Solar and Stellar Magnetic Activity

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This timely volume provides the rst comprehensive review and synthesis of thecurrent understanding of the origin, evolution,and effects of magnetic elds in the Sunand other cool stars. Magnetic activity results in a wealth of phenomena including

starspots, nonradiatively heated outer atmospheres, activity cycles, deceleration of rotation rates,andeven, inclosebinaries, stellar cannibalism allof which arecoveredclearly and authoritatively.

This book brings together for the rst time recent results in solar studies, withtheir wealth of observational detail, and stellar studies, which allow the study of howactivity evolves and depends on the mass, age, and chemical composition of stars.The result is an illuminating and comprehensive view of stellar magnetic activity. Ob-servational data are interpreted by using the latest models in convective simulations,dynamo theory, outer-atmospheric heating, stellar winds, and angular momentum

loss. Researchers are provided with a state-of-the-art review of this exciting eld, andthe pedagogical style and introductory material make the book an ideal and welcomeintroduction for graduate students.


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Cambridge astrophysics series

Series editors

Andrew King, Douglas Lin, Stephen Maran, Jim Pringle and Martin WardTitles available in this series

7. Spectroscopy of Astrophysical Plasmasby A. Dalgarno and D. Layzer

10. Quasar Astronomyby D. W. Weedman

17. Molecular Collisions in the Interstellar Mediumby D. Flower

18. Plasma Loops in the Solar Coronaby R. J. Bray, L. E. Cram, C. J. Durrant and R. E. Loughhead

19. Beams and Jets in Astrophysicsedited by P. A. Hughes

20. The Observation and Analysis of Stellar Photospheresby David F. Gray

21. Accretion Power in Astrophysics 2nd Editionby J. Frank, A. R. King and D. J. Raine

22. Gamma-ray Astronomy 2nd Editionby P. V. Ramana Murthy and A. W. Wolfendale

23. The Solar Transition Regionby J. T. Mariska

24. Solar and Stellar Activity Cyclesby Peter R. Wilson

25. 3K: The Cosmic Microwave Background Radiationby R. B. Partridge

26. X-ray Binariesby Walter H. G. Lewin, Jan van Paradijs and Edward P. J. van den Heuvel

27. RR Lyrae Starsby Horace A. Smith

28. Cataclysmic Variable Starsby Brian Warner

29. The Magellanic Cloudsby Bengt E. Westerlund

30. Globular Cluster Systemsby Keith M. Ashman and Stephen E. Zepf

31. Pulsar Astronomy 2nd Editionby Andrew G. Lyne and Francis Graham-Smith

32. Accretion Processes in Star Formationby Lee W. Hartmann

33. The Origin and Evolution of Planetary Nebulaeby Sun Kwok

34. Solar and Stellar Magnetic Activity

by Carolus J. Schrijver and Cornelis Zwaan


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SOLAR AND STELLARMAGNETIC ACTIVITY

C . J . S C H R I J V E RStanford-Lockhead Institute for Space Research, Palo Alto

C . Z WA A N Astronomical Institute, University of Utrecht


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PUBLISHED BY CAMBRIDGE UNIVERSITY PRESS (VIRTUAL PUBLISHING)FOR AND ON BEHALF OF THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGEThe Pitt Building, Trumpington Street, Cambridge CB2 IRP40 West 20th Street, New York, NY 10011-4211, USA477 Williamstown Road, Port Melbourne, VIC 3207, Australia

http://www.cambridge.org

Cambridge University Press 2000This edition Cambridge University Press (Vi t rtual Publishing) 2003

First published in printed format 2000

A catalogue record for the original printed book is availablefrom the British Library and from the Library of CongressOriginal ISBN 0 521 58286 5 hardback

ISBN 0 511 00960 7 virtual (netLibrary Edition)


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Die Sonne t ont nach alter WeiseIn Brudersph aren Wettgesang,Und ihre vorgeschriebene ReiseVollendet sie mit Donnergang.

Ihr Anblick gibt den Engeln St arke,Wenn Keiner Sie ergrunden mag.Die unbegreiich hohen WerkeSind herrlich wie am ersten Tag.

Johann Wolfgang von Goethe


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Contents

Preface page xiii

1 Introduction: solar features and terminology 1

2 Stellar structure 102.1 Global stellar structure 102.2 Convective envelopes: classical concepts 142.3 Radiative transfer and diagnostics 192.4 Stellar classication and evolution 382.5 Convection in stellar envelopes 452.6 Acoustic waves in stars 602.7 Basal radiative losses 652.8 Atmospheric structure not affected by magnetic elds 70

3 Solar differential rotation and meridional ow 733.1 Surface rotation and torsional patterns 743.2 Meridional and other large-scale ows 773.3 Rotation with depth 79

4 Solar magnetic structure 824.1 Magnetohydrodynamics in convective envelopes 834.2 Concentrations of strong magnetic eld 924.3 Magnetohydrostatic models 984.4 Emergence of magnetic eld and convective collapse 1054.5 Omega loops and toroidal ux bundles 1084.6 Weak eld and the magnetic dichotomy 110

5 Solar magnetic congurations 1155.1 Active regions 1155.2 The sequence of magnetoconvective congurations 1265.3 Flux positioning and dynamics on small scales 1265.4 The plage state 1325.5 Heat transfer and magnetic concentrations 137

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x Contents

6 Global properties of the solar magnetic eld 1386.1 The solar activity cycle 1386.2 Large-scale patterns in ux emergence 1436.3 Distribution of surface magnetic eld 1556.4 Removal of magnetic ux from the photosphere 167

7 The solar dynamo 1737.1 Mean-eld dynamo theory 1747.2 Conceptual models of the solar cycle 1787.3 Small-scale magnetic elds 1827.4 Dynamos in deep convective envelopes 184

8 The solar outer atmosphere 1868.1 Topology of the solar outer atmosphere 1868.2 The lament-prominence conguration 1978.3 Transients 1998.4 Radiative and magnetic ux densities 2098.5 Chromospheric modeling 2178.6 Solar coronal structure 2208.7 Coronal holes 2278.8 The chromospherecorona transition region 2298.9 The solar wind and the magnetic brake 231

9 Stellar outer atmospheres 2389.1 Historical sketch of the study of stellar activity 2389.2 Stellar magnetic elds 2389.3 The Mt. Wilson Ca II HK project 2429.4 Relationships between stellar activity diagnostics 2469.5 The power-law nature of stellar uxux

relationships 252

9.6 Stellar coronal structure 25810 Mechanisms of atmospheric heating 266

11 Activity and stellar properties 27711.1 Activity throughout the HR diagram 27711.2 Measures of atmospheric activity 28111.3 Dynamo, rotation rate, and stellar parameters 28311.4 Activity in stars with shallow convective envelopes 291

11.5 Activity in very cool main-sequence stars 29411.6 Magnetic activity in T Tauri objects 29611.7 Long-term variability of stellar activity 299

12 Stellar magnetic phenomena 30312.1 Outer-atmospheric imaging 30312.2 Stellar plages, starspots, and prominences 305


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Contents xi

12.3 The extent of stellar coronae 31012.4 Stellar ares 31212.5 Direct evidence for stellar winds 31412.6 Large-scale patterns in surface activity 31812.7 Stellar differential rotation 319

13 Activity and rotation on evolutionary time scales 32413.1 The evolution of the stellar moment of inertia 32413.2 Observed rotational evolution of stars 32613.3 Magnetic braking and stellar evolution 329

14 Activity in binary stars 33614.1 Tidal interaction and magnetic braking 33614.2 Properties of active binaries 34014.3 Types of particularly active stars and binary systems 342

15 Propositions on stellar dynamos 344

Appendix I: Unit conversions 351 Bibliography 353 Index 375


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Image taken with TRACE in its 171-A passband on 26 July 1998, at 15:50:23 UT of Active Region 8,272 at the southwest limb, rotated over 90. High-arching loops arelled with plasma at 1 MK up to the top. Most of the material is concentrated nearthe lower ends under the inuence of gravity. Hotter 35 MK loops, at which the bulk of the radiative losses from the corona occur, do not show up at this wavelength. Theirexistence can be inferred from the emission from the top of the conductively heatedtransition region, however, where the temperature transits the 1-MK range, as seen inthe low-lying bright patches of moss. A lament-prominence conguration causesextinction of the extreme-ultraviolet radiation.


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Preface

This book is the rst comprehensive review and synthesis of our understanding of theorigin, evolution, and effects of magnetic elds in stars that, like the Sun, have convec-tive envelopes immediately below their photospheres. The resulting magnetic activityincludes a variety of phenomena that include starspots, nonradiatively heated outer at-mospheres, activity cycles, the deceleration of rotation rates, and in close binaries even stellar cannibalism. Our aim is to relate the magnetohydrodynamic processes inthe various domains of stellar atmospheres to processes in the interior. We do so by ex-ploiting the complementarity of solar studies, with their wealth of observational detail,and stellar studies, which allow us to study the evolutionary history of activity and thedependence of activity on fundamental parameters such as stellar mass, age, and chem-ical composition. We focus on observational studies and their immediate interpretation,in which results from theoretical studies and numerical simulations are included. We donot dwell on instrumentation and details in the data analysis, although we do try to bringout the scope and limitations of key observational methods.

This book is intended for astrophysicists who are seeking an introduction to the physicsof magnetic activity of the Sun and of other cool stars, and for students at the graduatelevel. The topics include a variety of specialties, such as radiative transfer, convectivesimulations, dynamo theory, outer-atmospheric heating, stellar winds, and angular mo-

mentum loss, which are all discussed in the context of observational data on the Sun andon cool stars throughout the cool part of the HertzsprungRussell diagram. Althoughwe do assume a graduate level of knowledge of physics, we do not expect specializedknowledge of either solar physics or of stellar physics. Basic notions of astrophysicalterms and processes are introduced, ranging from the elementary fundamentals of ra-diative transfer and of magnetohydrodynamics to stellar evolution theory and dynamotheory.

The study of the magnetic activity of stars remains inspired by the phenomena of solarmagnetic activity. Consequently, we begin in Chapter 1 with a brief introduction of the

main observational features of the Sun. The solar terminology is used throughout thisbook, as it is in stellar astrophysics in general.Chapter 2 summarizes the internal and atmospheric structure of stars with convective

envelopes, as if magnetic elds were absent. It also summarizes standard stellar termi-nology and aspects of stellar evolution as far as needed in the context of this monograph.

The Sun forms the paradigm, touchstone, and source of inspiration for much of stellar astrophysics, particularly in the eld of stellar magnetic activity. Thus, having

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xiv Preface

introduced the basics of nonmagnetic solar and stellar classical astrophysics in therst two chapters, we discuss solar properties in Chapters 38. This monograph is basedon the premise that the phenomena of magnetic activity and outer-atmospheric heatingare governed by processes in the convective envelope below the atmosphere and its in-terface with the atmosphere. Consequently, in the discussion of solar phenomena, muchattention is given to the deepest part of the atmosphere, the photosphere, where the mag-netic structure dominating the outer atmosphere is rooted. There we see the emergenceof magnetic ux, its transport across the photospheric surface, and its ultimate removalfrom the atmosphere. We concentrate on the systematic patterns in the dynamics of mag-netic structure, at the expense of very local phenomena (such as the dynamics in sunspotpenumbrae) or transient phenomena (such as solar ares), however fascinating these are.Page limitations do not permit a discussion of heliospheric physics and solarterrestrialrelationships.

Chapter 3 discusses the solar rotation and large-scale ows in the Sun. Chapters 48cover solar magnetic structure and activity. Chapter 4 deals with fundamental aspects of magnetic structure in the solar envelope, which forms the foundation for our studies of elds in stellar envelopes in general. Chapter 5 discusses time-dependent congurationsin magnetic structure, namely the active regions and the magnetic networks. Chapter 6addresses the global properties of the solar magnetic eld, and Chapter 7 deals with thesolar dynamo and starts the discussion of dynamos in other stars. Chapter 8 discussesthe solar outer atmosphere.

Chapters 9 and 1114 deal with magnetic activity in stars and binary systems. Thisset of chapters is self-contained, although there are many references to the chapters onsolar activity. Chapter 9 discusses observational magnetic-eld parameters and variousradiative activity diagnostics, and their relationships; stellar and solar data are compared.Chapter 11 relates magnetic activity with other stellar properties. Chapter 12 reviewsspatial and temporal patterns in the magnetic structure on stars and Chapter 13 discussesthe dependence of magnetic activity on stellar age through the evolution of the stellarrotation rate. Chapter 14 addresses the magnetic activity of components in binary systemswith tidal interaction, and effects of magnetic activity on the evolution of such interactingbinaries.

Two integrating chapters, 10 and 15, are dedicated to the two great problems in mag-netic activity that still require concerted observational and theoretical studies of the Sunand the stars: the heating of stellar outer atmospheres, and the dynamo action in starswith convective envelopes.

We use Gaussian cgs units because these are (still) commonly used in astrophysics.Relevant conversions between cgs and SI units are given in Appendix I.

We limited the number of references in order not to overwhelm the reader seeking an

introduction to the eld. Consequently, we tried to restrict ourselves to both historical,pioneering papers and recent reviews. In some domains this is not yet possible, so therewe refer to sets of recent research papers.

We would appreciate your comments on and corrections for this text, which we intendto collect and eventually post on a web site. Domain and computer names are, however,


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Preface xv

notoriously unstable. Hence, instead of listing such a URL here, we ask that you sende-mail to kschrijver at solar.stanford.edu with either your remarks or a request to let youknow where corrections, notes, and additions will be posted.

In the process of selecting, describing, and integrating the data and notions presentedin this book, we have greatly proted from lively interactions with many colleagues byreading, correspondence, and discussions, from our student years, through collaborationwith then-Ph.D. students in Utrecht, until the present day. It is impossible to do justice tothese experiences here. We can explicitly thank the colleagues who critically commentedon specic chapters: V. Gaizauskas (Chapters 1, 3, 5, 6, and 8), H. C. Spruit (Chapters 2and 4), R. J. Rutten (Chapter 2), F. Moreno-Insertis (Chapters 4 and 5), J. W. Harvey(Chapter 5), A. M. Title (Chapters 5 and 6), N. R. Sheeley (Chapters 5 and 6), P. Hoyng(Chapters 7 and 15), B. R. Durney (Chapters 7 and 15), G. H. J. van den Oord (Chapters 8and 9), P. Charbonneau (Chapters 8 and 13), J. L. Linsky (Chapter 9), R. B. Noyes(Chapter 11), R. G. M. Rutten (Chapter 11), A. A. van Ballegooijen (Chapter 10), K.G. Strassmeier (Chapter 12), and F. Verbunt (Chapters 2 and 14). These reviewers haveprovided many comments and asked thought-provoking questions, which have greatlyhelped to improve the text. We also thank L. Strous and R. Nightingale for their help inproof reading the manuscript. It should be clear, however, that any remaining errors andomissions are the responsibility of the authors.

The origin of the gures is acknowledged in the captions; special thanks are given toT. E. Berger, L. Golub and K. L. Harvey for their efforts in providing some key gures.C. Zwaan thanks E. Landr e and S. J. Hogeveen for their help with gure production andwith LaTeX problems.

Kees Zwaan died of cancer on 16 June 1999, shortly after the manuscript of this book had been nalized. Despite his illness in the nal year of writing this book, he continuedto work on this topic that was so dear to him. Kees research initially focused on the Sun,but he reached out towards the stars already in 1977. During the past two decades heinvestigated solar as well as stellar magnetic activity, by exploiting the complementarityof the two elds. His interests ranged from sunspot models to stellar dynamos, and fromintrinsically weak magnetic elds in the solar photosphere to the merging of binary sys-tems caused by magnetic braking. His very careful observations, analyses, solar studies,and extrapolations of solar phenomena to stars have greatly advanced our understandingof the sun and of other cool stars: he was directly involved in the development of theux-tube model for the solar magnetic eld, he stimulated discussions of ux storage andemergence in a boundary-layer dynamo, lead the study of sunspot nests, and stimulatedthe study of stellar chromospheric activity. And Kees always loved to teach. That wasone of the main reasons for him to undertake the writing of this book.


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xvi Preface

Kees Zwaan (24 July 192816 June 1999)


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1

Introduction: solar features

and terminology

The Sun serves as the source of inspiration and the touchstone in the study of stellarmagnetic activity. The terminology developed in observational solar physics is also usedin stellar studies of magnetic activity. Consequently, this rst chapter provides a brief illustrated glossary of nonmagnetic and magnetic features, as they are visible on theSun in various parts of the electromagnetic spectrum. For more illustrations and detaileddescriptions, we refer to Bruzek and Durrant (1977), Foukal (1990), Golub and Pasachoff (1997), and Zirin (1988).

The photosphere is the deepest layer in the solar atmosphere that is visible in whitelight and in continuum windows in the visible spectrum. Conspicuous features of the photosphere are the limb darkening (Fig. 1.1a ) and the granulation (Fig. 2.12), atime-dependent pattern of bright granules surrounded by darker intergranular lanes .These nonmagnetic phenomena are discussed in Sections 2.3.1 and 2.5.

The magnetic structure that stands out in the photosphere comprises dark sunspotsand bright faculae (Figs. 1.1a and 1.2b). A large sunspot consists of a particularly dark umbra , which is (maybeonly partly) surrounded by a lessdark penumbra . Small sunspotswithout a penumbral structure are called pores . Photospheric faculae are visible in whitelight as brighter specks close to the limb.

The chromosphere is the intricately structured layer on top of the photosphere; it is

transparent in the optical continuum spectrum, but it is optically thick in strong spectrallines. It is seen as a brilliantly purplish-red crescent during the rst and the last fewseconds of a total solar eclipse, when the moon just covers the photosphere. Its color isdominated by the hydrogen Balmer spectrum in emission. Spicules are rapidly changing,spikelike structures in the chromosphere observed beyond the limb (Fig. 4.7 in Bruzek and Durrant, 1977, or Fig. 9-1 in Foukal, 1990).

Chromospheric structure can always be seen, even against the solar disk, by meansof monochromatic lters operating in the core of a strong spectral line in the visiblespectrum or in a continuum or line window in the ultraviolet (see Figs. 1.1 b, 1.1c, 1.2c

and 1.3). In particular, ltergrams recorded in the red Balmer line H display a wealthof structure (Fig. 1.3). Mottle is the general term for a (relatively bright or dark) detailin such a monochromatic image. A strongly elongated mottle is usually called a bril .

The photospheric granulation is a convective phenomenon; most other features ob-served in the photosphere and chromosphere are magnetic in nature. Sunspots, pores,and faculae are threaded by strong magnetic elds, as appears by comparing the magne-tograms in Figs. 1.1 and 1.2 to other panels in those gures. On top of the photospheric

1


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Introduction: solar features and terminology 3

1.1 b

1.1 c

Figs. 1.2b and 1.2c. All active regions, except the smallest, contain (a group of) sunspotsor pores during the rst part of their evolution.

Active regions with sunspots are exclusively found in the sunspot belts on either sideof the solar equator, up to latitudes of 35; the panels in Fig. 1.1 show several largeactive regions. In many young active regions, the two magnetic polarities are found ina nearly EW bipolar arrangement, as indicated by the magnetogram of Fig. 1.1 e, andbetter in the orientations of the sunspot groups in Fig. 1.1 a . Note that on the northernsolar hemishere in Fig. 1.1 e the western parts of the active regions tend to be of negative


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6 Introduction: solar features and terminology

1.2 c

1.2 d


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When a large active region decays, usually rst the sunspots disappear, and then theplages crumble away to form enhanced network. One or two stretches of enhancednetwork may survive the active region as a readily recognizable bipolar conguration.Stretches of enhanced network originating from several active regions may combine intoone large strip consisting of patches of largely onedominant polarity, a so-called unipolarregion. On the southern hemisphere of Fig. 1.1 e, one such strip of enhanced network of positive (white) polarity stands out. Enhanced network is a conspicuous congurationon the solar disk when activity is high during the sunspot cycle.

Outside active regions and enhanced network, we nd a quiet network that is bestvisible as a loose network of small, bright mottles in Ca II K ltergrams and in the UVcontinuum. Surrounding areas of enhanced network and plage in the active complex, thequiet network is indicated by tiny, bright mottles; see Fig. 1.2 c. Quiet network is alsovisible on high-resolution magnetograms as irregular distributions of tiny patches of magnetic ux of mixed polarities. This mixed-polarity quiet network is the congurationthat covers the solar disk everywhere outside active regions and their enhanced-network remnants; during years of minimum solar activity most of the solar disk is dusted withit. The areas between the network patches are virtually free of strong magnetic eld inthe photosphere; these areas are often referred to as internetwork cells. Note that in largeparts of the quiet network, the patches are so widely scattered that a system of cellscannot be drawn unambiguously.

The distinctions between plages, enhanced network, and quiet network are not sharp.Sometimes the term plagette is used to indicate a relatively large network patch or acluster of faculae that is too small to be called plage.

Bright chromospheric mottles in the quiet network are usually smaller than faculaein active regions and mottles in enhanced network, but otherwise they appear similar.Historically, the term facula has been reserved for bright mottles within active regions;we call the bright mottles outside active regions network patches . (We prefer the termpatch over point or element, because at the highest angular resolution these patches andfaculae show a ne structure.)

The comparison between the magnetograms and the photospheric and chromospheric

images in Figs. 1.1 and 1.1 shows that near the center of the solar disk there is anunequivocal relation between sitesof strong,verticalmagneticeld andsunspots, faculae,and network patches. As a consequence, the adjectives magnetic and chromospheric areused interchangeably in combination with faculae, plages, and network.

In mostof the magnetic features, the magnetic eld isnearlyverticalat the photosphericlevel, which is one of the reasons for the sharp drop in the line-of-sight magnetic signalin plages and network toward the solar limb in Fig. 1.1 e. Markedly inclined photosphericelds are found within tight bipoles and in sunspot penumbrae.

Filtergrams obtained in the core of H are much more complex than those in the

Ca II H and K lines (see Fig. 1.3, and Zirins 1988 book, which is full of them). Inaddition to plages and plagettes consisting of bright mottles, they show a profusion of elongated dark brils. These brils appear to be directed along inclined magnetic eldlines in the upper chromosphere (Section 8.1); they are rooted in the edges of plages andin the network patches. The brils stand out particularly well in ltergrams obtained at

0.5 A from the line core (see Fig. 1.3 b).


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100"

F1

EN+ F2

pl

P+ S

P

72,000 km

a

100"

FC

N

WS

72,000 km

b

Fig. 1.3. Nearly simultaneous H ltergrams of active complex McMath 14,726 on 18 April1977, observed in the line core (panel a) and at = +0.65 A in the red wing (panel b).The letter symbols indicate the following: S, sunspot; P, plage; pl, plagette; F, lament; FC,lament channel; EN, enhanced network cell. Signs are appended to indicate the magneticpolarities. Fibrils are prominent in both panels. Exceptionally long and well-ordered brilsare found in the northwestern quadrants. Several features are discussed in Sections 8.1 and8.2. The chirality of lament F1 is sinistral (gure from the archive of the Ottawa River Solar

Observatory, National Research Council of Canada, courtesy of V. Gaizauskas.)


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The longest dark structures visible in the core of the H line are the laments (Figs. 1.1cand 1.3). Many laments are found at borders of active regions and within activecomplexes, but there are also laments outside the activity belts, at higher latitudes. Mostlaments differ from brils by their length and often also by their detailed structure.Small laments can be distinguished from brils by their reduced contrast at distances

| | > 0.5 A from the line core. Large laments are visible outside the solar limb as prominences that are bright against a dark background.The corona is the outermost part of the Sun, which is seen during a total eclipse as a

pearly white, nely structured halo, locally extending to several solar radii beyond thephotospheric limb; see Figs. 8.4 and 8.11, Fig. 1.2 in Golub and Pasachoff (1997), orFig. 9-10 in Foukal (1990). Thecoronal plasma is extremely hot ( T 11065106 K)and tenuous. The radiation of the white-light corona consists of photospheric light,scattered by electrons in the corona and by interplanetary dust particles; the brightnessof the inner corona is only 106 of the photospheric brightness. The thermal radiationof the corona is observed in soft X-rays, in spectral lines in the ultraviolet and opticalspectrum,and in radio waves. Thecorona is optically thin throughout the electromagneticspectrum, except in radio waves and a few resonance lines in the extreme ultraviolet andin soft X-rays.

The coronal structure in front of the photospheric disk can be observed from satellitesin the EUV and in X-rays; see Figs. 1.1 d and 1.2d . In these wavelength bands, the coronalplasma, however optically thin, outshines the much cooler underlying photosphere. Thefeatures depend on the magnetic eld in the underlying photosphere. The corona isparticularly bright in coronal condensations immediately above all active regions inthe photosphere and chromosphere. Coronal loops trace magnetic eld lines connectingopposite polarities in the photosphere. Note that in Fig. 1.1 d therearealso long, somewhatfainter, loops that connect magnetic poles in different active regions. The nest coronalstructure is displayed in Fig. 1.2 d , where the passband reveals radiation from bottomparts of loops with T < 1106 K, without contamination by radiation from hotter loopswith T > 2 106 K.Coronal holes stand out as regions that emit very little radiation; these have beenidentied as regions where the magnetic eld is open to interstellar space. Usually largecoronal holes are found over the polar caps; occasionally smaller coronal holes areobserved at low latitudes.


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2

Stellar structure

This chapter deals with the aspects of stellar structure and evolution that are thought tobe independent of the presence of magnetic elds. In this classical approach to globalstellar structure, the effects of stellar rotation are also ignored. Rather than summarizethe theory of stellar structure, we concentrate on features that turn out to be important inunderstanding atmospheric structure and magnetic activity in Sun-like stars, that is, starswith convective envelopes. For more comprehensive introductions to stellar structure werefer to Chapter 4 in Unsold and Baschek (1991), and to B ohm-Vitense (1989a, 1989b,1989c).

We present a brief synopsis of the transfer of electromagnetic radiation in order toindicate its role in the structuring of stellar atmospheres and to sketch the possibilitiesand limitations of spectroscopic diagnostics, including Zeeman diagnostics of magneticelds.

In addition, in this chapter we summarize the convective and purely hydrodynamicwave processes in stellar envelopes and atmospheres. In this framework, we also dis-cuss the basal energy deposition in outer atmospheres that is independent of the strongmagnetic elds.

2.1 Global stellar structure

2.1.1 Stellar time scalesStars are held together by gravity, which is balanced by gas pressure. Their

quasi-steady state follows from the comparison of some characteristic time scales.The time scale of free fall t ff is the time scale for stellar collapse if there were no

pressure gradients opposing gravity. Then the only acceleration is by gravity: d 2r / dt 2 =G M / r 2, where r is the radial distance to the stellar center, G is the gravitationalconstant, and M and R are the stellar mass and radius, respectively. This leads to theorder-of-magnitude estimate:

t ff R3

G M

1/ 2

=1,600 M M 1/ 2

R R3/ 2

(s), (2.1)

where M and R are the solar mass and radius, respectively.For a star virtually in hydrostatic equilibrium, local departures from equilibrium are

restored at the speed of sound:

cs =[( p)/ ]1/ 2, (2.2)

10


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2.1 Global stellar structure 11

where is the mass density, p is the gas pressure, and c p / cV is the ratio of thespecic heats at constant pressure and constant volume. Using the order-of-magnitudeestimate from Eq. (2.7) for hydrostatic equilibrium, p/ R G M / R2, we nd thehydrodynamic time scale t hy:

t hy Rcs =

R2 p

1/ 2

R3

G M

1/ 2

= 1/ 2 t ff , (2.3)

which is of the same order of magnitude as the free-fall time scale t ff .The KelvinHelmholtz time scale t KH estimates how long a star could radiate if there

were no nuclear reactions but the star would emit all of its present total potential gravi-tational energy E g at its present luminosity L:

t KH | E g| L G M 2

RL 3 107 M M 2

R R 1

L L 1

(yr). (2.4)

From the virial theorem (see Sections 2.6.4 and 4.12.4 in Uns old and Baschek, 1991or Section 2.3 in Bohm-Vitense, 1989c) applied to a star in hydrostatic equilibrium, itfollows that the internal (thermal) energy E i is half | E g|. Hence the KelvinHelmholtztime scale is of the order of the thermal time scale , which a star would need to radiateall its internal energy at the rate of its given luminosity L.

The nuclear time scale t nu , the time that a star can radiate by a specic nuclear fusion

process, is estimated from stellar evolution calculations. The time scale for hydrogenfusion is found to be

t nu 1 1010 M M

L L

1(yr). (2.5)

The comparison of the stellar time scales shows

t ff t hy t KH t nu . (2.6)Consequently, a star is in both mechanical (that is, hydrostatic) and thermal equilibriumduring nearly all of its evolutionary phases.

2.1.2 Shell model for Sunlike starsIn classical theory, the stellar structure is approximated by a set of spherical

shells. The stellar interior of the Sun and Sunlike stars consists of the central part, theradiative interior, and the convective envelope.

The central part is the section where nuclear fusion generates the energy ux thateventually leaves the stellar atmosphere. In the Sun and other main-sequence stars,

hydrogen is fused into helium in the spherical core , on the time scale t nu [Eq. (2.5)].In evolved stars, the central part consists of a core, in which the hydrogen supply isexhausted, which is surrounded by one or more shells, which may be dead (and hencein a state of gravitational contraction), or which may be in a process of nuclear fusion.

In the Sun and in all main-sequence stars, except the coolest, the core is surroundedby the radiative interior , which transmits the energy ux generated in the core as elec-tromagnetic radiation.


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12 Stellar structure

The convective envelope (often called the convection zone) is the shell in which theopacity is so high that the energy ux is not transmitted as electromagnetic radiation;there virtually the entire energy ux is carried by convection.

The atmosphere is dened as the part of the star from which photons can escapedirectly into interstellar space. In the Sun and other cool stars, the atmosphere is sit-uated immediately on top of the convective envelope. It consists of the following do-mains, which are often conveniently, but incorrectly, pictured as a succession of sphericalshells:

1. The photosphere is the layer from which the bulk of the stellar electromagneticradiation leaves the star. This layer has an optical thickness < 1 in the near-ultraviolet, visible, and near-infrared spectral continua, but it is optically thick (Section 2.3.1) in all but the weakest spectral lines.

2. The chromosphere is optically thin in the near-ultraviolet, visible, and near-infrared continua, but it is optically thick in strong spectral lines. The chromo-sphere can be glimpsed during the rst and the last few seconds of a total solareclipse as a crescent with the purplish-red color of the Balmer spectrum.

3. The corona is optically very thin over the entire electromagnetic spectrum ex-cept for the radio waves and a few spectral lines (see Section 8.6). Stellar andsolar coronae can be observed only if heated by some nonradiative means, be-cause they are transparent to photospheric radiation. The chromosphere and thecorona differ enormously both in density and in temperature; the existence of an intermediate domain called the transition region , is indicated by emissionsin specic spectral lines in the (extreme) ultraviolet.

In the present Sun, with its photospheric radius R of 700 Mm, the core has a radiusof 100 Mm. The radiative interior extends to 500 Mm from the solar center, and theconvective envelope extends from 500 to 700 Mm. The photosphere has a thickness of no more than a few hundred kilometers; the chromosphere extends over nearly 10 Mm.The appearance of the corona is of variable extent and complex shape; see Figs. 1.1 d and 1.2d . The corona merges with the interplanetary medium.

2.1.3 Stellar interiors: basic equations and modelsThe models for the quasi-static stellar interiors are determined by four rst-order

differential equations.The force balance is described by hydrostatic equilibrium :

d p(r )dr = g(r ) (r ) =

G m (r ) (r )r 2

, (2.7)

where p(r ) is the gas pressure, (r ) is the mass density, and g(r ) is the acceleration bygravity, all at a radial distance r from the stellar center; m(r ) is the mass contained in asphere of radius r . The contribution of the radiation pressure prad =4 T 4/ c to the totalpressure balance is negligible in Sun-like stars.

In the conditions covered in this book, the perfect gas law is applicable:

= pR T

, (2.8)


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where c p is the specic heat at constant pressure and cV is the specic heat at constantvolume. Section 2.2 discusses an approximate procedure to determine the mean temper-ature gradient in convective envelopes. There it is shown that for the largest part of such a zone the adiabatic gradient

ad is a good approximation, with the exception of

the very top layer immediately below the photosphere.In order to compute p(r ), T (r ), L(r ), m(r ) and the other r -dependent parameters that

determine the time-dependent chemical composition, the dependences of , , , , R,and on p, T and the chemical composition are needed.

The set of equations are to be completed by boundary conditions. There are twoconditions at the stellar center:

L(0) =0 and m(0) =0. (2.16)

The two boundary conditions at the outer edge r = R concern T and p. For the temper-ature we haveT ( R) =T eff

L( R)4 R2

1/ 4

, (2.17)

where T eff is the stellar effective temperature, sometimescalled the surface temperature.The corresponding gas pressure p( R), at the same photospheric plane where T = T eff ,is best obtained from a model atmosphere for the star of interest. The location of thisstellar surface is discussed in Section 2.3.1.

Although the parameters of the stellar interiors are well dened by the set of equa-tions, their solution requires sophisticated numerical methods, which are discussed inspecialized texts; see, for example, in Chapter 11 of Kippenhahn and Weigert (1990).

The models indicate a strong concentration of mass toward the stellar center. In thepresent Sun, nearly half of the mass is contained within r = 0.25 R . The convectiveenvelope contains more than 60% of the solar volume but less than 2% of the solar mass.2.2 Convective envelopes: classical concepts

2.2.1 Schwarzschilds criterion for convective instabilityWhen a blob of gas is lifted from its original position, it may be heavier or

lighter than the gas in its new environment. In the rst case, the blob will move back toits original location, and the medium is called convectively stable. In the second case, itcontinues to rise, and the gas is said to be convectively unstable . The criterion for this(in)stability was derived by K. Schwarzschild in 1906.

Consider a blob lifted by some disturbance over a height r . Let us assume that initiallythe thermodynamic conditions within the blob, indexed in, were equal to those outside,for the mass density: in(r )

= (r ). It is further assumed that the rise is sufciently fast, so

that the blob behaves adiabatically (without heat exchange), yet so slow that the internalpressure pin adjusts to balance the ambient pressure p during the rise. These assumptionsare plausible in stellar envelopes because of the high opacity and the short travel timefor sound waves across the blob, respectively. The blob continues to rise if the internaldensity remains smaller than the external density, so the condition for instability is

in(r +r ) (r +r ) = r ddr ad

ddr

< 0, (2.18)


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2.2 Convective envelopes: classical concepts 15

where (d/ dr )ad is the density gradient under adiabatic conditions. With substitutionof the perfect gas law [Eq. (2.8)] and use of pin(r ) = p(r ), the instability condition inEq. (2.18) becomes

dT dr ad T ddr ad > dT dr T ddr . (2.19)The mean molecular weight is a function of p and T because it depends on the degreeof ionization of the abundant elements hydrogen and helium. If the adjustment of theionization equilibrium in therising blob is instantaneous, the function ( p, T )isthesamefor the plasma inside and outside the blob. Hence, the gradients d / dr and (d/ dr )ad inEq. (2.19),

d/ dr

=(/ p)T

d p/ dr

+(/ T ) p

dT / dr and

(d/ dr )ad =(/ p)T (d p/ dr )ad +(/ T ) p (dT / dr )ad ,differ only in the second term containing the gradient d T / d r , because from pressureequilibrium it follows that (d p/ dr )ad = d p/ dr . Hence the Schwarzschild criterion for convective instability is:

dT dr

>dT dr ad

. (2.20)

In the case

|dT / dr

| ad . (2.21)Stellar models computed under the assumption of radiative equilibrium (indicated by thesubscript RE) are consistent in layers where the computed temperature gradient

RE

turns out to be smaller than the adiabatic gradient ad , and there wehave = RE < ad .If stability does not apply, we have ad in < < RE , (2.22)

where the index in refers to the interior of a moving gas blob and quantities without anindex refer to a horizontal average over the ambient medium. The rst inequality allowsfor a departure from adiabatic conditions in the moving blob by radiative exchangewith its surroundings; this difference is very small except in the very top layers of the

convective envelope. The second inequality accounts for the driving of the motion of theblob relative to its environment by the convective instability. The third inequality standsfor a relatively large difference; see Fig. 2.1.

The adiabatic temperature gradient ad [Eq. (2.15)] dependson the degreeof ionizationin the plasma. In a strictly monoatomic gas that is either completely neutral or completelyionized, =5/ 3, and hence ad =2/ 5. It is easy to see that in a partly ionized gas ad


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Fig. 2.1. Run of parameters through the solar convection zone, as a function of depth z belowthe photosphere (from tables in Spruit, 1977b).

is not ionized at all or that remains completely ionized. The quantities and ad dependparticularly on the ionization equilibria in the most abundant elements, H and He; forthe formulas, see Chapter 14 in Kippenhahn and Weigert (1990).

In the solar envelope, ad is found to drop to nearly one quarter of the monoatomicvalue of 2/ 5 (see Fig. 2.1), but this dip is restricted to a shallow layer immediately belowthe photosphere. For the convective instability, the extremely large value of the radiative-equilibrium gradient RE is much more important: in the solar convective envelope, REexceeds the comparison value of 2 / 5 by several orders of magnitude; see Fig. 2.1. Thelarge value of

RE is caused primarily by the large opacity R, which is explained in

Section 2.3.1.Note that stellar atmospheres, in which the characteristic optical depth (based on the

mean Rosseland opacity) is smaller that unity, are stable against convection because fromthere photons can escape readily. Indeed, classical models for stellar atmospheres showthat = RE < ad .In O- and B-type stellar envelopes there is no convective instability. There the ion-ization of H and He is so complete that R remains small so that everywhere = RE < ad = 2/ 5. Nor is convective instability important in the envelopes of A-typemain-sequence stars.

Cooler stars of spectral types F, G, K, and M are convectively unstable in the envelopesbelow their photospheres. The term cool stars is often used to indicate stars that haveconvective envelopes.

2.2.2 The mixing-length approximationIn layers where the Schwarzschild criterion Eq. (2.21) indicates instability, con-

vective motions develop. There are two commonly used approaches to the modeling


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2.2 Convective envelopes: classical concepts 17

of convection extending over many pressure scale heights. In hydrodynamic modeling,numerical simulations are used, as discussed in Section 2.5 this ab initio method islimited by computational resources. The alternative is the classical mixing-length for-malismwhich attempts to derive some properties of convective envelopes from a stronglysimplied description of the convective processes. We rst discuss some results obtainedfrom mixing-length modeling because nearly all quantitative models of stellar convec-tive envelopes are based on this approach. Moreover, many current ideas about processesin convective envelopes are still dressed in terms of the mixing-length theory or somesimilar theory of turbulent convection.

The mixing-length (ML) concept was developed by physicists, including L. Prandtl,between 1915 and 1930. The mixing length ML is introduced as the distance over whicha convecting blob can travel before it disintegrates into smaller blobs and so exchangesits excess heat, which can be positive or negative, depending on whether the blob ismoving up or down. In some applications, this ML is taken to equal the distance to thenearest boundary of the convective layer, butgenerally some smaller characteristic lengthscale of the medium is used. After 1930, the concept was introduced in astrophysics;many applications have followed the formalism put forward by Vitense (1953) see alsoBohm-Vitense (1958). In such applications, the mixing length is assumed to be a localquantity related to the pressure scale height [Eq. (2.9)]

ML ML H p , (2.23)where

ML is introduced as an adjustable parameter. From the extremely small viscosity

of the gas in stellar envelopes it was inferred that the convective ow is very turbulent;hence ML was expected to be small, of the order of unity.

Only the mean value of the upward and downward components in the velocity eld areconsidered. Other quantities are also represented by their mean values that vary only withdistance r from the stellar center. The chief aim is to determine the mean temperaturegradient (r ) in order to complete the model of the internal thermodynamic structure of the star.

A basic equation for the stellar envelope is the continuity equation for the energy ux:

L4 r 2 F (r ) =F R +F C, (2.24)

where L is the stellar luminosity. For the radiative energy ux density F R(r ) we have

F R = 16 T 3

3RdT dr =

16 T 4

3R H p (2.25)[see Eq. (2.14)]. The mean gradient adjusts itself so that whatever part of the total uxF cannot be carried as radiative energy ux F R is carried by convection. The convectiveenergy ux F C is estimated from assumed mean properties of rising and sinking blobs,depending on the local conditions; for reviews, we refer the reader to Section 6.2 inStix (1989) or to Chapter 6 in B ohm-Vitense (1989c). Eventually a relation is foundthat expresses the mean gradient in terms of local quantities. This relation can benumerically solved in the framework of the equations determining stellar structure givenin Section 2.1.

The principal adjustable parameter is the mixing length; in the classical, strictly lo-cal theory, values of 1 < ML < 2 have been preferred, because in solar models these


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parameter values are found to correctly reproduce the measured solar radius. This as-sumption of a strictly local mixing length ML is not consistent close to the boundariesof the convection zone; hence models invoking the distance z to the nearest boundaryhave been constructed, for instance, with ML

= min( z

+ z0, ML H p), where the depth

z0 allows for convective overshoot (explained below).Mixing-length models provide a rst-order estimate for the superadiabaticy ad ,which for the Sun turns out to be extremely small for all depths except in the thin, truly

superadiabatic top layer at depths z < 1,000 km. The extremely small values of adin the bulk of the convective envelope follow from the relatively high mass density :near the bottom, the convective heat ux can be transported at a very small temperaturecontrast T / T 106. Even though the computed ad is not accurate, the mean ad is fairly well dened; hence so is the temperature run T (r ). Because of hydro-staticequilibrium, thevalues of theother local parameters that depend on thermodynamic

quantities are also reasonably well established.Figure 2.1 shows the proles of several parameters as computed for the convective

envelopeof the Sun. Thisgure shows that the extent of the convection zone isdeterminedby the radiative-equilibrium gradient RE . The depth dependence of RE(r ) reectsthat of the Rosseland mean opacity R(r ). The mean-free path of photons R dropsfrom approximately 10 km near the surface (where R 1) to values < 1 mm for z > 10,000 km.

Dynamic quantities, such as the mean convective velocity vML , depend critically onthe crude assumptions of ML theory. Nonetheless, the mean convective velocity vML hasbeen used to estimate the kinetic energy density in the convection by

E kin 12

v 2ML (2.26)

(see Fig. 2.1). Despite the uncertainty in vML , there is little doubt that even in the top of the convection zone the mean kinetic energy in the convective ow E kin is smaller thanthe internal thermal energy density E th:

E th

=

32

p

+[nH+

H

+nHe+

He

+nHe++

( He

+ He+)], (2.27)

where ni and i represent the number density and the ionization energy of particles of species i. (Note that the ionization energy contributes appreciably to the total thermalenergy E th in the layers where H and He are partially ionized.) In the deep layers of the convection zone, E kin is many orders of magnitude smaller than E th. The statement E kin E th is equivalent to vML cs, where cs is the sound velocity [Eq. (2.2)].Within the ML approximation, there is an estimate for the characteristic velocity v of the turbulent convection in terms of the stellar energy ux density. From ML formulasgiven in Chapter 6 in Bohm-Vitense (1989c), one nds that the convective ux density

can be estimated byF c

5 ML

(r ) v3(r ). (2.28)

Throughout the convective envelope, convection carries the entire energy ux; hence

5 ML

(r ) v3(r ) T 4eff R2

r 2 . (2.29)


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2.3 Radiative transfer and diagnostics 19

Application to the top of the solar convection zone, assuming ML = 1.6, leads to v 3.9 km/s, which is a substantial fraction of the local sound velocity cs =8.2 km/s. Nearthe bottom of the convection zone, the characteristic velocity is approximately 60 m/s:there the energy ux is carried by small convectivevelocities (and very small temperaturecontrasts) in large-scale ows.

In discussions of large-scale ows in convective envelopes, often a convective turnovertime t c of convective eddies near the bottom of the convection zone is introduced. Inturbulent convection, this time scale is dened as the typical length scale of the convectiveeddies, divided by the typical velocity in the eddies; hence in the ML approximation,

t c ML

vML. (2.30)

Near the bottom of the solar convection zone, one nds values for t c between approxi-mately one week and one month.Although the Schwarzschild criterion predicts sharp boundaries between the convec-

tive zone and the adjacent stably stratied layers, convective overshoot is to be expectedbecause of the inertia of the convecting matter. Overshooting blobs suddenly nd them-selves in a subadiabatic domain, hence the force acting on the blob is reversed: the blobis decelerated. Section 2.5 discusses observational data on the convective overshoot intothe atmosphere in connection with the convective dynamics and the radiative exchange.

Theoretical studies of the overshoot layer at the base of the convection zone that

consider the nonlocal effects of convective eddies on the boundary layer (see Van Bal-legooijen, 1982a; Skaley and Stix, 1991) indicate that this layer is shallow. In the solarcase, it is only 104 km thick, that is, no more than 20% of the local pressure scale height.The temperature gradient is found to be only slightly subadiabatic: ad 106.Basic assumptions in the ML approach are not conrmed by the observed solar granu-lation andnumerical simulations of thestructureof the top layersof convectiveenvelopes.In Section 2.5 we discuss the resulting change in the picture of the patterns in stellarconvective envelopes.

2.3 Radiative transfer and diagnostics

2.3.1 Radiative transfer and atmospheric structureFor the description of a radiation eld, the monochromatic specic intensity

I ( x , y, z, , , t ) is the fundamental parameter. It is dened as the proportionality factorthat quanties the energy d E owing during d t through an area d A within a solid angled about the direction l(, ) within a frequency interval d around (see Fig. 2.2):

d E

= I ( x , y, z, , , t ) d A cos d d dt . (2.31)

I is measured in units [erg s 1 cm2 Hz1 ster1]. Often I is just called the monochro-matic intensity, without the qualier specic.

The intensity I (s ) along a pencil of light remains constant unless there is emission orextinction along the path s :

d I (s) = I (s +ds ) I (s ) = j (s ) ds (s ) (s ) I (s ) ds , (2.32)


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a b

Fig. 2.2. Panel a : the denition of the specic intensity and related quantities. Panel b: theconcept of a planeparallel atmosphere.

where j (s ) [erg s1 cm3 Hz1 ster1] is the monochromatic emission coefcient, (s )[cm2 g1] is the monochromatic extinction coefcient, and (s ) [g cm3] is the massdensity.

The optical thickness of a layer of geometrical thickness D follows from the denitionof the dimensionless monochromatic optical path length d across a layer of thicknessds , d

(s) (s ) ds :

( D) = D0 (s ) (s ) ds , (2.33)where the tilde indicates that the optical thickness is measured along the propagationdirection of the beam.

If there is no emission within the layer, it transmits a fraction exp[ ( D)] of theincident intensity. The layer is called optically thick when ( D) 1 and optically thinwhen ( D) 1. The local mean-free path for photons is

(s ) =1

(s ) (s ). (2.34)

With the use of optical path length d and the introduction of the source function S ,dened as the emission per optical path length is

S (s ) j (s )

(s ) (s ), (2.35)

and the transfer equation (2.32) takes the formd I d = S I . (2.36)

The source function has the same dimension as the specic intensity. If local thermo-dynamic equilibrium (LTE; discussed in the following paragraphs) applies, the sourcefunction is given by the local Planck function B (T ). In the case of pure scattering that


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is both coherent and isotropic, the source function equals the angle-averaged monochro-matic intensity :

J

I d/ (4 ). (2.37)

The Planck function is

B (T ) =2h 3

c21

eh/ kT 1, (2.38)

where h is the Planck constant. For h/ kT 1, the Planck function simplies to theRayleighJeans function,

B (T )2 2kT

c2 , (2.39)

which describes the source function in the far-infrared and radio domain of the spectrum.In radio astronomy, intensity is usually expressed as (brightness) temperature, which isa linear measure for radiative emission.

The StefanBoltzmann function is the total LTE source function, integrated over theentire spectrum B(T )

0 B (T ) d :

B(T ) = T 4, (2.40)where is the StefanBoltzmann constant.

Consider a beam of intensity I (0) passing through a homogeneous layer characterizedby an optical thickness ( D) and a source function S . The emergent intensity followsfrom Eq. (2.36):

I ( D)

= I (0)e ( D)

+ ( D)

0

S e[ ( D)t ] dt

= I (0)e ( D) +S 1 e

( D) . (2.41)

In the optically thin case, the result is

I ( D) = I (0) +[S I (0)] ( D) : (2.42)the emergent intensity I ( D) is larger or smaller than the incident I (0), depending onwhether S is larger or smaller than I (0). The emission or extinction of an optically thin

layer is proportional to its optical thickness.When the layer is optically thick, Eq. (2.41) leads to

I ( D) = S . (2.43)Themathematicalformulation of radiative transfer in optically thick mediasuch as stel-

lar atmospheres is greatly simplied if the structure of the atmosphere is approximated by


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a planeparallel horizontal stratication in which the thermodynamicquantities vary onlywith the height z perpendicular to the layers (Fig. 2.2). In this geometry, the monochro-matic optical depth is dened by d ( z) ( z) ( z) d z, measured against the height zand the direction of the observed radiation, from the observer at innity into the atmo-sphere:

( z0) z0 ( z) ( z) d z. (2.44)The symbol cos is then used for the perspectivity factor, so that the radiativetransfer equation, Eq. (2.36) becomes in planeparallel geometry

d I ( , )

d = I ( , ) S ( ). (2.45)The radiative energy ow is described by the monochromatic ux density F ( ):

F ( ) I ( , ) d =2 +11 I ( , ) d, (2.46)measured in [erg s 1 cm2 Hz1]. Note that F is the net outwardly directed ux density:the downward radiation is counted as negative because of the perspectivity factor .

The formal solution of Eq. (2.45) for the intensity that emerges from the atmosphereis

I ( =0, ) = 0 S ( )e / d / (2.47)

and yields the EddingtonBarbier approximation ,

I ( =0, ) S ( = ), (2.48)which is exact if S is a linear function of . It implies that solar limb darkening (seenin Fig. 1.1a ) is a consequence of the decrease of the source function with height in thephotosphere.

The corresponding approximation for the radiative ux density emerging from a stellaratmosphere is

F ( =0) S ( =2/ 3), (2.49)which is also exact if the source function varies linearly with optical depth.

Spectroscopic studies of the center-to-limb variation of the intensity emerging fromthe solar disk and application of the Schwarzschild criterion (Section 2.2) have shownthat stellar photospheres are close to radiative equilibrium, that is, the outward energyow is transmitted almost exclusively as electromagnetic radiation. The condition for

radiative equilibrium in an optically thick medium is that through the chain of extinction

F is often called just ux. In many texts on radiative transfer and stellar atmospheres, the net ux density F is written as F .

The so-called astrophysical ux density F corresponds to the mean intensity as averaged over thestellar disk, as seen from innity.


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and emission processes the total radiative ux density F is transmitted unchanged; henceat all depths z,

F ( z) 0 F ( z) d T

4

eff = L/ (4 R2

), (2.50)

where L is the luminosity of the star, R is its radius, and T eff is its effective temperature .An alternative formulation of the condition of radiative equilibrium is

0 (S J ) d =0, (2.51)which is a continuity equation: a volume element emits as much radiative energy as it

absorbs.Equation (2.50) completely determines the variation of the source function with depth z, and hence the temperature stratication, but in a very implicit manner. The determi-nation of an accurate model atmosphere satisfying the RE condition of Eq. (2.50) fora specic T eff requires a sophisticated numerical technique. The main characteristics of such a model in RE can be illustrated by approximations, however.

The simplest approximation is that of a gray atmosphere, that is, the assumption thatthe extinction is independent of frequency: ( z) ( z). A single optical depth scale ( z) =

z

( z) ( z) d z then applies to all frequencies, and all monochromatic quantities

Q may be replaced by the corresponding integral over the spectrum: Q 0 Q d .The integrated source function S ( ) in a gray atmosphere in RE is found to be a nearly

linear function of (the MilneEddington approximation ):

S ( )3F 4

( +2/ 3), (2.52)

where F = T 4eff is the depth-independent radiative ux density. In the case of LTE, thespectrum-integrated source function equals the StefanBoltzmann function, Eq. (2.40),so that in a gray atmosphere in RE and LTE the temperature stratication is given by

T ( ) T eff 34

+12

1/ 4

. (2.53)

In actual stellar atmospheres, however, the extinction is far from gray; see Fig. 2.3.Numerical modeling shows that the nongrayness of the extinction steepens the temper-ature dependence on optical depth. In the deepest layers, both the temperature and thetemperature gradient must be somewhat higher than in the gray case in order to push

the radiative ux through the atmosphere, using the restricted spectral windows of lowopacity. The outermost layers radiate more efciently in the opaque parts of the spectrumthan in the gray case, so that the temperature settles at a lower level. The detailed tem-perature stratication depends on the variation of the extinction through the spectrum,the wavelengths of the extinction peaks and valleys, and the measure of the departurefrom LTE in the outer part of the atmosphere.


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Fig. 2.3. Continuum extinction coefcient [cm2

] per heavy particle in the solar photosphereat the optical depth of 5 = 0.1 at 5,000 A, where T = 5,040K and log pe = 0.5, plottedagainst wavelength . The spectral lines (not shown) add to the nongrayness in the form of numerous spikes on top of the continuous extinction curve. The dashed line species theRosseland mean opacity ( R in the text) dened by Eq. (2.54), for continuum extinction only(from Bohm-Vitense, 1989b).

In the subsurface layers, below the depth =1 at the extinction minimum, radiativetransfer can be handled simply by Eq. (2.25), using the Rosseland mean opacity R:1

R 0 1 d B / dT

d B/ dT d. (2.54)

Note that Eq. (2.25) describes radiative transfer as a process of photon diffusion, withthe photon mean-free path

R (R )1, (2.55)

which is representative for the entire ensemble of photons. This diffusion approx-imation for radiative transport holds when the bulk of the photons are locked up within

the local environment. In that case LTE applies as well, so that the source function S equals the Planck function B (T ).The Rosseland opacity is a harmonic mean, thus favoring the lower values of ;

Fig. 2.3 shows a solar example. The weighting function between the parentheses inEq. (2.54) selects the ux window tting the local temperature; its shape resemblesthat of the Planck function for the same temperature, but its peak is shifted to higherfrequency .


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The Rosseland optical depth R is based on the Rosseland opacity as the extinctioncoefcient:

R( z0)

z0

R d z. (2.56)

As another reference optical depth scale in solar studies, the monochromatic opticaldepth at = 5,000 A is often used. In this book, it is indicated by 5. The level R =1corresponds approximately to 5 =1.In Section 2.2, it is mentioned that the Rosseland opacity is particularly high forconditions in which themost abundant elements,hydrogen and helium, arepartly ionized.This may be understood as follows. At temperatures less than 10 4 K, the elements Hand He are not ionized and can only absorb efciently in their ultraviolet ground-statecontinua (for H: the Lyman continuum). At such temperatures, these continua contributevery little to the mean opacity R, because they fall outside the ux window, which is inthe near-ultraviolet, visible, and infrared. The main contribution to R then comes fromthe H ions, whose concentration is very low (typically less than 10 6 nH) because freeelectrons, provided by the low-ionization metals, are scarce. At higher temperatures, aslong as H is only partly ionized, the energy levels above the ground state (which are closeto the ionization limit) become populated, so that the resulting increased extinction inthe Balmer, Paschen, and other bound-free continua boosts the mean opacity R. At yethigher temperatures, the bound-free continua of He and He + add extinction. The meanopacity remains high until H is virtually completely ionized and He is doubly ionized.Then R drops because the remaining extinctions (Thomson scattering by free electronsand free-free absorption) are very inefcient.

The approximation of radiative transport by photon diffusion with the Rosselandopacity as a pseudogray extinction coefcient holds to very high precision in the stellarinterior, including the convective envelope. It holds approximately in the deepest partof the photosphere, for R > 1. Hence, Eq. (2.53) is a reasonable approximation of thetemperature stratication in the deep photosphere, provided that is replaced by R.The location of the stellar surface, dened as the layer where T equals the effectivetemperature, is then at

T ( R 2/ 3) =T eff ; (2.57)see Eq. (2.49).

Higher up, for optical depths R < 1, this approximation of radiative transportby diffu-sion with some mean opacity breaks down. Numerical modeling of radiativeequilibrium,in which the strong variation of the extinction coefcient with wavelength is taken intoaccount, fairly closely reproduces the empirical thermal stratication throughout thesolar photosphere; small departures from this stratication caused by overshooting con-

vection are discussed in Section 2.5. In other words, the thermal structure of the solarphotosphere is largely controlled by the radiative ux emanating from the interior.Radiative-equilibrium models for stellar atmospheres predict that the temperature

decreases steadily with height until it attens out where the medium becomes opticallythin to the bulk of the passing radiation. Yet spectra of the solar chromosphere andcorona indicate temperatures above and far above the temperature T min 4,200 K foundat the top of the photosphere. Hence some nonthermal energy ux must heat the outer


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Table 2.1. Mean, integrated radiative losses from variousdomains in the solar atmosphere

Domain F (erg cm2 s1) Ref.a

Photosphere 6 .4 1010 1Chromosphere 26 106 1Balmer series 5 105 1H 4 105 1Ly 3 105 1metal lines (Mg II, Ca II) 34 105 1Transition region 46 105 1Corona, quiet Sun 6 105 2Coronal hole 104 2a References: 1, Schatzman and Praderie (1993); 2, Section 8.6.

solar atmosphere. The radiative losses from its various parts (Table 2.1) indicate that thenonthermal heating uxes required to balance the losses are many orders of magnitudesmaller than the ux of electromagnetic radiation leaving the photosphere. The natureof these nonthermal heating uxes is discussed at some length in subsequent sectionsand chapters; here, we conne ourselves to the observation that undoubtedly some sort

of transmission of kinetic energy contained in the subsurface convection is involved.Even though electromagnetic radiation is not involved in the heating and thus the

creation of coronal structure, radiative losses are essential as one of the mechanismsfor cooling the corona and also for coronal diagnostics. The corona has temperaturesT > 1 106 K in order to make the emission coefcient per unit volume j equal to thelocal heating rate, which is small per unit volume but extremely high per particle.

In the extreme ultraviolet and softX-rays, the coronal emission consists of the superpo-sition of free-bound continua and spectral lines from highly ionized elements (mainly Fe,Mg, and Si).The radiativetransfer is simple for the many transitions inwhich the corona isoptically thin, although the conditions are very far from LTE. The excitations and ioniza-tions are induced exclusivelyby collisions, predominantly by electrons.Thephotosphericradiation eld at T 6,000 K is much too weak in the extremeultraviolet and softX-raysto contribute to the excitations and ionizations. Each collisional excitation or ionizationis immediately followed by a radiative de-excitation or recombination. If the corona isoptically thin for the newly created photon, then that photon escapes (or is absorbedby the photosphere). Hence, in any spectral line or continuum, the number of emittedphotons per unit volume is proportional to neny, where ne is the electron density and nyis the number density of the emitting ion. The ion density ny is connected with the protondensity n p throughtheabundanceof theelement relative to hydrogen and thetemperature-dependent population fraction of the element in the appropriate ionization stage.

The total power Pi, j emitted per unit volume in a spectral line i , j of an element Xmay be written as

Pi , j AXG X(T , i , j ) hc i , j

n enH , (2.58)


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where AX is the abundanceof element X relative to H, and temperature-dependent factorsare assembled in the contribution function GX(T , i , j ). A similar expression applies tothe power emitted per unit wavelength in a continuum.

Coronal spectral emissivities P (, T ) are dened such that P (, T )n enH is the poweremitted per unit volume per unit wavelength at . This emissivity is obtained by addingthe contributions of all relevant spectral lines and continua; see Fig. 3.11 in Golub andPasachoff (1997) for such theoretical spectra.

The total radiative loss function P (T ) 0 P (, T ) d determines the total radiativeloss rate : E rad P (T )n enH. (2.59)

Since the proton density approximately equals the electron density, in many applicationsof Eqs. (2.58) and (2.59) the factor nenH is replaced by n2

e.

Figure 2.4 shows P (T ) for one spectral code. The bumps in the curve are caused bycontributions from the strongest spectral lines emitted by a small number of elements.For T > 2107 K the emission in spectral lines drops below the level of the continuum.Recall that the function P (T ) is valid only for optically thin plasmas! For a more detaileddiscussion of radiation from hot plasmas, see Section 3.3 in Golub and Pasachoff (1997)and references given there.

At the other end of the electromagnetic spectrum, radio waves present another im-portant diagnostic on coronal physics. Throughout the corona, the propagation of radio

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0log(T)

1

10

100

P ( T ) [ 1 0 - 2 3

e r g c m

3 s - 1

]Solar abund.C+N+OMg+Si+NeFe

Fig. 2.4. The radiative loss function P (T ) for an optically thin plasma [Eq. (2.59)], forthe passband from 0.1keV up to 100keV, using the spectral code by Mewe, Kaastra,and colleagues (private communication). The curve is computed for solar photosphericabundances; the contributions by some of the major elements are also shown. The dottedstraight line is an approximation to the radiative loss function between 0.3 MK and 30 MK:P (T ) 1.5 1018T 2/ 3 erg cm3s1.


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waves is signicantly affected by refraction for all frequencies smaller than 300 MHz( > 1 m). The index of refraction n at frequency is given by

n2

=1

p

2

, (2.60)

where

p =n ee2

me

1/ 2

=9 103 n 1/ 2

e (Hz) (2.61)

is the plasma frequency with which electrons can oscillate about the relatively stationaryions. Consequently, only waves with frequency > p can propagate the plasma fre-quency determines a low-frequency cutoff. With the electron density n e , the plasma fre-quency decreases with height. The consequence is that meter waves, with < 300 MHz,can escape from the corona but not from deeper layers, decimeter waves ( < 3 GHz)can reach us from the transition region, and microwaves ( < 30 GHz) can leave thechromosphere.

Ray paths generally are curved because the refractive index varies throughout thecorona. A ray is bent away from a higher-density region when its frequency is close tothe local plasma frequency. Hence, in a hypothetically spherically symmetric, hydro-static corona, an inward propagating ray is reected upward, with the turning pointsomewhat above the level where its frequency equals the local plasma frequency. In theactual corona, density uctuations affect the ray path. More complications in the wavepropagation are caused by the magnetic eld. For more comprehensive introductions tosolar radio emission, we refer to Sections 3.4, 7.3, and 8.5 in Zirin (1988); Dulk (1985)provides a broad introduction to solar and stellar radio emission.

The emergent intensity equals the integral along the ray path B exp( ) d , whered = ds is an optical path length. The optical depth of the corona reaches unityin the meter range; it increases with wavelength because the extinction coefcient is determined by free-free absorption, which is proportional to 2. Most of the ob-served emission originates from just above the level where the frequency correspondsto the local plasma frequency, because there the free-free opacity, proportional to n2e ishighest.

Radiative transfer in the chromosphere and photosphere presents a complicated prob-lem because the optical thickness of the order of unity in some continuum windows andmany spectral lines necessitate a detailed treatment, with proper allowance for the strongvariation of the extinction coefcient throughout the spectrum. In detailed spectroscopicdiagnostics, theLTE approximation of the SahaBoltzmann population partitioning mustbe replaced by the equations for statistical equilibrium that relate the populations of the

stages of ionization and of the discrete energy levels in the atoms and ions to the localtemperature and to the radiation elds of nonlocal origin that may inuence the transitionof interest. If the latter is a bound-bound transition, its spectral-line source function isgiven by

S =bubl

B 1 blbu

eh/ kT , (2.62)


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line formed in a magnetized plasma, the monochromatic intensity I describing an un-polarized radiation eld must be replaced by four parameters characterizing a polar-ized radiation eld; for this, the four Stokes parameters { I , Q , U , V }are frequentlyused. Below, we suppress the subscripts or in the indications for the monochro-matic Stokes parameters. The parameter I is the total intensity, as dened in Eq. (2.31),the parameter V [Eq. (2.64)] species the magnitude of the circular polarization, andthe parameters Q and U together specify the magnitude and the azimuth of the linearpolarization.

Thesingletransferequation for I mustbereplacedbya set of fourdifferential equationsthat relate the Stokes parameters. The corresponding matrix equation is

dIds = K (I S), (2.63)

where I ( I , Q , U , V )T is the Stokes vector and K is the propagation matrix con-taining the extinction coefcients for the various Stokes parameters and the coefcientsfor the magneto-optical effects producing cross talk between the Stokes parameters.The vector S (S I , S Q , S U , S V )T stands for the source function vector in the Stokesparameters and s measures the geometrical length along the chosen line of sight. Thesuperscript T denotes the transpose of the matrix. We do not discuss the theory here,but we refer to Rees (1987) for a ne introduction to the treatment of polarized radi-ation and the Zeeman effect. Other references are Semel (1985), Steno (1985), and

Solanki (1993).In this brief introduction we consider possibilities and limitations of some standard

procedures of magnetic-eld measurements that take into account only the Stokes pa-rameters V and I . We consider a simple two-component model: magnetic ux tubes of strength B with an inclination angle with respect to the line of sight, which ll a frac-tion f of the photospheric surface, with the rest of the atmosphere being nonmagnetic.Except for the magnetic eld, the other plasma conditions inside and outside the mag-netic structures are considered to be identical, as far as the formation of the spectral lineis concerned. Only the spectra I L( ) and I R( ) in the left-handed and the right-handeddirections of circular polarization are used. From these spectra, the total intensity I ( )and Stokes parameter V ( ) follow:

I ( ) = I L( ) + I R( ) and V ( ) = I L( ) I R( ). (2.64)Consider a spectral line of the simplest magnetic splitting (Zeeman triplet), yielding

one central, undisplaced component and two components equidistant from the component (Fig. 2.5). The wavelength displacement of the components is proportionalto Bg 2,where g istheLand e factor, whichdepends on theatomictransition that produces

the line.If the Zeeman splitting is so large that the and components do not overlap signif-icantly (Fig. 2.5 a ), then the wavelength separation V between the extrema in the V prole simply equals the separation between the components, which is proportionalto Bg 2. In this case, the amplitude AV of the V prole scales with f cos and witha factor depending on the formation of the spectral line; it does not depend on B. In


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for developments in vector eld measurements, see November (1991). For an applicationof vector spectropolarimetry, see Skumanich et al. (1994).

The determination of the part of the atmosphere that contributes to the magneticsignal is more involved than the determination of the contribution functions in the caseof spectral line formation outside magnetic elds (Section 2.3.1) the result dependssensitively on the magnetic ne structure in the atmosphere. Consequently, quantitativedeterminations of height-dependent effects in the magnetic structure require caution. Werefer to Van Ballegooijen (1985) for a convenient method to treat line formation in amagnetic eld including the determination of contribution functions for the individualStokes parameters.

2.3.3 Stellar brightness, color, and sizeThis section is a synopsis of stellar terminology and two-dimensional classica-

tion. For general introductions to stellar terminology, data, and classication, we refer toChapter 4 of Uns old and Baschek (1991), to Bohm-Vitense (1989a), and to Chapters 13in Bohm-Vitense (1989b).

Traditionally, stellar brightnesses are expressed in logarithmic scales of stellar mag-nitudes m Q, which decrease with increasing stellar brightness,

mQ 2.5log R2

d 2 0 F Q d +C , (2.66)where Q refers to a specic photometric lter combination (which includes the Earthatmosphere) with transmission function Q , R is thestellar radius, d is thestellar distance,and F is the radiative ux density at the stellar surface. The normalization constant C is settled by a convention.

The absolute magnitude M Q is the magnitude of the star if it were placed at a distanceof 10 pc; hence:

mQ M Q =5log d 10

, (2.67)

where d is the distance in parsecs (1 pc equals 3 .09 1018 cm, or 3.26 light years). Bolometric magnitudes m bol are set by the total stellar luminosity L, that is, takingQ 1. For the absolute bolometric magnitude we have

M bol = 2.5log L L +4.72, (2.68)

where L is the solar luminosity.The UBV photometric system is often used for stellar brightnesses and colors, with U

for ultraviolet, B for blue, and V for visual. The corresponding magnitudes are usuallyindicated as U =mU , B =mB and V =mV, and the absolute magnitudes are indicatedby M U, M B and M V .The color index B V describes the color of the star. It is a measure of the spectral

energy distribution, andhenceof theeffective temperature T eff ; BV alsodepends, thoughto a lesser extent, on gravity g and chemical abundances. Note that stellar magnitudesand color indices are affected by interstellar extinction and reddening, respectively.


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The bolometric correction , BC , is dened by

BC m V mbol . (2.69)By visual inspection of their low-resolution spectra, the majority of stars are readilyarranged in a one-dimensional spectral sequence, which consists of the spectral types(Sp): OBAFGKM (with a decimal subdivision). In this order, these spectraltypes form a temperature sequence, the O-type stars being the hottest and the M-typestars being the coolest. As a historical artifact, the hot stars are often described as early,and the cool stars as late; spectral type A is earlier than spectral type G. Somewhatmore appropriately, the stars are sometimes indicated by their color impression: A- andearly F-types are called white, G-types yellow, K-types orange, and M-types red.

In addition to temperature characteristics, stellar spectra also contain more subtle in-formation bearing on the pressure in the atmosphere, and thus on the stellar radius andluminosity. Such characteristics are used in the MorganKeenan (MK) two-dimensionalspectral classication, which labels a star with a spectral type O through M and a lumi-nosity class (LC) which is indicated by Roman numerals IVI and the following terms:I, supergiant; II, bright giant; III, giant; IV, subgiant; V, dwarf or main-sequence star; VI,subdwarf. For instance, the MK classication of the Sun is G2 V.

Occasionally, a one-letter code is added to the spectral type. The appendix p, forpeculiar, indicates that the spectrum differs in some aspects from the standard spectrumof the corresponding type, usually because the stellar chemical composition differs from

that of the majority of the stars in the solar neighborhood. The addition e, for emission,refers to emission seen in some of the strongest lines in the spectrum.For some stars that have not yet been classied in the MK system, an earlier classi-

cation is indicated by a letter preceding the spectral type, with d for dwarf, g for giant,and sg for supergiant. For instance, dM4e indicates a dwarf M4-type star, with emissionlines in its spectrum.

2.3.4 Radiative diagnostics of outer atmospheresThe nonradiativeheating of the outer atmospheres of the Sun and other cool stars

leads to emission in a multitude of spectral lines and continua. Table 2.2 summarizes thediagnostics that are used in this book. Throughout this book, we dene chromosphere,transition-region, and corona in terms of the associated temperatures as follows:

chromosphere : T < 20,000K ,transition region : 50 ,000K < T < 5 105 K,

corona : T > 106 K

(see 14 for the radiative properties of these domains).

The Ca II H and K lines are the strongestand broadest lines in the visible solar spectrum.The strength of the lines and the double ionization of calcium limits the lines formationto the chromosphere. In solar observations, the linecores of this doublet locally display anemission reversal, with a central absorption minimum; the naming convention for thesefeatures is shown in Fig. 2.6. Clearly, the source function S [Eq. (2.35)] correspondingto such a line shape cannot be a monotonically decreasing function of height in theatmosphere, indicative of (intermittent) deposition of nonradiative energy.


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Table 2.2. Frequently used radiative diagnostics of atmospheric activity,together with the characteristic temperatures of maximum contribution a

Diagnostic (A) Characteristic Formed inTemperature (K)

TiO bands < 4,000 SpotsUV cont. 1,600 4,500 Temp. minimumCO bands < 4,400 Temp. minimum

CaIIH and K 3,9673,933 (47) 103 ChromosphereH (core) 6,563 ( 510) 103 ChromosphereMgIIh and k 2,8032,796 (420) 103 ChromosphereLy 1,215 . . . High Chrom.C II 1,335 1.5 104 low TRSiII 1,8081,871 2 104 low TRCIII 977 4 104 TRC IV

1,5481,551 10

5 TR

SiIV 1,394

1,403 105 TR

O IV 554 2 105

TRO VI 1,032 3 105 TRMg X 625 1 .5 106 CoronaX-rays 10200 > 106 Coronaa Some diagnostics, particularly the hydrogen Lyman line and the soft X-ray emis-sion, are formed over a broad range in temperature. The temperature domains (seeVernazza et al. , 1981, for the chromospheric diagnostics) are crude indications only.TR indicates the transition region.

The detailed formation of the H and K lines

Solar and Stellar Magnetic Activity

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