscillating spatially, may decay exponentially (solar global f /p/g-mode oscillations); waveguide modes - a discontinuity (magnetic, density, ...) in the medium may give rise to waves guided by the structure (e.g. surface or body modes in a flux tube).

2.3.1. MHD waves at magnetic interface Let us first ignore gravity and focus interest on the effect of magnetic structuring on MHD wave propagation. Assume that in the basic state the plasma is permeated by a magnetic field B0 (x)ˆz, working in Cartesian coordinates x, y, z (see Fig. 4b). Then the pressure and density are structured by the x-dependence of the magnetic field and the basic state is found

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

73

to be p0 = p0 (x),

ρ0 = ρ0 (x),

d dx

  B02 p0 + = 0, 2µ

(31)

for the pressure, density and total (magnetic + gas) pressure. Linearised perturbations from this state are taken and the equations of continuity, momentum, the induction and energy equations in ideal MHD give: ∂ρ + divρ0 v = 0, ∂t   ∂v 1 1 1 ρ0 = −∇ p + B0 · b + (B0 · ∇) b + (b · ∇) B0 , ∂t µ µ µ

(32)

(33)

∂b = curl (v × B0 ) , ∂t

(34)

  ∂p 2 ∂ρ + v · ∇p0 = c0 + v·∇ρ0 . ∂t ∂t

(35)

After Fourier analysis, e.g., (vx = vˆx (x)ei(ωt+ly+kz) , p = pˆ(x)ei(ωt+ly+kz) ) a single ODE for vˆx (x) is obtained. Consider the magnetic interface defined by  Be , x > 0, B0 (x) = (36) B0 , x < 0, with B0 and Be both constants. Pressure continuity at x = 0 gives pe +

Be2 B2 = p0 + 0 . 2µ 2µ

(37)

It is then found that the equation governing vx (x) is simply d2 vˆx − m20 vˆx (x) = 0 dx2

for x < 0,

(38)

d2 vˆx − m2e vˆx (x) = 0 dx2

for x > 0,

(39)

and

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

74

Ch5

R. Erd´ elyi

where m20 (x) =

2 (k 2 c20 (x) − ω 2 )(k 2 vA (x) − ω 2 ) 2 (x))(k 2 c2 (x) − ω 2 ) , (c20 (x) + vA T 2 c20 (x)vA (x) 2 2 (x) , c0 (x) + vA way to m20 except that

c2T (x) =

(40) (41)

and m2e is defined in a similar the Alfv´en and sound speeds appropriate to x > 0 are taken. It is the presence of the discontinuity in B0 (x) that is responsible for the existence of surface waves which may arise if m20 and m2e are both real and positive. Solving (38) and (39) for vˆx (x) gives  αe e−me x , x > 0, vˆx (x) = (42) α0 em0 x , x < 0.

In writing this solution we are excluding laterally propagating waves, so only the modes with non-vanishing amplitudes (end energy) near the discontinuity arise. Then, using the continuity of vˆx (x) and of the total pressure perturbation pˆT (x) across the interface at x = 0 the general dispersion relation for the magnetic interface is found: 2 2 ρ0 (k 2 vA − ω 2 )me + ρe (k 2 vAe − ω 2 )m0 = 0,

(43)

valid for m20 and m2e both positive and for l = 0. From the inspection of the dispersion relation (43) one is able to conclude that the existence of a magnetic interface supports the propagation of surface waves as shown by, e.g. Ref. 41, and what we follow here closely. In particular, slow MHD surface waves exist if one side of the interface is field-free. If the gas in the magnetised region is cooler than the field-free medium the discontinuity may also support fast MHD surface waves. In general slow and fast MHD surface waves may arise if there is a sharp transition in the background slow and/or Alfv´en speeds. 2.3.2. Waves in magnetic slab First consider a magnetic slab with zero field surrounding it so that  B0 , |x| < x0 , B0 (x) = 0, |x| > x0 ,

(44)

with pressure p0 and density ρ0 inside the slab, pe and ρe outside. The two regions are related by  2 1 2  c0 + 2 γvA B2 p e = p 0 + 0 , ρe = ) ρ0 (45) 2µ c2e

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

75

where co and ce are the sound speeds inside and outside the slab and vA is the Alfv´en speed in the slab. Again, attention is confined to two-dimensional disturbances so that the velocity perturbation component vy and wavenumber l are assumed to be zero. Analogue to the case of the magnetic interface it is found that d2 vˆx − m20 vˆx = 0 f or |x| < x0 , (46) dx2 and d2 vˆx − m2e vˆx = 0 f or |x| > x0 , (47) dx2 for m0 as above and me by analogous comparison. Again, the boundary conditions, vˆx (x) and pˆT (x) being continuous across the boundary x = ±x0 are used to obtain the general dispersion relation, as shown by, e.g. Refs. 42-43:     tanh ρe 2 − ω 2 )me = ω 2 m0 m0 x0 , (48) (k 2 vA ρ0 coth

valid for ω 2 < k 2 c2e . Dispersion relation (48) describes the existence of slow and fast magneto-acoustic waves that could be either body or surface waves, depending on their structure within the slab (i.e. evanescent for surface and oscillatory for body waves, respectively). Perturbations that are symmetric about the vertical axis of the slab are called sausage oscillations, while perturbations that are anti-symmetric are called kink oscillations. When the slab is considered thin in comparison to the wavelength (long wavelength approximation that is of interest for photospheric and coronal conditions) the kink mode vibrates as a single thin string and the sausage mode vibrates as both surface and body waves. 2.3.3. MHD waves in magnetic cylinder The magnetic building blocks in the solar atmosphere are the magnetic flux tubes. In a pioneering work,44 using cylindrical coordinates, it was derived the dispersion relations of MHD waves propagating in cylindrical magnetic flux tubes. The main obstacle to be overcome when introducing the concept of flux tubes is the conversion from Cartesian to cylindrical coordinates. This change results involving Bessel functions in the dispersion relation which are not yet possible to be solved analytically without simplification, e.g. through incompressibility or long and short wavelength approximations. Let us summarise here the key steps.44 Consider a uniform magnetic cylinder of magnetic field B0ˆ z confined to a region of radius a,

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

76

Ch5

R. Erd´ elyi

Fig. 4. Left: Magnetic flux tube showing a snapshot of Alfv´ en wave perturbation propagating in the longitudinal z-direction along field lines at the tube boundary. At a given height the Alf´ enic perturbations are torsional oscillations, i.e. oscillations are in the ϕ-direction, perpendicular to the background field. Right: Snapshot showing Alfv´ en waves propagating along a magnetic discontinuity. Again, the key feature to note is that Alfv´ enic perturbations are within the magnetic surface (yz-plane) at the discontinuity, perpendicular to the background field (y-direction), while the waves themselves propagate along the field lines (z-direction).

surrounded by a uniform magnetic field Beˆ z (see Figure 4a). To simplify the MHD equations we assume again zero gravity, there are no dissipative effects and all the disturbances are linear and isentropic. Pressure (plasma and magnetic) balance at the boundary implies that p0 +

B02 B2 = pe + e . 2µ0 2µ0

(49)

Linear perturbations about this equilibrium give the following pair of equations valid inside the tube, ∂2 ∂t2

! 2 ∂2 2 2 2 2 2 ∂ − (c + v )∇ ∆ + c v ∇2 ∆ = 0, 0 A 0 A ∂t2 ∂z 2

(50)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

! 2 ∂2 ∂ 2 − vA Γ = 0, ∂t2 ∂z 2

Ch5

77

(51)

where ∇2 is the Laplacian operator in cylindrical coordinates (r, θ, z) and ∆ ≡ divv,

Γ=ˆ z · curlv

(52)

for velocity v = (vr , vθ , vz ). A similar pair of equations to (50) and (51) are valid outside the tube. Fourier analysing we let ∆ = R(r)exp[i(ωt + nθ + kz).

(53)

Then equations (50) and (51) give Bessel’s equation satisfied by R(r) as follows d2 R 1 dR + − dr2 r dr

! 2 n m20 + 2 R = 0, r

(54)

where m20 =

2 (k 2 c20 − ω 2 )(k 2 vA − ω2) 2 )(k 2 c2 − ω 2 ) . (c2e + vAe Te

(55)

We have used the notation cT for the the characteristic tube speed (sub2 −1/2 Alfv´enic), where cT = c0 vA /(c20 + vA ) . To obtain a solution to (54) bounded at the axis (r = 0) we must take R(r) = A0



In (m0 r), m20 > 0 Jn (n0 r), n20 = −m20 > 0



(r < a),

(56)

where A0 is an arbitrary constant and In , Jn are Bessel functions45 of order n. For a mode locked to the waveguide it is required that no energy propagates to or from the cylinder in the external region, i.e. the waves are evanescent outside the flux tube. Therefore we take R(r) = A1 Kn (me r),

r > a,

(57)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

78

Ch5

R. Erd´ elyi

where A1 is a constant and m2e =

2 (k 2 c2e − ω 2 )(k 2 vAe − ω2) 2 )(k 2 c2 − ω 2 ) , (c2e + vAe Te

(58)

which is taken to be positive (no leaky waves). Since we must have continuity of velocity component vr and total pressure at the cylinder boundary r = a, this yields the dispersion relations 2 ρ0 (k 2 vA − ω 2 )me

Kn′ (me a) I ′ (m0 a) 2 = ρe (k 2 vAe − ω 2 )m0 n , Kn (me a) In (m0 a)

(59)

for surface waves (m20 > 0) and 2 ρ0 (k 2 vA − ω 2 )me

Kn′ (me a) J ′ (m0 a) 2 = ρe (k 2 vAe − ω 2 )m0 n , Kn (me a) Jn (m0 a)

(60)

for body waves (m20 = −n20 < 0). The well-observed axisymmetric sausage mode is given by n = 0, while the kink mode (non-axisymmetric) is given by n = 1. Modes with n > 1 are called flute modes. Although the dispersion relations (59) and (60) are complicated, finding the phase speed for e.g. kink waves with coronal parameters simplifies matters considerably. In the corona one can assume B0 ≈ Be , vAe , vA > ce , c0 and ρ0 > ρe . This means that only fast and slow body waves may occur and there are no longer surface waves present (see Fig. 5). Also, for coronal loops the thin flux tube/long wavelength approximation (| k | a ≪ 1) is a good approximation since the observed loop length is always much larger than loop width. For instance, for fundamental standing mode kink oscillations the wavelength is twice the loop length. In this limit, also called the slender tube limit, the fast sausage mode does not even exist, while the fast kink wave propagates with the kink phase speed ck , given by ck =

2 2 ρ0 vA + ρe vAe ρ0 + ρe

!1/2

.

(61)

Fast kink modes, when vAe > vA , are sustained in dense loops with periods on an Alfv´enic timescale and it is found that these body waves have a low wavenumber cut off implying that only wavelengths shorter than the diameter of a loop can propagate freely. The sausage mode, however, has a much shorter period, approximately one tenth of that of the kink mode. Sausage and kink fast body modes exist only in high density loops. However, the slow modes appear in both high and low density cylinders.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

79

Fig. 5. The solution of the dispersion relations (59)-(60) in terms of phase speed (ω/k) of modes under coronal conditions vAe > vA > c0 > ce (all speeds are in km/s). The slow band is zoomed, and only the two first harmonics of a mode are shown (lower panel).

Under photospheric conditions, characterised by ce > vA > c0 , representative for sunspots or pores both the slow and fast bands have surface and body modes, respectively. The slow waves are in a narrow band since c0 ≈ cT . The slow body waves are almost non-dispersive, whereas the almost identical slow surface sausage and kink modes are weakly dispersive (bottom zoomed out panel in Fig. 6). 2.3.4. MHD waves in magnetically twisted cylinder Granular shear motions, differential rotation or meridional circulation in the photosphere can introduce a twist to the flux tubes from pores to sunspots. Erupting prominences or CMEs, with their footpoints anchored in the dense sub-photosphere, often appear to have twisted field lines. It is natural and practical to extend the investigations of MHD wave modes in twisted magnetic flux tubes. Twisted tubes have been studied before but mainly in terms of stability. Here we briefly summarise the current status on results

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

80

Ch5

R. Erd´ elyi

Fig. 6. The solution of the dispersion relations (59)-(60) in terms of phase speed (ω/k) of modes under photospheric conditions ce > vA > c0 (all speeds are in km/s). The slow band is zoomed (lower panel).

on MHD waves in magnetically twisted solar atmospheric tubes, see, e.g. Refs. 46-48. Let a uniformly twisted flux tube embedded in a straight magnetic field (Fig. 7) given by: B=



(0, Ar, B0 ), (0 , 0 , Be ),

r < a, r > a.

(62)

In cylindrical equilibrium the magnetic field and plasma pressure satisfy the pressure-balance equation in the radial direction: d dr

2 2 B0ϕ + B0z p0 + 2µ

!

+

2 B0ϕ = 0. µr

(63)

Here, the second term in the brackets represents magnetic pressure and the third term derives from magnetic tension due to the azimuthal component of the equilibrium magnetic field, B0ϕ . For the sake of simplicity the plasma is taken incompressible, with the field and plasma pressure being structured in the radial direction only. Again, using continuity of total pressure pT and perturbation velocity vr across r = a and seeking for a bounded solution,

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

81

Fig. 7. Straight, vertical, uniformly twisted magnetic flux tube in an ambient magnetic field. At the boundary of the flux tube there is a jump in the magnetic twist.

at r = 0 and r → ∞, leads to the dispersion relation ′

2 (ω 2 −ωA0 )

m0 aIm (m0 a) A −2mωA0 √ Im (m0 a) µρ0

2 )2 −4ω 2 (ω 2 −ωA0 A0

A2 µρ0

′

=

|kz |aKm (|kz |a) Km (|kz |a)

′

ρe 2 A2 |kz |aKm (|kz |a) 2 (ω −ωAe )+ ρ0 µρ0 Km (|kz |a)

.

(64) In this equation the dash denotes derivative with respect to the argument of the Bessel function and 1 ωA0 = √ (mA + kz B0 ), µρ0  2 2 m0 = kz 1 −

kz Be ωAe = √ , µρe

2 4A2 ωA0 2 )2 µρ0 (ω 2 − ωA0



(65) .

Eq. (64) is the dispersion relation for waves in an incompressible flux tube with uniform magnetic twist, embedded in a straight magnetic environment. For an incompressible tube without magnetic twist there are no body waves. However, magnetic twist introduces an infinite band of body waves. A dual nature of the mode is also discovered where a body wave exist for long wavelengths but surface wave characteristics are displayed when the eigenfunction is plotted for shorter wavelengths.

September 11, 2008

82

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

2.3.5. MHD oscillations in annular magnetic cylinders Sub-resolution flux tube structure is still more a matter of some speculations. It is anticipated, that Hinode/SOT may advance the research on the internal structure of solar magnetic flux tubes. As a specific example, let as recall an interesting earlier observation:49 combined data of May 13 1998 from both the EIT instrument on SOHO and from TRACE show the simultaneous observation of two slow magnetosonic waves propagating along a perceived coronal loop with speeds of 95 and 110 km s−1 . This observation was interpreted by the authors as temperature differences within the observed loop hinting at a substructure of perhaps either concentric shells of different temperatures or of thin strands within the same loop at different temperatures. There is no conclusive proof disputing these possible flux tube structures nor preference given towards one in particular. Here let us shall assume that a flux tube consists of a central core surrounded by a shell or annulus layer, all embedded in uniform magnetic field (Fig. 8). For details see, e.g. Refs. 50-53. We restrict our investigation

Bi

B0 Be

a R

Fig. 8. The equilibrium configuration of a magnetic cylinder consisting of a core, annulus and external regions, all with straight magnetic field.

to an incompressible plasma for which the phase speeds of the slow and Alfv´en waves, in the limit of incompressibility, become indistinguishable, the modes are discernable through direction of perturbation only (Alfv´en wave perturbations are perpendicular to both magnetic field and propagation vector while slow wave perturbations are in the same plane). The fast waves are removed from the system. We consider a longitudinal magnetic

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

field in each region of a magnetic annulus so that  r < a,  Bi = (0, 0, Bi ), B = B0 = (0, 0, B0 ), a ≤ r ≤ R,  Be = (0, 0, Be ), r > R,

83

(66)

where Bi , B0 , Be are constant. We take the densities in the core, annulus and external regions as ρi , ρ0 , ρe , respectively, and similarly denote the pressure in each region as pi , p0 , pe . The pressure balance at the boundaries r = a and r = R gives the relations Bi2 B2 = p0 + 0 , 2µ 2µ

pi +

p0 +

B02 B2 = pe + e , 2µ 2µ

(67)

and we denote vAi = Bi /(µρi )1/2 , vA0 = B0 /(µρ0 )1/2 and vAe = Be /(µρe )1/2 as the Alfv´en speeds in the internal, annulus and external regions, respectively. Taking linear perturbations of the ideal MHD equations about this equilibrium and Fourier-decomposing the total Lagrangian pressure pT (= p + B.b/µ for perturbed field b and plasma pressure p) and normal component of Lagrangian displacement ξr like (pT , ξr ) ∼ (ˆ pT (r), ξˆr (r)) ei(mθ+kz z−ωt) ,

(68)

and omitting the hat of the Fourier decomposed perturbations for the sake of simplicity of notation, we find that pT (r) satisfies the Bessel equation   1 dpT m2 d2 pT 2 + − kz + 2 pT = 0, (69) dr2 r dr r where m is the azimuthal wavenumber (see Eq. 24). To obtain a dispersion relation for the configuration of a magnetic annulus we require the solutions for pT in Eq. (69) to satisfy the continuity of total Lagrangian pressure pT and the continuity of the normal displacement perturbation, ξr , across the boundaries r = a and r = R. After some algebra and using these boundary conditions we arrive at: ′ ′ Qi0 Km (kz a) − (Im (kz a)Km (kz a)/Im (kz a)) ′ (k a)(Qi − 1) Im z 0

=

′ Km (kz R)(Qe0 − 1) . ′ (k R) − (K ′ (k R)I (k R)/K (k R)) Qe0 Im z m z m z m z

(70)

in which we have defined Qi0 =

2 ρi (ω 2 − ωAi ) , 2 2 ρ0 (ω − ωA0 )

Qe0 =

2 ρe (ω 2 − ωAe ) . 2 2 ρ0 (ω − ωA0 )

(71)

September 11, 2008

84

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

Eq. (70) is the dispersion relation for wave propagation in an incompressible straight magnetic flux tube structure consisting of a core tube and an annulus of radii a and R, respectively, embedded in an ambient magnetic environment with straight field lines. For the limiting case of no core a → 0 (or equally a → R) we recover a dispersion relation derived earlier44 for a single straight monolithic tube with external Alfv´en speed vAe and internal Alfv´en speed vA0 (vAi ). There are two surface mode solutions to the disper-

Fig. 9. Dispersion curves for m = 0 sausage (dotted) and m = 1 kink modes (solid) giving the phase speed ω/k as function of the dimensionless wavenumber ka for a magnetic annulus with a/R = 0.8 modelling left photospheric conditions, and right the a flux tube with a dense core.

sion relation (70) arising for each the sausage and kink modes, respectively, for the annulus-core model. These modes propagate along the two natural surfaces of the system, i.e. at r = a and r = R. In the incompressible approximation, for a straight magnetic field, body modes are removed from the system. The modifications to the phase speeds, introduced by the annulus, as compared to a monolithic loop are significant and dependent not only on the Alfv´en speed in each region but also on the ratio a/R of the core and annulus radii. 2.4. Magneto-Seismology: Inhomogeneous Magnetic Field Post-flare transversal coronal loop oscillations have been observed many times using the high-resolution EUV imager onboard TRACE, see, e.g. Refs. 54-58. These oscillations were identified as the fundamental mode of the standing fast kink wave from MHD wave theory.44 The basic theory models a coronal loop as a straight magnetic cylinder with different external and internal plasma densities, both of which are taken to be constants (see the previous sections).

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

85

Using emission measures, there is observational evidence that there is density stratification in coronal loops. In younger active regions there have been measurements of super-hydrostatic density scale heights that are up to four times higher than expected.59,60 On the other hand, loops have been observed in older active regions that are close to hydrostatic equilibrium61 with density scale heights that can be explained by gravitational stratification. The implications of this for coronal loop oscillations have been considered, e.g. Refs. 62 and 63 through numerical modelling of damping of loop oscillations in the framework of dissipative and radiative MHD. To complicate matters further, significant dynamical behaviour has also been observed in loops, e.g., flows64 -66 and cooling events.67,68 This is an important point because background flows can cause complex interactions between MHD waves. Theoretically, the effect of steady state flows on MHD waves in a uniform magnetic slab-geometry was investigated by, e.g. Refs. 69-71. They found the dispersion relation for such steady states and also have shown the presence of negative energy waves. Refs. 72-74 generalised the slab studies to flux tubes but their derivation is valid only for limited parameters. A detailed and comprehensive derivation of steady flow effects on uniform MHD waveguides in cylindrical geometry (with stratification due to gravity ignored) can be found, e.g. Ref. 75. In light of the exciting observations from TRACE, much work has been done developing more realisitic theory of fast kink waves in coronal loops. E.g., models have been developed with inhomogeneous plasma density equilibria. Firstly, spatial variation of density in the radial direction has been included in the analysis leading to a change in period and damping of the MHD waves.76 -80 Secondly, spatial variation of density in the longitudinal direction has been included in the analysis leading to changes in the ratios of the periods of the overtone modes to that of the fundamental mode and to deviations of the eigenfunctions from a single sine term in the longitudinal direction.81 -89 To develop a more complete theory of fast kink waves coronal loops, here we quantify the effects of both inhomogeneous plasma density and magnetic equilibria. At present, the structure of the magnetic field along coronal loops is probably even less well understood from observation than plasma density stratification.90 The indirect observational evidence so far has been rather puzzling. A study of TRACE loops91 has shown that the cross-sectional width remains relatively constant with increasing height above the photosphere. The flux tube interpretation suggests that magnetic field is therefore almost constant along loops but this contradicts potential and force-free

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

86

Ch5

R. Erd´ elyi

field extrapolations using data from the Michelson Doppler Imager (MDI) onboard SOHO, where the field lines always diverge with height. It was also suggested92 that by twisting a loop this could reduce the amount of width expansion with height. It is therefore crucial that theoretical models are developed which can predict how different magnetic field structures in loops affect the properties of loop oscillations. It was summarised above the work that has already been done, particularly regarding the effect of magnetic twist on loop oscillations, e.g. Refs. 46-48 and 93. Furthermore, the effect of a twisted shell, i.e., tube within a tube, was studied by Refs. 51-53. It is hoped that models of this type that have more complex magnetically structured tubes can be tested against observations and help further advance the field of magneto-seismology.

2.4.1. Magnetic field and plasma density equilibrium Using cylindrical coordinates (r, θ, z), a magnetic flux tube of length 2L is modelled with arbitrary external and internal plasma densities ρe (z) and ρi (z). To model a magnetic field equilibrium that decreases in strength with height above the photosphere, we construct an expanding flux tube with rotational symmetry (see Fig. 10). To do this one must have ~ = Br (r, z)~er + Bz (r, z)~ez B

(72)

so that the solenoidal and force-free (potential) conditions are satisfied.

2.4.2. Governing equation and analysis By Fourier analysing the linear MHD equation and using the thin flux tube approximation in cold plasma and the fact the Br ≪ Bz , it can be shown that the governing equation of radial motion at the tube boundary (where all quantities can be expressed as functions of z only) for the observed fast kink mode (m = 1) is ′′

(Bz vr ) +

+

 Br ′ + 4ro′ (Bz vr ) Bz # 2  ′  ′ 2 ω 1 Br ro ro′′ + + + Bz vr = 0. ck 2ro Bz ro ro

1 2ro "



(73)

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

Fig. 10.

87

The equilibrium plasma density and expanding magnetic field.

Assuming constant densities, ρe and ρi , Eq. (73) is equivalent to h z  z i z h z i a1 cosh2 + a2 cosh + a3 vr′′ + sinh a4 cosh + a5 vr′ L L L L z i h z + a6 cosh2 + a7 cosh + a8 + a9 ω 2 vr = 0, (74) L L where an are constants. Unfortunately, we know of no analytical solution to equations of general type Eq. (74). However, Eq. (74) is trivial to solve numerically by e.g., the shooting method. Solving Eq. (74) for the fundamental mode and first harmonic, the observable signatures of magnetic stratification are plotted in Fig. 11 for a small expansion Γ ∈ [1, 2]. In contrast to the case of density stratification with constant magnetic field, see, e.g. Ref. 89, the anti-node shift of the first harmonic is towards the loop apex (see Fig. 11a). It was argued94 that for Γ approximately less than 1.5 there is almost a linear relationship with node and anti-node shifts. In further contrast to the case of density stratification with constant magnetic field, see, e.g. Ref. 95, the frequency ratio of the first harmonic to the fundamental mode, ω2 /ω1 is greater than 2 (see Fig. 11b). There have been various studies to calculate the value of Γ for coronal loops in both soft X-ray and EUV. Using Yohkoh data, it was found96 that the mean value of Γ for a sample of 43 soft X-ray loops was 1.30.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

88

Ch5

R. Erd´ elyi

A

anti−node shift

Γ=1

(a)

Amplitude

Γ = 1.5

0

−A anti−node shift

−L

0 z

L

4 (c)

2

ω /ω

1

3

2

1

0 1

1.2

1.4

Γ

1.6

1.8

2

Fig. 11. (a) Comparison of 1st harmonic amplitude profiles with constant magnetic field (Γ = 1) and magnetic stratification (Γ = 1.5). (b) Frequency ratio of 1st harmonic and fundamental mode ω2 /ω1 against Γ.

In another study using EUV TRACE data91 it was also found that the mean Γ value for post-flare loops was 1.13. However there may have been large uncertainties in these results. Errors could also have been introduced by e.g., incorrect background subtraction and line of sight effects. Even allowing for a relatively small expansion factor of Γ = 1.13, this should give measurable observable effects. E.g., a loop half length L = 100 Mm and fundamental mode period 5 minutes, Γ = 1.13 will give an anti-node shift of 3.5 Mm and a change in the period of the first harmonic of -6.23 seconds. Certainly, spatial changes to the amplitude profile of a few Mm is within the current resolution of TRACE. Measuring changes in frequency down to the order of seconds may be possible with the fastest time cadences of the planned EUV imagers onboard SDO and SO (signal to noise ratio permitting). 2.5. Mechanism of Resonant Absorption Let us consider an ideal inhomogeneous vertical magnetic flux tube embedded in a magnetic free plasma such that the Alfv´en speed has a maximum

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

89

at the axis of the tube and the Alfv´en speed is monotonically decreasing to zero as a function of the radial coordinate (Figure 12a). Suppose that there

Fig. 12. Left: Schematic sketch of resonant absorption. Right: Phase-mixing of surface waves caused by gradients in the background magnetic field (or Alfv´ en speed) where the footpoints of the field lines are shaken in the y-direction.

is a sound wave continuously impinging at the boundary of this flux tube. If the phase speed of this impinging (or driving) wave matches the local Alfv´en speed at a given location of the radius, say at rA , the driving wave is in resonance with the local Alfv´en waves at the magnetic surface at rA . In ideal MHD this would result in infinite amplitudes of the perturbations resulting in large gradients. However, once the gradients of perturbations become large, one cannot assume any longer the plasma is ideal, i.e. dissipative effects (e.g. resistivity, viscosity) have to be considered at least within the vicinity of such resonant location leading to energy dissipation. Such dissipation, i.e. energy absorption of the driving wave, will result in heating of the plasma converting the energy of the driving wave into localised thermal heating.97 -99 Resonant absorption, originally considered by plasma physicists as means of excess heating source for thermonuclear fusion, seems to work very well when modelling e.g. the interaction of solar global oscillations with sunspots;101,102 when applied to explain the damping of coronal loop oscillations,31,103 resonant flow instabilities, e.g. Ref. 70, etc. 2.6. Process of Phase Mixing There was proposed104 another interesting mechanism that is in a way fairly similar to resonant absorption. There is a magnetised plasma that is inhomogeneous in the x-direction of the xz-plane where the magnetic field lines are parallel to the z-axis (Figure 12b). We perturb each field line in a

September 11, 2008

90

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

coherent (e.g. sinusoidal) way in the y-direction. Along each of the field lines an Alfv´en wave will develop and will propagating in the z-direction with a speed characteristic to that field line. Since the plasma is inhomogeneous the Alfv´en speed at two adjacent field lines is different and neighbouring oscillating field lines will be soon out of phase after some time resulting in large gradients of perturbations. At a given point when the gradients reach a critical value it is not correct anymore to assume that the plasma is ideal and dissipative effects have to be included in the analysis (just like in the case of resonant absorption) resulting in local heating. This dissipation of the initial perturbations is called phase mixing. Phase mixing is an excellent candidate for MHD wave energy dissipation in open magnetic regions like coronal funnels, plumes, solar wind. 3. MHD Waves in the Lower Solar Atmosphere It is a natural question, whether from the intrinsic periodic motions of the magnetic structures one would be able to derive crucial diagnostic information, e.g. density stratification both along and across a coronal loop; derive constrain on fine structure (single monolithic vs multi-thread magnetic structuring); obtain information about the geometry (curvature, inclination, expansion rate, cross-sectional formation) and topology (magnetic connectivity), etc. An application of the above approach to the solar corona was first put forward105,106 introducing the term of coronal seismology, what was reviewed for standing waves in the previous sections. The method of coronal seismology, however can be further extended and generalised to the entire solar atmosphere from the partially magnetised photosphere to magnetically dominated corona. In order to reduce the number of various and sometimes confusing labelling, but at the same time to express the strong relation and overlap of the methodology of helioseismology and coronal seismology, the unifying term of solar magneto-seismology is introduced. The terminology serves well the rapid emergence of seismic and diagnostic studies of the magnetised solar atmosphere, and, at the same time expresses the major future direction of developments in local helio(and astero)seismology. An interesting and promising application of magneto-seismology is to specifically investigate oscillations in the lower part of the solar atmosphere, where periodic motions maybe mostly but not exclusively in the form of propagating waves. The various and by large concentrated magnetic structures at photospheric to low-TR and coronal heights serve as excellent waveguides to the propagation of perturbations excited at foot-

September 17, 2008

17:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

91

point regions. A typical and characteristic lower atmospheric waveguide, in a first approximation, could possibly be considered as an isolated magnetic flux tube with practically no magnetic field in their environment, resulting in a hat-like functional dependence of the Alfv´en speed variation across the MHD waveguide. Another feature that distinguishes these lower atmospheric waveguides from their coronal counterparts is that the density, pressure and magnetic strength scale heights inside the waveguide (and in the surrounding environment except for magnetic fields) may be comparable and of the order to the horizontal (i.e. radial) dimension of the magnetic structures, and/or, also comparable to the characteristic wavelength of the periodic motions they support. Even for the simplest mathematical approximation this latter feature leads to the introduction of a linear term to the governing equation of wave perturbations resulting in a Klein-Gordon-type of equation.26,107 Before we dwell on the recent developments made on observations of wave leakage and progressive waves in the (lower) solar atmosphere, we refer the reader to related reviews.8,11,33,108 -115 3.1. Magneto-Seismology in the Lower Boundary Layer To carry out magneto-seismology in the lower partially magnetised solar atmosphere, one of the first tasks is to understand what is the role of the presence of the boundary layer between the solar interior (β ≫ 1) and the magnetically dominated corona (β ≪ 1). The transition between the solar interior and the corona occurs in a rather narrow layer. This boundary layer, that includes the photosphere, chromosphere and TR is around 2-3 Mm thick and contains coherent and random magnetic and velocity fields, giving a very difficult task to describe even in simple approximate terms the wave perturbations. Random flows (e.g. turbulent granular motion), coherent flows (meridional flows or the near-surface component of the differential rotation), random magnetic fields (e.g. the continuously emerging small-scale magnetic flux, often referred to as magnetic carpet) and coherent fields (large loops and their superposition of the magnetic canopy region) each have their own effect on wave perturbations. Some of these effects may be more relevant and deterministic than the others. From practical helioseismology perspectives the main task here is to estimate the magnitude of these corrections one by one. It is strongly suspected that, both coherent and random magnetic and velocity fields may contribute to line widths or frequency shifts of the global acoustic oscillations on a rather equal basis. In helioseismology the corrections from this boundary layer

September 11, 2008

92

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

are summarised in the surface term34,111,116 and in many helioseismologic diagnostic modelling the surface term is taken in some ad-hoc functional form. Now we dwell on the question: how photospheric motions (e.g. coherent internal acoustic oscillations or casually generated pulses) penetrate into the solar atmospheric boundary layer, and, what observable dynamic consequences this leakage and penetration may have on the magnetic structures of the solar atmosphere. 3.2. Wave Leakage from Photosphere Spicules and moss oscillations, detected by TRACE and by SUMER on board SOHO may bring us closer to the origin of the running (propagating) waves of coronal loops. The correlations on arcsecond scales between chromospheric and transition region emission in active regions were studied.118 The discovery of active region moss,117 i.e, dynamic and bright upper transition region emission at transition region heights above active region (AR) plage, provides a powerful diagnostic tool to probe the structure, dynamics, energetics and coupling of the magnetized solar chromosphere and transition region. It was also studied118 the possibility of direct interaction of the chromosphere with the upper TR, by searching for correlations (or lack thereof) between emission at varying temperatures using concurrently taken EUV lines emitted from the low chromosphere (Ca II K-line), the middle and upper chromosphere (Hα), the low transition region (C IV4 1550 ˚ A at 0.1 MK), and from the upper transition region (Fe IX/X 171 ˚ A at 1 MK and Fe XII 195 ˚ A at 1.5 MK). The high cadence (24 to 42 seconds) data sets obtained with the Swedish Vacuum Solar Telescope (SVST, La Palma) and TRACE allowed to them find a relation between upper transition region oscillations and low-laying photospheric oscillations. Intensity oscillations were analysed24 in the upper TR above AR plage. They suggested the possible role of a direct photospheric driver in TR dynamics, e.g. in the appearance of moss (and spicule) oscillations. Wavelet analysis of the observations (by TRACE) verifies strong (∼ 5 - 15%) intensity oscillations in the upper TR footpoints of hot coronal loops. A range of periods from 200 to 600 seconds, typically persisting for about 4 to 7 cycles were found. A comparison to photospheric vertical velocities (using the Michelson Doppler Imager onboard SOHO) revealed that some upper TR oscillations show a significant correlation with solar global acoustic p-modes in the photosphere. In addition, the majority of the upper TR oscillations are directly associated with upper chromospheric oscillations observed in

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

93

Hα, i.e., periodic flows in spicular structures. The presence of such strong oscillations at low heights (of order 3,000 km) provides an ideal opportunity to study the direct propagation of oscillations from photosphere and chromosphere into the TR32 and low magnetic corona see, for example Ref. 23. These type of measurements can also help us to (i) understand atmospheric magnetic connectivity that is so crucial for diagnostic reconstruction in the chromosphere and TR, and, shed light on the dynamics of the lower solar atmosphere, e.g. the source of chromospheric mass flows such as spicules;32 (ii) explore the dynamic and magnetised lower solar atmosphere using the method of magneto-seismology.33,34 3.2.1. Global resonant acoustic waves in a stratified atmosphere Acoustic waves have often been invoked as possible candidates for the heating of solar and stellar chromospheres and coronae (see, e.g. Ref. 27 for the latest review on observations; and Refs. 14, 100 on theory). Until recently it was thought that high frequency waves could be responsible for the heating of the non-magnetic chromosphere. On the other hand, low frequency waves were believed to play little role as far as the dynamics and energetics of the atmosphere are concerned due to reflection from regions with steep temperature gradients. Recent works have changed these views. It was established that the power of the observed high frequency propagating (> 5 mHz) acoustic waves is not enough to balance the radiative losses in the chromosphere.119 On the other hand, new observations have shown that the energy flux carried by the low frequency (< 5 mHz) acoustic waves into the chromosphere is about a factor of 4 greater than that carried by high frequency waves.120 It was argued that these low frequency waves could propagate and carry their energy into the higher layers of the atmosphere through portals formed by the inclined magnetic field lines.23,32 These and other results have prompted renewed strong interest in the theory of low frequency acoustic wave propagation in stratified media. The vertical propagation of acoustic waves in a stratified atmosphere (either plasma or gaseous) can be demonstrated in a two-layer model (Fig. 13). The waves are described by the Klein-Gordon (KG) equation. The key point here to note is the resonance occurring at low frequencies which extends into the entire unbounded atmosphere as was first shown by Refs. 121 and 122. This previously unknown resonance may be responsible for the transfer of wave energy which could have dynamic consequences and heat the higher atmospheric layers. The KG equation is widely used in a range of fields such as atmospheric physics, cosmology, quantum field

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

94

Ch5

R. Erd´ elyi

theory, solid state physics, solar/stellar physics. Therefore, the results121,122 also have wider applicability in distinct areas of physics and astrophysics. A

Fig. 13. Two-layer model depicting a stratified solar atmosphere. The lower part of the atmosphere (index 1) is separated from the upper part (index 2) by a density and temperature discontinuity at z = L. Waves are launched at z = 0 and propagate in the vertical z-direction.

necessary condition for the existence of a resonance is L/2Λ2 > 1 in a stratified atmosphere. The waves are resonantly amplified when

ω=

c2 L

r

L − 1. Λ2

(75)

The physical mechanism responsible for wave amplification is the following: the decreasing temperature results in an increasing acoustic cut-off frequency Ω = Ω(z) which forms a potential barrier similar to the one in quantum mechanics.123 Low frequency waves with Ω0 < ω < Ω2 driven at z = 0 are reflected back from the barrier and trapped in the lower layer 1. When the driver frequency matches the natural frequency of the cavity where the waves are trapped a standing wave is set up and amplified resonantly. In the case of a thin layer, only the fundamental mode is present with a frequency given by Eq. (75). The frequency of the fundamental mode decreases and higher harmonics appear as the thickness L increases. The resonance affects the evanescent tail of the waves in the upper atmosphere leading to a global resonance.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

95

3.3. Propagating Waves into Corona In the pre-SOHO/TRACE era probably, the first observations of MHD waves in the corona were reported124 with GSFC extreme-ultraviolet spectroheliograph on OSO-7 (the spatial resolution was few arcsec, the cadence time was 5.14 s). In Mg VII, Mg IX and He II emission intensity periodicity at about 262s was detected. The importance of this early work is that within the range of low-frequencies an analogy to photospheric and chromospheric oscillations was found, and, it was further speculated that the photospheric and chromospheric evanescent waves become vertically propagating, gravity-modified acoustic waves at that height in the chromosphere where temperature rise admits propagation again. Using Harvard College Observatory EUV spectroheliometer on Skylab,125 oscillations were detected in the C II, O IV, and Mg X emission intensity with periods of 117s and 141s. They suggested that the intensity fluctuation of the EUV lines is caused by small amplitude waves, propagating in the plasma confined in the magnetic loop, and that size of the loop might be important in determining its preferential heating in the active region. A final example from that era, though in a much shorter wavelength, is the observations,126 who detected with the Hard X-ray Imaging Spectrometer on-board SMM soft X-ray (3.5-5.5 keV) pulsations of period 24 min lasting for six hours. The periodicity was thought to be produced by a standing wave or a traveling wave packet which exists within the observed loop. It was concluded the candidates for the wave are fast or Alfv´en MHD modes of Alfv´enic surface waves. The situation by the launches of SOHO and TRACE have considerably changed our views since abundant evidences merged for MHD wave phenomena, in particular for propagating waves. In what follows we give an account of the observed propagating waves, and, we overview the attempts made to link these progressive waves to solar global (photospheric) motions, where the latter is accounted for being the driver behind these periodic coronal motions.

3.3.1. Observations of progressive waves Progressive waves may propagate in open (e.g. plumes) and closed (e.g. loops) coronal magnetic structures. The first undoubtable detection of progressive slow MHD waves was made by UVCS/SOHO (Ultraviolet Coronagraph Spectrometer). Observations of slow waves in an open magnetic structure, i.e. high above the limb

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

96

Ch5

R. Erd´ elyi

of coronal holes127,128 were carried out. Analysing EIT/SOHO (Extremeultraviolet Imaging Telescope) data of polar plumes similiar compressive distrubances were detected129 with linear amplitudes of the order of 1020% and periods of 10-15 minutes. The observed compressive longitudinal distrubances were identified130,131 as progressive slow MHD waves. The detected damping of slow progressive waves was attributed to compressive viscosity. In closed structures, using EIT/SOHO132 it was reported first on slow modes. Following the success of SOHO, observers using TRACE also searched successfully for quasi-periodic disturbances of coronal loops.133 -135 A detailed overview of the observed properties of these propagating intensity perturbations is given by, e.g. Refs. 136 and 137 and we here only summarise the main fetures.110 An overview of the periods and propagation speeds found by various authors is given in Table 3. In all the reported Table 3. Overview of the periodicities and propagation speeds of propagating slow MHD waves detected in coronal loops. Adopted from Ref. 110. Nightingale et al. (1999) Schrijver et al. (1999) Berghmans & Clette (1999) De Moortel et al. (2000) Robbrecht et al. (2001) Berghmans et al. (2001) De Moortel et al. (2002a) De Moortel et al. (2002b) King et al. (2003)

Period (s) 300 ∼600 180–420 (282 ± 93) 172 ± 32 (sunsp.) 321 ± 74 (plage) 120–180 & 300-480

Speed (km/s) 130–190 70–100 75–200 70–165 65–150 ∼300 122 ± 43 25–40

Wavelength 171 & 195 195 195 171 171 & 195 SXT 171 171 171 171 & 195

cases the phase speed is of the order of the coronal sound speed. Since the progressive waves are observed as intensity oscillations, hence they are likely to be candidates of compressive disturbances. No significant acceleration or deceleration was observed. The combination of all these facts leads to the most plausible conclusion that the observed progressive waves are indeed slow MHD waves. 3.3.2. Source of progressive waves In order to answer the question of what is the source of progressive coronal waves, and, inspired by the observational findings of similarities between photospheric and TR oscillations, there was developed32 the general framework of how photospheric oscillations can leak into the atmosphere along inclined magnetic flux tubes. In a non-magnetic atmosphere p-modes are

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

97

evanescent and cannot propagate upwards through the temperature minimum barrier since their period P (∼ 200 − 450 s) is above the local acoustic cut-off period Pc ≈ 200 s. However, in a magnetically structured atmosphere, where the field lines with some natural inclination θ -where θ is measured between the magnetic guide channelling the oscillations and the √ vertical- the acoustic cut-off period takes the form Pc ∼ T / cos θ with the temperature T . This inclination will allow to tunnel some non-propagating evanescent wave energy through the temperature minimum into the hot chromosphere of a the waveguide, where propagation is once again possible because of higher temperatures (Pc > 300 s). The authors have shown that inclination of magnetic flux tubes (well applicable e.g. to plage regions) can dramatically increase tunnelling, and can even lead to direct propagation of p-modes along inclined field lines as plotted in Fig. 14. a

b

Fig. 14. Leakage of evanescent photospheric p-mode power into chromosphere. Distribution of wavelet power (in arbitrary units, independent for each height)(for cases a and b, resp. θ = 0◦ and 50◦ ) as a function of wave period for different heights above the photosphere. Vertical flux tubes (a) allow minimal leakage of p-modes with periods of 300 s (> Pc ∼ 220 s), so that only oscillations with lower periods (< 250 s) can propagate and grow with height to dominate chromospheric dynamics. Inclined flux tubes (b) show an increased acoustic cut-off period Pc , allowing enhanced leakage and propagation of normally evanescent p-modes. Adapted from Ref. 32.

A perfectly natural generalisation of the above idea was put forward,23 who proposed that a natural consequence of the leakage of photospheric oscillations is that the spicule driving quasi-periodic shocks propagate into the low corona, where they may lead to density and thus intensity oscillations with properties similar to those observed by TRACE in 1 MK coronal

September 11, 2008

98

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

loops. In other words, the origin of the propagating slow MHD waves detected in coronal loops (see a recent review on their properties by e.g. Ref. 110) was linked to wave energy leakage of solar global standing oscillations. It was highlighted23 that oscillations along coronal loops associated with AR plage have many properties that are similar to those of the moss oscillations: (i) the range of periods is from 200 to 600 seconds, with an average of 350 ± 60 s and 321 ± 74 s, for moss and coronal oscillations, respectively; (ii) the spatial extent for coherent moss oscillations is about 1-2′′ , whereas for coronal waves, the spatial coherence is limited to ∼′′ in the direction perpendicular to that of wave propagation. They also point out that although the oscillations in moss and corona have similar origins, they are results of different physical mechanisms: moss oscillations occur because of periodic obscuration by spicules, and coronal oscillations arise from density changes associated with the propagating magneto-acoustic shocks that drive the periodic spicules. 4. Where Magneto-Seismology and the Coronal Heating Enigma Meet The coronal heating enigma has challenged solar physicists for over half a century. None of the theories proposed so far has yet been the answer to teh coronal heating problem. The real task now is to establish which of these theories (if any) represents the the actual heating mechanism or perhaps a combination of various models. It is now becoming increasingly evident that predicting observational signatures and footprints of various heating models through forward modelling, and their direct comparisons with currently available high resolution observations is the way to proceed. Forward modelling and inversion of observed data represent two complementary aspects of solving the heating enigma. A combination of these two approaches will, however, provide an integrated and most efficient currently available route of tackling coronal. Section 4 briefly outlines this concept. In particular, we focus on the problem of inversion and plasma heating diagnostics using MHD waves. 4.1. Inversion and Diagnostics with MHD Waves Forward modelling is used to derive observational signatures and footprints predicted by a theoretical model. The reverse procedure of deriving from observations the actual footprints and physical background quantities characterising the magnetised plasma is a difficult task in general.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

99

A well-known example of the controversy in solar data inversion is the analysis of a single Yohkoh data set from which three different temperature and heating profiles along a magnetic loop were derived by three different authors: uniform,138 footpoint60 and apex139 heating. The discrepancy between the results was mainly attributed to the different ways in which the background was subtracted. Other authors have criticized the assumption of isothermal approximation which is implicit in the conventional filter ratio analysis.140,141 An alternative approach to the inversion problem is the use of MHD waves. MHD waves are an excellent diagnostic tool and their full potential has yet to be explored. We have already demonstrated that MHD waves could be used for a number of purposes: measuring the magnetic field strength, the fine structure, the transport coefficients in the solar atmosphere, understanding the processes responsible for the damping of these waves and so on.47,51,52,58,76,77,81,83,87,89,95,135,142 -146 For more details on the observed MHD waves, their damping, etc.27,113 Standing acoustic waves were detected147,148 in hot active region loops could be quite useful in quantifying the heating function. The damping of these oscillations was studied by Refs. 6, 62, 63, 149, 150. Observations show that these waves are usually preceded by footpoint brightenings. It haa been determined151 the mathematical form of the heating pulse required to rapidly set up a standing wave: the duration of the heating pulse must match approximately the period of the standing oscillations. The results were applied152 to an active region. An 86 Mm long loop underwent heating to T ≈ 7 MK followed by cooling to T ≈ 2 MK in less than two hours. The heating was followed by rapidly damped standing longitudinal oscillations with a period of about 8 mins. The maximum initial Doppler shift, observed by SUMER, was about 30 km s−1 . Forward modelling was carried out, using the parameters of the observed Yohkoh SXT loop, in dissipative hydrodynamics including thermal and radiative losses. The numerical results were then converted into observable quantities by applying spectral line synthesis. Figure 15 compares the simulated line profiles with the actual SUMER measurements from 3 to 7 MK. Note, the oscillations only appear in the Doppler shift as opposed to intensity. The initial negative blue shift of about 32 km/s is followed by damped oscillations. It was concluded152 that the observed oscillations are the fundamental mode standing acoustic wave; the standing waves are excited by a microflare occurring at one of the loop footpoints; they also determined the evolution of the heating rate along the loop.

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

100

Ch5

R. Erd´ elyi

(a)

red

blue

black

red

blue

(b) blue black red

Time on 2000 Sep 17 (UT)

Fig. 15. Left - SUMER observations: the top panel represents the time series of the Fe xix (black) line intensity along the slit. The overlaid contours represent the Ca xv (λ 1098, red) and Ca xiii (blue) intensity time series. The contour levels are 70, 80, and 90% of the peak intensity; the bottom panel is the average time profile of the line-integrated intensity along the cut in the top panel. Right - synthesized observations: the top panel shows the forward modelled time profiles of the line intensities along the slit cut. The black, red and blue lines correspond to Fe xix , Ca xv (λ 1098), and Ca xiii intensities; the bottom panel shows the corresponding Doppler shift from Ref. 152.

The next task now is to show an example of data inversion. A reliable inversion may come from the analysis of Doppler shift time series. The idea, proposed by Ref. 153, is borrowed from helioseismology where the use of such time series has become a routine method for one of the most precise diagnostic measurements in astrophysics. The new method does not require the presence of coherent standing waves. The only underlying assumption is that loops (or their individual and unresolved components, called strands) are heated randomly both in time and in space (an argument supported strongly by solar observations). A linear ideal 1D loop, heated by random pulses, will respond to energy perturbations (depositions) by an infinite number of peaks in the velocity power spectrum corresponding to the frequencies of standing waves. The inclusion of dissipative processes (e.g. radiative losses, thermal conduction) and nonlinearity introduces noise in the power spectrum. The most prominent peak corresponding to the fundamental mode is always present regardless of the random heating function and the heliographic position of the loop on the solar disk. This peak could therefore be used to determine the average temperature of the plasma inside

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

Waves and Oscillations in the Solar Atmosphere

101

Line Shift (km s-1)

the loop! The peak corresponding to the second harmonic only appears in the case of uniformly random heating. Peaks corresponding to higher harmonics do not show up due to their small amplitudes, losses and nonlinearity. The results of the wavelet analysis for such a loop are displayed in Figure 16. Forward modelling shows that loops heated near their footpoints a) Doppler Shift Time Series for Fe X WAVELET ANALYSIS

5 0 -5 -10 0

500

1000

1500 Time (min)

2000

2500

3000

red

b) Wavelet Power Spectrum

c) Global

Frequency (mHz)

10

10

1

1

90% 99% 0

500

blue

1000

1500 Time (min)

2000

2500

3000

0

40 80 120 160 200 Power (km2s-2)

Fig. 16. Wavelet analysis of temporally randomly but spatially evenly heated 30 Mm long loop. The top panel (a) shows the Doppler shift time series in the Fe x line. The bottom left panel (b) displays the wavelet power spectrum. The red color represents high power and the blue color corresponds to low power. The bottom right panel (c) is the global wavelet spectrum. In the wavelet spectrum diagram (b), regions with 90% significance level are outlined in black. In the global wavelet diagram, the dotted lines indicate 99% significance level.

only display the fundamental mode. This is mainly due to the large values of thermal conduction around the maximum of the second harmonic which results in strong damping. The phenomenon is explained in more details by Ref. 153. On the other hand the power spectra (i.e. inversion) are also sensitive to the temporal distribution of heating and, as a result, they could be used to estimate the average amount of energy involved in a single heating event. Interestingly, the power peaks do not show up when the same analysis is applied to the intensity time series. This effect as well as the phase shifts in the synthesized EIS observations are analytically explained by Ref. 154. Finally, it is now concluded that the power spectrum analysis

September 11, 2008

10:26

102

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

(i.e. data inversion) could therefore be used to establish the distribution of the random heating, i.e. determine the nature of the heating process. 5. Summary Let us now summarise the key points made in this chapter. MHD waves and oscillations serve as excellent diagnostic tools in gaining diagnostic information about the structure of the solar atmosphere. MHD waves are important in the energy balance of the atmosphere from photosphere to corona. Forward modeling of wave propagation and inversion of wave observations are two complementary approaches which allow the heating problem to be tackled in an integrated and comprehensive manner. The new generation satellites (TRACE, Hinode, STEREO, SDO, SOLO) have (or will) provided new constraints on the forward modelling of the signatures and observable consequences of the feasible theoretical heating scenarios. Coordinated efforts of the diagnostic capabilities of these missions are most likely to improve our understanding of the solar atmospheric magnetised plasma from the photosphere to the corona, and beyond, in the interplanetary space. Analogue to helioseismology, standing MHD waves in magnetic structures are important for quantifying the unknown heating function and the internal structure of the MHD waveguide. The analysis of Doppler shift and Doppler width time series is a new and very efficient tool for determining the spatial and temporal distribution of the heating function, magnetic field geometry, dissipative coefficients, etc. The method does not require the presence of individual coherent waves. It is strongly anticipated that new inversion methods, based on MHD waves, must be developed in order to make fundamental progress. MHD inversion methods will have the potential to compete and serve as real alternative to more conventional techniques, and, hopefully will close a chapter adequately in modern astrophysics. Acknowledgments Solar physics research at the Dept. of Applied Mathematics, Univ. of Sheffield is supported by the Science and Technology Facilities Council (STFC) of the UK. The author would like to thank M.S. Ruderman and Y. Taroyan for giving advise and help when preparing this chapter. He is

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

103

grateful to V. Fedun for his immense readiness in preparing some of the figures, and also is grateful to B. Carter, M. Douglas and G. Verth for their help putting together this Chapter. RE also acknowledges M. K´eray for patient encouragement and is grateful to NSF Hungary (OTKA K67746). References 1. Golub, L. & Pasachoff, J.M. The Solar Corona, CUP (1997). 2. Cargill, P. in (eds.) J.L. Birch & J.H. Waite, Jr., Solar System Plasma Physics: Resolution of Processes in Space and Time, (1993). 3. Zirker, J.B., Solar Phys., 148, 43 (1993). 4. Mendoza-Brice˜ no, C.A., Erd´elyi, R. & Sigalotti, L.D.G. Astrophys. J., 579, 49 (2002). 5. Mendoza-Brice˜ no, C.A., Sigalotti, L.D.G., Erd´elyi, R. AdSpR, 32, 995 (2003). 6. Mendoza-Briceno, C.A., Sigalotti, L.D.G. & Erd´elyi, R. Astrophys. J., 624, 1080 (2005). 7. Mendoza-Briceno, C.A. & Erd´elyi, R. Astrophys. J., 648, 722 (2006). 8. Aschwanden, M. Review of Coronal Oscillations, in (eds.) R. Erd´elyi et al., Turbulence, Waves and Instabilities in the Solar Plasma, NATO Science Ser., 124, 215 (2003). 9. Ballai, I., Erd´elyi, R. & Ruderman, M.S. Phys. Plasmas, 5, 2264 (1998). 10. Erd´elyi, R., Doyle, J.G., Perez, E.P. & Wilhelm, K. Astron. Atrophys., 337, 213 (1998). 11. Taroyan, Y. in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-seismology, IAU Symposium, 247, 186 (2008). 12. Narain, U. & Ulmschneider, P. Space Sci. Rev., 75, 453 (1996). 13. Ulmschneider, P. in (eds.) J.-C. Vial et al., Lect. Notes in Phys., 507, 77 (1998). 14. Erd´elyi, R. Astron. & Geophys., 45, p.4.34 (2004). 15. Erd´elyi R., & Ballai, I. Astron. Nacht., 328, 726 (2007). 16. Browning, P.K. Plasma Phys. and Controlled Fusion, 33, 539 (1991). 17. Gomez, D.O. Fund. Cosmic Phys., 14, 131 (1990). 18. G´ omez D., & Dmitruk P., in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magnetoseismology, IAU Symposium, 247, 271 (2008). 19. Hollweg, J.V. in (eds.) P. Ulmschneider et al., Mechanisms of Chromospheric and Coronal Heating, Springer-Verlag, Berlin p.423 (1991). 20. Priest, E.R. & Forbes, T. Magnetic Reconnection, CUP (2000). 21. Roberts, B. & Nakariakov, V.M., in (eds.) R. Erd´elyi et al., Turbulence, Waves and Instabilities in the Solar Plasma, NATO Science Ser., 124, 167 (2003). 22. Walsh, R.W. & Ireland, J. Astron. Astrophys. Rev., 12, 1 (2003). 23. De Pontieu, B., Erd´elyi, R. & De Moortel, I. Astrophys. J., 624, 61 (2005). 24. De Pontieu, B., Erd´elyi, R., de Wijn, A.G.: 2003, Astrophys. J., 595, 63.

September 11, 2008

104

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

25. Roberts, B. Solar Phys., 193, 139 (2000). 26. Roberts, B. in (sci. eds.) R. Erd´elyi, J.L. Ballester & B. Fleck SOHO 13 Waves, Oscillations and Small-Scale Transient Events in the Solar Atmosphere: A Joint View from SOHO and TRACE, ESA-SP, 547, 1 (2004). 27. Banerjee, D., Erd´elyi, R., Oliver, R. & O’Shea, E. Solar Phys., 246, 3 (2007). 28. Gomez, D.O., Dmitruk, P.A. & Milano, L.J. Solar Phys., 195, 299 (2000). 29. Hollweg, J.V. Adv. Space Res., 30, 469 (2002). 30. van Ballegooijen, A.A. Astrophys. J., 311, 1001 (1986). 31. Erd´elyi, R. in (eds.) J.L. Ballester & B. Roberts, MHD Waves in Astrophysocal Plasmas, UIB press, p.69 (2001). 32. De Pontieu, B., Erd´elyi, R. & James, S.P. Nature, 430, 536 (2004). 33. De Pontieu, B. & Erd´elyi, R. Phil. Trans. Roy. Soc. A., 364, 383 (2006). 34. Erd´elyi, R. Phil. Trans. Roy. Soc. A., 364, 351 (2006a). 35. Landau, L.D., Lifshitz, E.F. & Pitaevskii, L.P. Electrodynamics of Continuous Media Pergamon Press, Oxford (1984). 36. Cowling, T.G. Magnetohydrodynamics Interscience (1960). 37. Alfv´en, H. & Falthammar, G.G. Cosmical Electrodynamics, Oxford U.P. (1962). 38. Priest, E.R. Solar magnetohydrodynamics, D. Reidel, Dordrecht (1982). 39. Goedbloed, J.P. Lecture notes on ideal magnetohydrodynamics, Rijnhuizen Report 83-145 (1983). 40. Choudhuri, A.R. The physics of fluids and plasmas, Cambridge Univ. Press (1998). 41. Roberts, B. Solar Phys., 69, 27 (1981a). 42. Roberts, B. Solar Phys., 69, 39 (1981b). 43. Edwin, P.M. & Roberts, B. Solar Physics, 76, 239 (1982). 44. Edwin, P.M. & Roberts, B. Solar Physics, 88, 179 (1983). 45. Abramowitz, M. & Stegun, A. Handbook of Mathematical Functions, (New York : John Wiley and Sons), 1967. 46. Bennett, K., Roberts, B. and Narian, U. Solar Phys., 185, 41 (1999). 47. Erd´elyi, R. & Fedun, V. Solar Phys., 238, 41 (2006). 48. Erd´elyi, R. & Fedun, V. Solar Phys., 246, 101 (2007). 49. Robbrecht, E., Verwichte, E., Berghmans, D., Hochedez, J.F., Poedts, S., et al. Astron. Astrophys., 370, 591 (2001). 50. Mikhalyaev, B.B. and Solov’ev, A.A. Solar Phys. 227, 249 (2005). 51. Erd´elyi, R. and Carter, B.K., Astron. Astrophys., 455, 361 (2006). 52. Carter, B.K. and Erd´elyi, R., Astron. Astrophys., 475, 323 (2007). 53. Carter, B.K. and Erd´elyi, R., Astron. Astrophys., 481, 239 (2008). 54. Aschwanden, M.J., Fletcher, L., Schrijver, C.J. & Alexander, D., 1999a, ApJ 520, 880. 55. Aschwanden, M.J., De Pontieu, B., Schrijver, C.J. & Title, A.M. 2002, Solar Phys. 206, 99. 56. Nakariakov, V.M., Ofman, L., DeLuca, E.E., Roberts, B. & Davila, J.M. Science, 285, 862 (1999). 57. Verwichte, E., Nakariakov, V.M., Ofman, L. & Deluca, E.E. Solar Phys., 223, 77 (2004).

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

105

58. Jess, D.B., Mathioudakis, M., Erd´elyi, R., Verth, G., McAteer, R. T. J. & Keenan, F. P. ApJ, 680, 1523 (2008). 59. Aschwanden, M.J., Nightingale, R.W. & Alexander, D. ApJ, 541, 1059 (2000). 60. Aschwanden M. J. A&A, 560, 1035 (2001). 61. Aschwanden, M.J., Newmark, J.S., Delaboudini`ere, J., Neupert, W.N., Klimchuk, J.A., et al. ApJ, 515, 842 (1999b). 62. Ofman, L. & Wang, T. ApJ, 530, L85 (2002). 63. Mendoza-Brice˜ no, C.A., Erd´elyi, R. & Sigalotti, L.D.G. Astrophys. J., 605, 493 (2004). 64. Brekke, P., Kjeldseth-Moe, O. & Harrison, R.A. Solar Phys., 175, 511 (1997). 65. Winebarger, A.R., DeLuca, E.E. & Golub, L. ApJ, 553, L81 (2001). 66. Winebarger, A.R., Warren, H., van Ballegooijen, A., DeLuca, E.E. & Golub, L. ApJ, 567, L89 (2002). 67. Winebarger A. R., Warren H. P., Mariska J. T. ApJ, 587 439 (2003a). 68. Schrijver, C.J. Solar Phys., 198, 325 (2001). 69. Nakariakov, V.M. & Roberts, B. Solar Phys., 159, 213 (1995). 70. Tirry, W., Cadez, V.M., & Erd´elyi, R. & Goossens, M. Astron. Astropys., 332, 786 (1998). 71. Joarder, P.S., Nakariakov, V.M., Roberts, B. Solar Phys., 176, 285 (1997). 72. Joarder, P.S. & Narayanan, A.S. A&A, 359, 1211 (2000). 73. Somasundaram, K., Venkatraman, S., & Sengottuvel, M.P. Plasma Phys. Control. Fusion, 41, 1421 (1999). 74. Narayanan, A.S. Plasma Phys. Control. Fusion, bf 33, 333 (1991). 75. Terra-Homem, M., Erd´elyi, R. & Ballai, I. Solar Phys., 217, 199 (2003). 76. Ruderman, M.S. & Roberts, B. ApJ, 577, 475 (2002). 77. Goossens, M., Andries, J. & Aschwanden, M.J. A&A, 394, L39 (2002). 78. Aschwanden, M.J., Nightingale, R.W., Andries, J., Goossens, M. & Van Doorsselaere, T. ApJ, 598, 1375 (2003b). 79. Van Doorsselaere, T., Andries, J., Poedts, S. & Goossens, M. ApJ, 606, 1223 (2004). 80. Arregui, I., Andries, J., Van Doorsselaere, T., Goossens, M., Poedts, S. A&A, 463, 333 (2007). 81. D´iaz, A.J., Oliver, R. & Ballester, J.L. ApJ, 580, 550 (2002). 82. Andries, J., Goossens, M., Hollweg, J.V., Arregui, I. & Van Doorsselaere, T. A&A, 430, 1109 (2005b). 83. Dymova, M.V. & Ruderman, M.S. Solar Phys., 229, 79 (2005). 84. Arregui, I., Van Doorsselaere, T., Andries, J., Goossens M. & Kimpe, D. Phil. Trans. R. Soc. A, 384, 529 (2006). 85. Goossens, M., Arregui, I. & Andries, J. Phil. Trans. R. Soc. A., 364, 433 (2006). 86. McEwan, M.P., Donnelly, G.R., D´iaz, A.J. & Roberts, B. A&A, 460, 893 (2006). 87. Erd´elyi, R. & Verth, G. A&A, 462, 743 (2007). 88. Verth, G. & Erd´elyi, R. in: E. Marsch, K. Tsinganos, R. Marsden & L. Conroy (eds.), The 2nd Solar Orbiter Workshop, ESA-SP 641 (2007).

September 11, 2008

106

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

89. Verth, G, Van Doorsselaere, T., Erd´elyi, R. & Goossens, M. A&A, 475, 341 (2007). 90. L´ opez Fuentes, M.C., Klimchuk, J.A. & D´emoulin, P. Apj, 639, 459 (2006). 91. Watko, J.A. & Klimchuk, J.A. Solar Phys., 193, 77 (2000). 92. Klimchuk, J.A., Antiochos, S.K. & Norton, D. ApJ, 542, 504 (2000). 93. Ruderman, M.S. Solar Phys., 246, 119 (2007). 94. Verth, G. in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-seismology, IAU Symposium, 247, 125 (2008). 95. Andries, J., Arregui, I. & Goossens, M. ApJ, 624, L57 (2005a) 96. Klimchuk, J.A. Solar Phys., 193, 53 (2000). 97. Ionson, J.A., Astrophys. J., 226, 650 (1978). 98. Erd´elyi, R., Goossens, M. Ruderman, M.S. Solar Phys., 161, 123 (1995). 99. Erd´elyi, R. Solar Phys., 171, 49 (1997). 100. Erd´elyi, R. AIP-CP, 919, 122 (2007). 101. Erd´elyi, R. & Goossens, M. Astrophys. Space Sci., 213, 273 (1994). 102. Erd´elyi R., & Goossens, M. A&A, 313, 664 (1996). 103. Erd´elyi R., & Goossens, M. A&A, 294, 575 (1995). 104. Heyvaerts, J. & Priest, E.R. Astron. Atrophys., 117, 220 (1983). 105. Uchida, Y.: 1970, PASJP, 22, 341. 106. Roberts, B. Edwin, P. M. & Benz, A. O.: 1984, Astrophys. J., 279, 957. 107. Lamb, H.: 1909, Proc London Math. Soc., I, 122. 108. Aschwanden, M.J.: Physics of the Solar Corona: An Introduction, 2004, Berlin: Springer-Verlag. 109. De Moortel, I.: 2005, Phil. Trans. Roy. Soc. A., 363, 274. 110. De Moortel, I.: 2006, Phil. Trans. Roy. Soc. A., 364, 461. 111. Erd´elyi, R.: 2006b, in K. Fletcher (ed) Beyond the Spherical Sun: A New era for Helio- and Asteroseismology, Phil. Trans. Roy. Soc. A. 624, 15.1. 112. Nakariakov, V.M., Roberts, B.: 2003, in (eds) R. Erd´elyi et al., Turbulence, Waves and Instabilities in the Solar Plasma, NATO Sci Ser., 124, 167. 113. Nakariakov, V.M., Verwichte, E.: 2005, Liv. Rev. Sol. Phys., 2, 3. 114. Nakariakov, V.M.: 2006, Phil. Trans. Roy. Soc. A., 364, 473. 115. Wang, T.J.: 2004, in (eds.) R. Erd´elyi et al., ‘Waves, oscillations and small scale transient events in the solar atmosphere: A joint view of SOHO and TRACE’, ESA-SP 547, 417. 116. Basu, S.: 2002, in A. Wilson (ed.) From solar minimum to maximum: Half a solar cycle with SOHO, ESA-SP, 508, 7. 117. Berger, T.E., De Pontieu, B., Schrijver, C.J., Title, A.M.: 1999, Astrophys J., 519, 97. 118. De Pontieu, B., Tarbell, T. & Erd´elyi, R. Astrophys J., 590, 502 (2003b). 119. Fossum, A. & Carlsson, M. Nature, 435, 919 (2005). 120. Jefferies, S.M., McIntosh, S.W., Armstrong, J.D. et al. ApJ, 648, L151 (2006). 121. Taroyan, Y. & Erd´elyi, R. in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magnetoseismology, IAU Symposium, 247, 318 (2008a).

September 11, 2008

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Waves and Oscillations in the Solar Atmosphere

Ch5

107

122. Taroyan, Y. & Erd´elyi, R. Solar Phys., in press (2008b). 123. Landau, L.D. & Lifshitz, E.M. Quantum Mechanics, Pergamon Press, Oxford, 1977. 124. Chapman, R.D., Jordan, S.D., Neupert, W.M., Thomas, R.J.: 1972, Astrophys J., 174, 97. 125. Antonucci, E., Gabriel, A.H., Patchett, B.E.: 1984, Sol. Phys,, 93, 85. 126. Harrison, R.A.: 1987, Astron. Astrophys., 182, 337. 127. Ofman, L., Romoli, M., Poletto, G., Noci, C., Kohl, J.L.: 1997, Astrophys. J., 491, 111. 128. Ofman, L., Romoli, M., Poletto, G., Noci, C., Kohl, J.L. Astrophys. J., 529, 592 (2000a). 129. DeForest, C.E., Gurman, J.B.: 1998, Astrophys. J., 501, 217. 130. Ofman, L., Nakariakov, V.M., DeForest, C.E.: 1999, Astrophys. J., 514, 441. 131. Ofman, L., Nakariakov, V.M., Seghal, N. Astrophys. J., 533, 1071 (2000b). 132. Berghmans, D., Clette, F.: 1999, Sol. Phys., 186, 207. 133. Schrijver, C.J., Title, A.M., Berger, T.E., Fletcher, L., Hulburt, N.E. et al.: 1999, Solar Phys., 187, 261. 134. Nightingale, R.W., Aschwanden, M.J., Hurlburt, N.E.: 1999, Sol. Phys., 190, 249. 135. De Moortel, I., Ireland, J., Walsh, R.W.: 2000, Astron. Astrophys., 355, L23. 136. De Moortel, I., Ireland, J., Walsh, R.W., Hood, A.W.: 2002a Sol. Phys., 209, 61. 137. De Moortel, I., Ireland, J., Hood, A.W., Walsh, R.W.: 2002b,Sol. Phys., 209, 89. 138. Priest E. R., Foley C. R., Heyvaerts J., et al. Nature, 393, 545 (1998). 139. Reale F. ApJ, 580, 566 (2002). 140. Landi, E. & Landini, M. ApJ, 618, 1039 (2005). 141. Schmelz, J. T. & Martens, P. C. H. ApJ, 636, L49 (2006). 142. Nakariakov V. M., & Ofman L. A&A, 372, L53 (2001). 143. Zaqarashvili T. V. A&A, 399, L15 (2003). 144. Ruderman, M. S. & Erd´elyi, R. in (sci-eds) R. Erd´elyi, J. L. Ballester, B. Fleck, Waves, Oscillations and Small-Scale Transient Events in the Solar Atmosphere: A Joint View from SOHO and TRACE, ESA-SP 547, 507 (2004). 145. Ballai I., Erd´elyi R., Pint´er B. ApJ, 633, L145 (2005). 146. O’Shea E., Srivastava A. K., Doyle J. G., Banerjee, D. A&A, 473, L13 (2007). 147. Kliem B., Dammasch I. E., Curdt W., & Wilhelm K. ApJ, 568, L61 (2002). 148. Wang, T. J., Solanki, S. K., Curdt, W., Innes, D. E., & Dammasch, I. E. ApJ, 574, L101 (2002). 149. Selwa M., Murawski K., Solanki S. K. A&A, 436, 701 (2005). 150. Luna-Cardozo, M., Erd´elyi, R. & Mendoza-Brice˜ no, C.A., in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-Seismology, IAU Symposium, 247, 318 (2008).

September 11, 2008

108

10:26

WSPC - Proceedings Trim Size: 9in x 6in

Ch5

R. Erd´ elyi

151. Taroyan Y., Erd´elyi, R., Doyle, J. G., & Bradshaw, S. J. A&A, 438, 713 (2005). 152. Taroyan Y., Erd´elyi R., Wang T. J., Bradshaw S. J. ApJ 659, L173 (2007a). 153. Taroyan Y., Erd´elyi R., Doyle J. G., Bradshaw S. J. A&A, 462, 331 (2007b). 154. Taroyan Y., & Bradshaw, S. J. A&A, 481, 247 (2008). 155. James, S. P. & Erd´elyi R. A&A, 393, L11 (2002). 156. Kudoh, T., in R. Erd´elyi and C.A. Mendoza-Brice˜ no (eds.) Waves & Oscillations in the Solar Atmosphere: Heating and Magneto-Seismology, IAU Symposium, 247, 197 (2008). 157. Moriyasu S., Kudoh T., Yokoyama T., & Shibata K. ApJ, 601, L107 (2004). 158. Ofman, L., Davila, J.M.: 1997, Astrophys. J., 476, 51. 159. Parker E. N. ApJ, 330, 380 (1988). 160. Patsourakos S., & Klimchuk J. A. ApJ, 628, 1023 (2005). 161. Roussev, I., Doyle, J.G., Galsgaard, K., Erd´elyi, R. Astron. Astrophys., 380, 719 (2001). 162. Sarro, L. M., Erd´elyi, R., Doyle, J. G., P´erez, M. E. A&A, 351, 721 (1999). 163. Teriaca, L., Doyle, J. G., Erd´elyi, R., & Sarro L. M. A&A, 352, L99 (1999b).

CHAPTER 6 VUV SPECTROSCOPY OF SOLAR PLASMA A. MOHAN Department of Applied Physics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

6.1. Introduction VUV spectroscopic diagnostics make use of information contained in the measured intensities and profiles of the spectral lines. Access to images and spectra of hot plasma in UV, EUV, and X- ray regions have provided basic tools to address the fundamental problems in the outer solar atmosphere. Observations of coronal lines have been recognized as important means of investigating the physical properties of coronal plasma and addressing the questions, which are in the forefront of space physics, such as coronal heating and solar wind acceleration. Diagnostics of solar plasma in the temperature range from 105 K to above 106 K carries the radiation signatures of chromosphere - corona transition region and the corona. Many solar ions belonging to various iso-electronic sequences have been systematically presented. Pre-SOHO era, the EUV spectra in the spectral range 170-450 Å observed by Solar Extreme Ultraviolet Research Telescope and Spectrograph (SERTS) have been used, and later, the high resolution ultraviolet observations obtained with Coronal Diagnostic Spectrometer (CDS, 150-600 Å) and Solar Ultraviolet Measurements of Emitted Radiation (SUMER, 465-1610 Å) onboard SOHO are used. A full description of the SUMER spectrograph and its performance are available1-3. 6.2. Atomic Processes For the atomic processes involved in hot (Te > 2×104 K) and low density (Ne < 1013 cm-3) plasma, we assume that the spectral lines are optically thin, which is valid for the outer atmosphere of the Sun and other stars. 109

110

A. Mohan

6.2.1. Emission lines Taking account of atomic excitation mechanisms, the line emissivity (per unit volume per unit time) for an optically thin spectral line is given by the expression:

ξ (λij ) = N j A ji

hc

λij

, ( j >i ), (erg cm-3 sec-1)

(6.1)

where Aji is spontaneous radiative transition probability, h is Planck’s constant, c is velocity of light, λij is wavelength for the transition i – j, and Nj is number density of level j. Thus the atomic physics problem reduces to the calculation of the population density of the upper excited level j. The case of low density optically thin plasmas are treated here. The number density Nj can be further parameterized as:

(

Nj X

+p

)

( ) ( ( )

)

N j X + p N X + p N ( X ) N (H ) = N e cm-3 N X + p N ( X ) N (H ) N e

where X+P is the pth ionization stage of the element; Nj(X+P)/N(X+P) is the population of level j relative to the total N(X+P) number density of the ion X+P and is a function of the electron temperature and density; N(X+P)/N(X) is the ionization ratio of the ion X+P which is predominantly a function of temperature; N(X)/N(H) is the element abundance relative to hydrogen which may vary in different astrophysical plasmas and also in different solar features; N(H)/Ne is the hydrogen abundance relative to electron density which is assumed to be 0.8 for a fully ionized plasma. The flux at the Earth of a spectral line is given by

I (λij ) =

1 ξ (λij )dV erg cm-2 sec-1 sr-1 4πR 2 ∫

where V is the volume of emission and R is Earth-to-object distance. The collisional excitation processes are generally faster than ionization and recombination time scales in low density optically thin plasmas. Therefore, the collisional excitation is dominant over ionization and recombination in producing excited states. Thus, the population density Nj of the upper excited level j must be calculated by solving the statistical equilibrium equations for a number of low-lying levels and taking account of all the important collisional and radiative excitation and de-excitation mechanisms.

VUV Spectroscopy of Solar Plasma

111

6.2.2. Coronal model approximation In coronal model approximation the assumption is made that the population of the upper level of transition j occurs mainly via collision excitation from the ground level g and the radiative decay overwhelms any other depopulation process. The statistical equilibrium equations can be solved as a two-level system for each transition

(

)

N g X + m N eC gje = N j Ajg If Ajg >> Ne Cgje, then the population of the upper level j is negligible in comparison with the ground level g, that means, Ng(X+m)/N(X+m) ≈ 1. Ni (X+P) Ne Cije = Nj Aji, where, Cgje and Cije are the electron collision excitation rate coefficient (cm3 sec-1) remembering that Ni (X+P)/N(X+P) = 1, and with substitutions in Eq. (6.1), we get

ξ (λij ) =

(

)

N X + p N ( X ) N (H ) e hc 2 Cij N N ( X ) N (H ) N e λij e

(6.2)

Thus the line intensity, which is the integration of the line emissivity over the emitting volume, is given by

I (λij ) = ∫ G (Te )N e2 dV

(6.3)

where G(Te) can be calculated from the ionization ratio, element abundance and atomic parameters. The electron density can be crudely deduced assuming that the spectral line is emitted over a homogeneous volume estimated from images in that line. 6.3. Plasma Diagnostics Without the knowledge of electron densities, temperatures and elemental abundances of space plasma, almost nothing can be said regarding the generation and transport of mass, momentum and energy. A fundamental property of hot solar plasmas is their inhomogeneity. The emergent radiances of spectral lines from optically-thin plasmas are determined by integrals along the line of sight through the plasma. Spectroscopic diagnostics of the density and temperature structures using emission-line intensities is described here. 6.3.1. Electron density diagnostics The first question which one might justifiably ask is why so much effort has been put into the development of electron density diagnostics? Obviously the electron

112

A. Mohan

pressure (NeTe) is an important parameter in any theoretical model for the plasma, but why not simply deduce the electron density from the total line emission and an estimate of the emitting volume? This can and has been done in numerous analyses. However, this assumes knowledge of the volume for the emission. If the spatial resolution for the observations was good enough and the emitting material was homogeneously distributed throughout the volume, then the electron density estimate would be reasonable. Solar plasma, however, is characterized by unresolved filamentary structures even with the best spatial resolution observations currently available. The determination of electron density from spectral line ratios from the same ion, makes no assumption about the size of the emitting volume, ionic fractions or the element abundance value, providing a powerful diagnostic for the plasma conditions. Line emissivity ratio of two spectral lines is expressed as,

ξ (λij ) Aji N j (X + p ) λkl = ξ (λkl ) Alk N l (X + p ) λij

(6.4)

The density-sensitive line ratios give the information about the density of the emitting region. 6.3.2. Electron temperature diagnostics The simplest but crudest method of deducing the plasma temperature is to assume ionization equilibrium. Since many ions are formed over the same range of temperature, line ratios can be plotted as a function of temperature of the emitting isothermal plasma. A more accurate determination of electron temperature can be obtained from the intensity ratio of two allowed lines excited from the ground level i but with significantly different excitation energy.

I (λij ) I (λik )

=

 ∆Eik − ∆Eij exp ∆Eik γ ik kTe  ∆Eij γ ij

  

(6.5)

The ratio is sensitive to the change in electron temperature if [(∆Eik - ∆Eij) / kTe] » 1, assuming that the lines are emitted by the same isothermal volume with the same electron density. 6.4. Nitrogen-Like Ions In this section the diagnostic applications of N–like ions are discussed. As shown in Fig. 6.1 the logarithmic temperatures of formation (TM) for the nitrogen-like

VUV Spectroscopy of Solar Plasma

113

coronal ions Al VII, Si VIII, P IX, S X, Ar XII, K XIII, and Ca XIV are 5.8, 5.9. 6.0, 6.1, 6.4, 6.4, and 6.5. respectively4. The lines from these ions can be used for plasma diagnostics in both quiet and active solar coronae. Here the densitydependence of these ions is presented, whose forbidden transitions within the levels in their ground configurations fall within the SUMER bandpass, and compare their emissivities and line ratios with a set of observations taken in the quiet-Sun and active regions. Some of the lines used in the present study are measured for the first time (Al VII 1056.77 Å and both the P IX line pairs). The higher resolution and enhanced sensitivity of the SUMER spectrograph allow for the identification and measurement of weaker lines than in any previous experiment, at positions higher off the solar limb. The Si VIII n = 3 → n = 3 lines observed by SUMER in the 900-1250 Å wavelength range, have been identified by SUMER for the first time in the solar spectrum. New calculations for n = 3 → n = 3 transitions of Si VIII have been performed by Bhatia & Landi5. These new atomic data allow to investigate the diagnostic potential of line ratios within the n = 3 configuration and in conjunction with ground forbidden n = 2 lines also observed by SUMER.

Fig. 6.1 Ion fractions of N-like ions as a function of log T

The availability of such a complete data set for the forbidden N-like lines allows, for the first time, a systematic assessment of the density diagnostic potential of these transitions along the N-like sequence. Version 4.0 of the CHIANTI database6,7 is used to calculate theoretical line ratios. In Version 4, CHIANTI data for the N-like sequence have been entirely renewed, by substituting more recent relativistic distorted wave (DW) calculations from Zhang & Sampson8 for the older DW calculations from Bhatia & Mason9. The

114

A. Mohan

former provide atomic data and DW transition probabilities for the first 15 finestructure levels, while the latter provide data only for the first 13 fine-structure levels. Moreover. R-matrix data have been used for the forbidden transitions within the ground configuration of Si VI and S X10,11. Si VIII and Ar XII levels include additional data for another 57 energy levels taken from other DW calculations5,13. CHIANTI, Version 4, also includes proton excitation rates for Si VIII transitions within the ground configuration, taken from Bhatia & Landi5. The effect of resonances in R-matrix transition probabilities, of radiative cascades from higher excitation levels, and of proton rates and photoexcitation from photospheric radiation may influence level populations of the ground configuration levels and alter intensity ratios from the forbidden transitions that originate from them and are considered here. They have been investigated in detail in order to assess their importance. This work also allows us to make a complete assessment of the quality of CHIANTI atomic data that are necessary to evaluate theoretical line emissivities for the considered lines. We study the effects of photoexcitation, proton collisional excitation, n = 3 levels, and resonances in collision rates on level populations. 6.4.1. Effects of different processes on level populations The populations of the Si VIII ground levels have been calculated with and without proton excitation rates, in order to check the effects of “proton rates” on the results12. The ratios between level populations of the ground 2D and 2P levels calculated in both ways have been plotted as a function of the electron density, for a few sample values of the electron temperature (log Te = 5.6, 5.9, and 6.1, Te in K). This shows that for the excited levels in the ground configuration, the proton rates are of moderate importance. At the temperature where Si VIII is most abundant (log Te = 5.9) the change in relative population is within 5%. However, the proton rate effects are larger at log Te = 6.2, so that for higher temperatures they need to be taken into account. Proton rates are more important at active region densities around l09 cm-3 and become negligible at very high densities. So, it is concluded that where quiet-Sun or moderate active region observations are used, the lack of proton excitation rates in CHIANTI and in the literature is not a source of significant uncertainty. Radiative cascades from higher energy levels can be an important process in the population of the lower lying levels. In order to assess the importance of the “n = 3” levels on level populations for N-like ions, we have compared level populations for the ground configuration calculated in two ways: by including

VUV Spectroscopy of Solar Plasma

115

the n = 2 levels only (corresponding to 15 energy levels), and by including the 2s22p33l (l = s,p,d) levels also (corresponding to 72 energy levels). It is important to note that the reduced atomic model (15 levels) is the one used for decades in plasma diagnostic studies involving N-like ions. The comparisons are made for Si VIII and Ar XII at three temperatures, log Te = 5.6, 5.9 and 6.2 for Si VIII and log Te = 6.1, 6.4 and 6.7 for Ar XII. It is found that the inclusion of n = 3 levels has greatly affected the level populations of the ground configuration and hence the N-like line emissivities. The effects are larger in Ar XII, where cascades from n = 3 levels cause alteration of the level population up to 35% at low densities. The effect is more marked at higher temperatures, while in cooler plasmas it is more limited. The inclusion of n = 3 levels in N-like atomic models provides two benefits: it takes into account the additional population process of cascades from higher levels, which can be non-negligible, and allows predictions of line emissivities for a number of n = 3 → n = 3 transitions that can be observed in the UV spectral range. Level populations are also calculated in the presence of a blackbody radiation field with temperature at 6000 K to see the effects of “photospheric radiations” on the level populations. This radiation field simulates the physical conditions in atmospheres of the Sun and late-type stars. This background radiation can potentially be an important process for radiative excitation within the ground levels. The geometrical dilution factor for the radiation field was taken to be 0.4, typical of the inner solar corona, where the N-like coronal lines are expected to be stronger. The differences in level populations and line emissivities are always smaller than 3%, so that this excitation mechanism has no real importance in the N-like ions and can be neglected. The effects of “resonances in collisional excitation” within the ground levels can be investigated by comparing populations calculated from collision rates derived using the R-matrix approximation14,15 and those obtained using the DW approximation16,17. R-matrix collision rates are available only for Si VIII and S X. A comparison is carried on their level populations. Results obtained using the Bell et al.10 for Si VIII and Bell & Ramshottom11 for S X effective collision strengths for transitions within the ground levels have been compared with populations obtained with the collisional data calculated using the DW approximation by Zhang & Sampson8. The ratios are plotted between level populations calculated using DW data and using R-matrix data, and no proton rates have been used in the calculation12. It is observed that the resonances play an important role in the level population process for the two ions considered.

116

A. Mohan

Effects are relevant for densities smaller than 1012 cm-3 and are larger for Si VIII. The study gives stress for the need of R-matrix data for the N-like sequence and demonstrates that their absence from the literature and from CHIANTI is a source of uncertainties in the calculation of N-like line emissivities. 6.4.2. Result and discussion Fig. 6.2 demonstrates that the (4S3/2 - 2P3/2 ) / (4S3/2 - 2P1/2) line intensity ratio can be effectively used as a tool for density diagnostics in the solar corona in both quiet and active regions, as ratios of ions from Mg VI to Si VIII are density sensitive in the density range typical of the solar corona. With the only exception of Mg VI, these line ratios are suggested for density diagnostics for the first time. Heavier elements (P IX, S X, and Ar XII), on the contrary, present nearly constant ratios up to 1010 cm-3 and are density sensitive only at higher densities. The (4S3/2 – 2D3/2) / (4S3/2 – 2D5/2) ratio, shown in Fig. 6.3 constitutes an even more useful tool for density diagnostics, as its density sensitivity is much more marked. Ratios from Si VIII can be used to infer electron density in low-density plasmas, such as those observed in off-disk coronal regions far from the solar surface. P IX and S X can also be used in coronal holes, quiet-Sun regions, and moderate active regions. Ar XII and K XIII can be excellent density indicators in active regions, while Ca XIV can diagnose even denser plasma. Line intensities and intensity ratios, along with measured electron densities from the SUMER

Fig. 6.2 Line intensity ratios involving the ground forbidden transitions (4S3/2 – 2P3/2) / (4S3/2 – 2P1/2) as a function of the electron density along the N-like sequence

Fig. 6.3 Line intensity ratios involving the ground forbidden transitions (4S3/2 – 2D3/2) / (4S3/2 – 2D5/2) as a function of the electron density along the N-like sequence

117

VUV Spectroscopy of Solar Plasma Table 6.1 Observed Line Intensities and Intensity Ratios Ion

log Te Wavelength Transition Å

Al VII

5.8

Si VIII

5.9

P IX

6.0

SX

6.1

Ar XII

6.4

Si VIII

5.9

P IX

6.0

SX

6.1

Ar XII

6.4

K XIII

6.4

Ca XIV

6.5

1053.84 1056.77 944.38 949.22 853 54 861.10 776.37 787.43 469.14 670.35 1440.49 1445.75 1307.57 1317.65 1196.2 1212.93 1018.89 1054.57 945.83 994.52 880.35 943.70

4

S3/2 – 2P3/2 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2P3/2 4 S3/2 – 2P1/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4 S3/2 – 2D5/2 4 S3/2 – 2D3/2 4

Intensity photons cm-2 s-1 arcsec-2

Ratio

0.089 ± 0.013 0.031 ± 0.005 4.9 ± 0.7 1.7 ± 0.3 0.069 ± 0.010 0.021 ± 0.003 2.2 ± 0.3 1.1 ± 0.2 2.7 ± 0.4 0.44 ± 0.07 0.91 ± 0.14 11.1 ± 1.7 0.027 ± 0.004 0.30 ± 0.05 4.1 ± 0.6 9.5 ± 1.4 3.6 ± 0.5 10.8 ± 1.6 0.98 ± 0.15 1.8 ± 0.3 19.2 ± 2.9 26.8 ± 4.0

2.9 ± 0.6 1.0 2.9 ± 0.7 2.9 ± 0.7 3.3 ± 0.7 1.0 2.0 ± 0.5 1.0 6.1 ± 1.3 1.0 1.0 12.2 ± 2.6 1.0 11.1 ± 2.5 1.0 2.3 ± 0.5 1.0 3.0 ± 0.6 1.0 1.84 ± 0.4 1.0 1.4 ± 0.3

log Ne

Solar Region

≤ 9.9

Quiet Sun Quiet Sun ≤ 9.7 Quiet Sun Quiet Sun 10.7 ± 0.5 Quiet Sun Quiet Sun ≤ 8.0 Quiet Sun Quiet Sun Obs. hig Active region Active region 8.2 ± 0.1 Quiet Sun Quiet Sun 8.95 ± 0.15 Quiet Sun Quiet Sun 8.10 ± 0.15 Quiet Sun Quiet Sun 9.6 ± 0.2 Active region Active region 9.6 ± 0.2 Active region Active region