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Astronomy: A Physical Perspective
This fully revised and updated text is a comprehen- worked examples and end-of-chapter problem sets. It
sive introduction to astronomical objects and phe- is suitable for undergraduate students taking a first
nomena. By applying some basic physical principles course in astronomy, and assumes a basic knowledge
to a variety of situations, students will learn how to of physics with calculus.
relate everyday physics to the astronomical world.
Marc L. Kutner obtained his doctorate in physics from
Starting with the simplest objects, the text contains
thorough explanations of how and why astronomi- Columbia University in 1972. He has been a Visiting
cal phenomena occur, and how astronomers collect Scientist in the Department of Astronomy at the
and interpret information about stars, galaxies and University of Texas at Austin since 1998, prior to
the Solar System. The text looks at the properties of which he was Professor in the Department of Physics
stars, star formation and evolution; neutron stars and Astronomy at the Rensselaer Polytechnic
and black holes; the nature of galaxies; and the Institute, New York, and Visiting Scientist at the
structure of the universe. It examines the past, pres- National Radio Observatory, Tucson, Arizona. His
ent and future states of the universe; and final chap- main area of research involves the use of radio
ters use the concepts that have been developed to astronomy to study of star formation in the Milky
study the Solar System and its formation; the possi- Way and other galaxies. He has also done some
bility of finding other planetary systems; and the research in cosmology. Professor Kutner has pub-
search for extraterrestrial life. This comprehensive lished three successful textbooks and over one hun-
text contains useful equations, chapter summaries, dred research papers.
Astronomy: A Physical
Perspective
Marc L. Kutner
©¤§ µ®©© °
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge  µ, United Kingdom
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521821964

© Marc L. Kutner 2003


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Contents


List of abbreviations used in the figure credits Page xv
Preface xvii


1 Introduction 1
1.1 An understandable universe 1
1.2 The scale of the universe 3


Part I Properties of ordinary stars 7

2 Continuous radiation from stars 9
2.1 Brightness of starlight 9
2.2 The electromagnetic spectrum 10
2.3 Colors of stars 12
2.3.1 Quantifying color 12
2.3.2 Blackbodies 13
2.4 Planck™s law and photons 15
2.4.1 Planck™s law 15
2.4.2 Photons 16
2.5 Stellar colors 17
2.6 Stellar distances 18
2.7 Absolute magnitudes 20
Chapter summary 21
Questions 21
Problems 22
Computer problems 23

3 Spectral lines in stars 25
3.1 Spectral lines 25
3.2 Spectral types 26
3.3 The origin of spectral lines 27
3.3.1 The Bohr atom 28
3.3.2 Quantum mechanics 31
3.4 Formation of spectral lines 32
3.4.1 Excitation 32
3.4.2 Ionization 33
3.4.3 Intensities of spectral lines 34
3.5 The Hertzprung“Russell diagram 36
Chapter summary 38
Questions 39
Problems 39
Computer problems 40

4 Telescopes 41
4.1 What a telescope does 41
4.1.1 Light gathering 41
4.1.2 Angular resolution 42
4.1.3 Image formation in a camera 43
vi CONTENTS



4.2 Refracting telescopes 45
4.3 Reflecting telescopes 46
4.4 Observatories 53
4.4.1 Ground-based observing 53
4.4.2 Observations from space 56
4.5 Data handling 58
4.5.1 Detection 58
4.5.2 Spectroscopy 60
4.6 Observing in the ultraviolet 62
4.7 Observing in the infrared 63
4.8 Radio astronomy 68
4.9 High energy astronomy 75
Chapter summary 78
Questions 78
Problems 79
Computer problems 81
5 Binary stars and stellar masses 83
5.1 Binary stars 83
5.2 Doppler shift 84
5.2.1 Moving sources and observers 84
5.2.2 Circular orbits 86
5.3 Binary stars and circular orbits 87
5.4 Elliptical orbits 91
5.4.1 Geometry of ellipses 91
5.4.2 Angular momentum in elliptical orbits 91
5.4.3 Energy in elliptical orbits 93
5.4.4 Observing elliptical orbits 93
5.5 Stellar masses 94
5.6 Stellar sizes 96
Chapter summary 97
Questions 98
Problems 99
Computer problems 100
6 The Sun: a typical star 101
6.1 Basic structure 101
6.2 Elements of radiation transport theory 101
6.3 The photosphere 105
6.3.1 Appearance of the photosphere 105
6.3.2 Temperature distribution 107
6.3.3 Doppler broadening of spectral lines 108
6.4 The chromosphere 109
6.5 The corona 110
6.5.1 Parts of the corona 110
6.5.2 Temperature of the corona 112
6.6 Solar activity 113
6.6.1 Sunspots 113
6.6.2 Other activity 116
Chapter summary 119
Questions 119
CONTENTS vii



Problems 120
Computer problems 121



Part II Relativity 123

7 Special relativity 125
7.1 Foundations of special relativity 125
7.1.1 Problems with electromagnetic radiation 125
7.1.2 Problems with simultaneity 127
7.2 Time dilation 128
7.3 Length contraction 129
7.4 The Doppler shift 131
7.4.1 Moving source 131
7.4.2 Moving observer 131
7.4.3 General result 132
7.5 Space-time 132
7.5.1 Four-vectors and Lorentz transformation 132
7.5.2 Energy and momentum 135
Chapter summary 136
Questions 136
Problems 137
Computer problems 137

8 General relativity 139
8.1 Curved space-time 139
8.2 Principle of equivalence 141
8.3 Tests of general relativity 143
8.3.1 Orbiting bodies 143
8.3.2 Bending electromagnetic radiation 144
8.3.3 Gravitational redshift 145
8.3.4 Gravitational radiation 147
8.3.5 Competing theories 148
8.4 Black holes 148
8.4.1 The Schwarzschild radius 148
8.4.2 Approaching a black hole 149
8.4.3 Stellar black holes 150
8.4.4 Non-stellar black holes 151
Chapter summary 152
Questions 152
Problems 152
Computer problems 153



Part III Stellar evolution 155

9 The main sequence 157
9.1 Stellar energy sources 157
9.1.1 Gravitational potential energy of a sphere 157
9.1.2 Gravitational lifetime for a star 158
9.1.3 Other energy sources 158
viii CONTENTS



9.2 Nuclear physics 159
9.2.1 Nuclear building blocks 159
9.2.2 Binding energy 160
9.2.3 Nuclear reactions 161
9.2.4 Overcoming the fusion barrier 162
9.3 Nuclear energy for stars 164
9.4 Stellar structure 168
9.4.1 Hydrostatic equilibrium 168
9.4.2 Energy transport 170
9.5 Stellar models 171
9.6 Solar neutrinos 172
Chapter summary 175
Questions 175
Problems 176
Chapter problems 176
10 Stellar old age 177
10.1 Evolution off the main sequence 177
10.1.1 Low mass stars 177
10.1.2 High mass stars 179
10.2 Cepheid variables 179
10.2.1 Variable stars 179
10.2.2 Cepheid mechanism 180
10.2.3 Period“luminosity relation 181
10.3 Planetary nebulae 183
10.4 White dwarfs 186
10.4.1 Electron degeneracy 186
10.4.2 Properties of white dwarfs 188
10.4.3 Relativistic effects 189
Chapter summary 190
Questions 190
Problems 190
Computer problems 191
11 The death of high mass stars 193
11.1 Supernovae 193
11.1.1 Core evolution of high mass stars 193
11.1.2 Supernova remnants 194
11.2 Neutron stars 197
11.2.1 Neutron degeneracy pressure 197
11.2.2 Rotation of neutron stars 198
11.2.3 Magnetic fields of neutron stars 199
11.3 Pulsars 199
11.3.1 Discovery 199
11.3.2 What are pulsars? 201
11.3.3 Period changes 203
11.4 Pulsars as probes of interstellar space 205
11.5 Stellar black holes 206
Chapter summary 206
Questions 206
CONTENTS ix



Problems 207
Computer problem 208

12 Evolution in close binaries 209
12.1 Close binaries 209
12.2 Systems with white dwarfs 211
12.3 Neutron stars in close binary systems 213
12.4 Systems with black holes 216
12.5 An unusual object: SS433 218
Chapter summary 219
Questions 220
Problems 220
Computer problems 220

13 Clusters of stars 221
13.1 Types of clusters 221
13.2 Distances to moving clusters 221
13.3 Clusters as dynamical entities 225
13.3.1 The virial theorem 225
13.3.2 Energies 227
13.3.3 Relaxation time 228
13.3.4 Virial masses for clusters 229
13.4 HR diagrams for clusters 231
13.5 The concept of populations 232
Chapter summary 233
Questions 233
Problems 233
Computer problem 234


Part IV The Milky Way 235

14 Contents of the interstellar medium 237
14.1 Overview 237
14.2 Interstellar extinction 237
14.2.1 The effect of extinction 238
14.2.2 Star counting 239
14.2.3 Reddening 240
14.2.4 Extinction curves 241
14.2.5 Polarization 242
14.2.6 Scattering vs. absorption 242
14.3 Physics of dust grains 243
14.3.1 Size and shape 243
14.3.2 Composition 243
14.3.3 Electric charge 244
14.3.4 Temperature 245
14.3.5 Evolution 246
14.4 Interstellar gas 246
14.4.1 Optical and ultraviolet studies 246
14.4.2 Radio studies of atomic hydrogen 247
x CONTENTS



14.5 Interstellar molecules 251
14.5.1 Discovery 251
14.5.2 Interstellar chemistry 253
14.5.3 Observing interstellar molecules 254
14.6 Thermodynamics of the interstellar medium 258
Chapter summary 259
Questions 260
Problems 261
Computer problems 262

15 Star formation 263

15.1 Gravitational binding 263
15.2 Problems in star formation 266
15.3 Molecular clouds and star formation 267
15.4 Magnetic effects and star formation 270
15.5 Protostars 272
15.5.1 Luminosity of collapsing clouds 272
15.5.2 Evolutionary tracks for protostars 273
15.6 Regions of recent star formation 274
15.6.1 HII regions 274
15.6.2 Masers 280
15.6.3 Energetic flows 282
15.6.4 T Tauri stars and related objects 285
15.7 Picture of a star forming region: Orion 287
Chapter summary 289
Questions 290
Problems 291
Computer problems 292

16 The Milky Way galaxy 293

16.1 Overview 293
16.2 Differential galactic rotation 294
16.2.1 Rotation and mass distribution 294
16.2.2 Rotation curve and Doppler shift 296
16.3 Determination of the rotation curve 300
16.4 Average gas distribution 302
16.5 Spiral structure in the Milky Way 304
16.5.1 Optical tracers of spiral structure 304
16.5.2 Radio tracers of spiral structure 304
16.6 The galactic center 306
16.6.1 Distribution of material near the center 306
16.6.2 A massive black hole? 308
Chapter summary 310
Questions 311
Problems 311
Computer problems 312
CONTENTS xi




Part V The universe at large 313

17 Normal galaxies 315
17.1 Types of galaxies 315
17.1.1 Elliptical galaxies 315
17.1.2 Spiral galaxies 317
17.1.3 Other types of galaxies 321
17.2 Star formation in galaxies 322
17.2.1 Star formation in the Large Magellanic Cloud 323
17.2.2 Star formation in spiral galaxies 324
17.3 Explanations of spiral structure 326
17.4 Dark matter in galaxies 330
Chapter summary 333
Questions 333
Problems 334
Computer problems 334

18 Clusters of galaxies 335
18.1 Distribution of galaxies 335
18.2 Cluster dynamics 335
18.3 Expansion of the universe 339
18.3.1 Hubble™s law 339
18.3.2 Determining the Hubble constant 341
18.4 Superclusters and voids 345
18.5 Where did all this structure come from? 347
18.6 The Hubble Deep Field 349
Chapter summary 350
Questions 351
Problems 351

19 Active galaxies 353
19.1 Starburst galaxies 353
19.2 Radio galaxies 355
19.2.1 Properties of radio galaxies 355
19.2.2 Model for radio galaxies 357
19.2.3 The problem of superluminal expansion 359
19.3 Seyfert galaxies 361
19.4 Quasars 362
19.4.1 Discovery of quasars 362
19.4.2 Properties of quasars 365
19.4.3 Energy“redshift problem 366
19.5 Gravitationally lensed quasars 368
19.6 A unified picture of active galaxies? 370
19.6.1 A common picutre 370
19.6.2 Black holes in galactic nuclei? 371
Chapter summary 373
Questions 374
xii CONTENTS



Problems 374
Computer problems 375

20 Cosmology 377
20.1 The scale of the universe 377
20.2 Expansion of the universe 378
20.2.1 Olbers™s paradox 378
20.2.2 Keeping track of expansion 380
20.3 Cosmology and Newtonian gravitation 381
20.4 Cosmology and general relativity 384
20.4.1 Geometry of the universe 384
20.4.2 Cosmological redshift 386
20.4.3 Models of the universe 388
20.5 Is the universe open or closed? 390
Chapter summary 392
Questions 393
Problems 394
Computer problems 394

21 The big bang 395
21.1 The cosmic background radiation 395
21.1.1 Origin of the cosmic background radiation 395
21.1.2 Observations of the cosmic background radiation 398
21.1.3 Isotropy of the cosmic background radiation 401
21.2 Big-bang nucleosynthesis 407
21.3 Fundamental particles and forces 410
21.3.1 Fundamental particles 410
21.3.2 Fundamental forces 411
21.3.3 The role of symmetries 412
21.3.4 Color 413
21.3.5 The unification of forces 416
21.4 Merging of physics of the big and small 417
21.4.1 Back to the earliest times 417
21.4.2 Inflation 419
21.4.3 Galaxy formation 420
21.4.4 Estimates of values of cosmological parameters 420
Chapter summary 422
Questions 423
Problems 424
Computer problems 425


Part VI The Solar System 427

22 Overview of the Solar System 429
22.1 Motions of the planets 430
22.2 The motion of the Moon 435
22.3 Studying the Solar System 438
22.4 Traveling through the Solar System 439
Chapter summary 443
Questions 444
CONTENTS xiii



Problems 444
Computer problems 445

23 The Earth and the Moon 447
23.1 History of the Earth 447
23.1.1 Early history 447
23.1.2 Radioactive dating 448
23.1.3 Plate tectonics 450
23.2 Temperature of a planet 452
23.3 The atmosphere 454
23.3.1 Pressure distribution 455
23.3.2 Temperature distribution 457
23.3.3 Retention of an atmosphere 462
23.3.4 General circulation 463
23.4 The magnetosphere 465
23.5 Tides 467
23.6 The Moon 469
23.6.1 The lunar surface 470
23.6.2 The lunar interior 473
23.6.3 Lunar origin 474
Chapter summary 475
Questions 476
Problems 477
Computer problems 478

24 The inner planets 479
24.1 Basic features 479
24.1.1 Mercury 479
24.1.2 Venus 479
24.1.3 Mars 480
24.1.4 Radar mapping of planets 481
24.2 Surfaces 483
24.3 Interiors 490
24.3.1 Basic considerations 490
24.3.2 Results 491
24.4 Atmospheres 491
24.5 Moons 494
Chapter summary 494
Questions 495
Problems 495
Computer problems 496

25 The outer planets 497
25.1 Basic features 497
25.2 Atmospheres 500
25.3 Interiors 506
25.4 Rings 506
25.4.1 Basic properties 507
25.4.2 Ring dynamics 509
xiv CONTENTS



25.5 Moons 512
Chapter summary 519
Questions 520
Problems 520
Computer problem 521

26 Minor bodies in the Solar System 523
26.1 Pluto 523
26.2 Comets 524
26.3 Meteoroids 530
26.4 Asteroids 532
Chapter summary 534
Questions 534
Problems 535
Computer problem 535

27 The origin of life 537
27.1 Origin of the Solar System 537
27.2 Chemistry on the early Earth 540
27.3 Origin of life on Earth 541
27.4 Life in the rest of the Solar System? 543
27.5 Other planetary systems? 544
27.6 Searches for extraterrestrial intelligence 547
Chapter summary 549
Questions 550
Problems 550
Computer problems 550


Appendix A Glossary of symbols 551
Appendix B Physical and astronomical constants 553
Appendix C Units and conversions 554
Appendix D Planet and satellite properties 555
Appendix E Properties of main sequence stars 558
Appendix F Astronomical coordinates and timekeeping 559
Appendix G Abundances of the elements 562


Index 565
Abbreviations used in the ¬gure credits


Figure credits are given in the captions. Abbreviations used are
as follows.

2MASS Two Micron All Sky Survey
AUI Associated Universities Inc.
AURA Association of Universities for Research in Astronomy
Caltech California Institute of Technology
CFA Center for Astrophysics
ESA European Space Agency
ESO European Southern Observatory
GSFC ADF Goddard Space Flight Center Astrophysics Data Facility
HST Hubble Space Telescope
IFA Institute for Astronomy
IRAM Institut de Radioastronomie Millim©trique
ISO Infrared Space Observatory
JCMT James Clerk Maxwell Telescope
MIT Massachusetts Institute of Technology
MPIFR Max Planck Institut für Radioastronomie
NASA National Aeronautics and Space Administration
NMTech New Mexico Institute of Mining and Technology
NOAA National Oceanographic and Atmospheric Observatory
NOAO National Optical Astronomy Observatory (operated by
AURA under contract with the NSF, all rights reserved)
NRAO National Radio Astronomy Observatory (operated by
AUI, under contract with the NSF)
NSF National Science Foundation
ONR Office of Naval Research
SCUBA Submillimeter Common User Bolometer Array
STScI Space Telescope Science Institute (operated by AURA
under contract with NASA)
UCLA University of California at Los Angeles
USGS US Geological Survey
Preface


This book is the successor to Astronomy: a Physical
The study of astronomy has blossomed in a vari-
Perspective, published by Wiley in 1986. I am grate-
ety of ways in the last decade of the 20th century.
Every part of the electromagnetic spectrum has ful to the loyal audience that book developed,
seen a revolution in observing techniques. While and for their encouragement to work on this new
much of this has been on the ground, space-based version.
observing has come into its own, as we are seeing I am grateful to Simon Mitton at Cambridge
the results of second and third generation space- University Press, who shared my view that a
based telescopes. These have provided sensitivity ˜higher level™ book could still be visually attrac-
and clarity that have revolutionized all subfields tive. I am also grateful to Jacqueline Garget, who
in astronomy and created some new ones. These believed in this project, seeing it through a few
observational developments have been supple- rough early reviews to its completion. At every
mented by massive improvements in computing stage, she always knew exactly how to answer my
power, allowing for the processing of large email questions to keep me going.
amounts of astronomical data, and the theoreti- Three professors, Stephen Boughn (Haverford),
cal modeling of the results. James Houck (Cornell) and Judith Pipher (Rochester)
The most amazing aspect of all of this class-tested various versions of this manuscript. I
progress is that we can still provide reasonable appreciate their patience and their feedback. I
answers to the naive question, ˜How does it all also appreciate their students taking the time to
work?™ As our astronomical horizon expands, we use a ˜book™ in a non-standard form, and to give
can still use familiar physics to explain the comments.
wealth of phenomena. Even when the explana- Special thanks go to Nadine Dinshaw, a friend/
tion at the research level requires a complex colleague, who read the whole manuscipt in an
application of certain physical laws, there is early form. Her comments and support were very
usually still a way of understanding the phe- helpful at that early stage.
nomena based on introductory level physics. At every stage the manuscript benefited greatly
Perhaps this is just the realization that the laws from the feedback from reviewers who read all or
of physics are small in number but apply uni- various parts of the manuscript. Some were anony-
versally. There are a few exceptions, where the mous, and others were: Imke DePater (University of
astronomical problems help drive back the fron- California at Berkeley), Debra Elmegreen (Vassar),
tiers of physics, but these can be explained in Andrea Ghez (UCLA), Steven Gottesman (University
more familiar terms. of Florida), Richard Griffiths (Carnegie Mellon),
This book is dedicated to the student who David Helfand (Columbia), Lee Mundy (University
would like more out of even a brief study of of Maryland), James Napolitano (Rensselaer), and
astronomy than a list of what there is. It is for the Heidi Newberg (Rensselaer).
student who wants to understand why certain Many astronomers and physicists have con-
phenomena occur, and how astronomical objects tributed data and illustrations which I have used
work. In addition, it addresses the question of directly. They are too numerous to mention here,
how we collect and interpret information about but are credited in the figure captions. My special
remote objects. thanks go to those who were anxious for me to
The primary audience of this book will be sci- have the most recent data or best pictures.
ence majors who have taken a year of college Gathering these figures proved to be frustrating
physics (classical) with calculus. We therefore pre- sometimes. However, the contact that I had with
sume that the student has seen the classical the vast majority was very rewarding. I would
physics needed for the astronomy course, but do also like to thank an extraordinary copy editor,
not presume a knowledge of ˜modern™ physics. Irene Pizzie, for always knowing what I meant to
xviii PREFACE



say, and production manager, Catherine Garland, On the personal level, I got my start in astron-
for keeping the project moving along, and always omy when my mother encouraged me to take
keeping me in the loop. courses at the Hayden Planetarium, in New York.
This project started during my three-year stay I am also grateful to my two sons, Eric and Jeff,
at the National Radio Astronomy Observatory, in who never stop asking questions.
Tucson. I am thankful to Paul Vanden Bout (NRAO Most important, at many levels, this book
director) for helping me settle into that position, would not be here without my best colleague and
and to all the people in Tucson who provided a best friend, Kathryn Mead. She encouraged me to
stimulating atmosphere and a view of the Santa tackle hard tasks, from running marathons, to
Catalina Mountains. The project has finished dur- biking centuries, to refereeing soccer, to writing
ing my stay at the University of Texas, Austin. I books. Her drive and curiosity led to our most
am grateful to Frank Bash (McDonald Observatory important discovery (molecular clouds in the
director) for arranging that position and always outer Milky Way). More immediately, she also
having an open door. I thank my colleagues here helped dress up those figures for this book that
in Austin for providing a stimulating environment needed it the most.
also.
Chapter 1




Introduction

for the remoteness of astronomical objects, we get
1.1 An understandable universe to study a large number of objects under a variety
of conditions.
Our curiosity about the world around us is most One of the most fascinating aspects of astron-
naturally manifested when we look up at the night omy is that many phenomena can be understood
sky. We don™t need any special instruments to tell in terms of relatively simple physics. This does not
us something interesting is going on. However, mean that we can explain every detail. However,
only with the scrutiny afforded by a variety of we can explain the basic phenomena. In this
instruments can these patches of light, and the book, we emphasize the application of a few phys-
dark regions between them, offer clues about their ical principles to a variety of situations. For this
nature. We have to be clever to collect those clues, purpose, some background in physics is needed.
and just as clever to interpret them. It is the total We assume that the reader has had an introduc-
of these studies that we call astronomy. tory course in classical physics (mechanics, elec-
We are fortunate to live in an era of extraordi- tricity and magnetism, thermodynamics). We
nary astronomical discovery. Some have even also use quite a bit of modern physics (relativity,
called this the ˜Golden Era of Astronomy™. For cen- atomic and nuclear physics). The modern physics
turies astronomers were restricted to making will be developed as we need it. In addition, a
visual observations from the surface of the Earth. familiarity with the concepts of calculus is
We can now detect virtually any type of radiation assumed. While most of the material can be mas-
given off by an astronomical object, from radio tered without actually taking derivatives and
waves to gamma rays. Where necessary, we can working out integrals, the concepts of derivatives
put observatories in space. For the Solar System, as representing changes and integrals as repre-
we can even visit the objects we are studying. senting sums are used. The reader may also note
For all of these capabilities, there is a major a variation in the mathematical level from sub-
drawback. We cannot do traditional experiments ject to subject. This is because the goal in writing
on remote astronomical objects. We cannot this book is to present each astronomical subject
change their environment and see how they at the simplest level that still provides for a rea-
respond. We must passively study the radiation sonable understanding.
that they give off. For this reason, we refer to In organizing an astronomy text, one impor-
astronomy as an observational science rather than tant question is where to put the material on the
an experimental one. It is because of this differ- Solar System. The traditional approach has been
ence that we must be clever in using the infor- to place the Solar System first. This allows the
mation that we do receive. In this book, we will student to start with familiar, nearby objects first
see what information we can obtain and how the and work out from there. The disadvantage is
clues are processed. We will see that, in exchange that we use techniques to study the Solar System
2 1 INTRODUCTION



that we cannot use on more distant objects. In In the final part, Part VI, we will study the
this book we place the Solar System last. This Solar System. We will see how the formation of
allows the student to form a better idea of how the Solar System can be fit into ideas already
astronomy is done on remote objects. We can also developed about star formation. We will encounter
use the physics that we develop in studying stars a variety of surfaces, atmospheres and rings that
and other astronomical objects to give us a better can be explained by using the physical ideas
appreciation for how the Solar System works. already developed. We will also look at the origin
Finally, putting the Solar System at the end of life on the Earth and the search for life else-
allows for a discussion of the formation of the where in the Solar System and in our galaxy.
Solar System, utilizing things that we learn about Although the organization of the book is
star formation. around astronomical objects, the presentation of
We start with stars, those points of light in the topics emphasizes the application of the
the night sky. This allows us to develop physical underlying physics. Almost all of the physical
ideas (radiation, gravity, etc.) that we will use tools will apply to several topics. A great strength
throughout the book. We will see how we obtain of physical theories is the great range of their
information about the basic properties of stars: applicability. For example, orbital mechanics can
temperatures, sizes, masses, compositions. The tell us about the masses of binary stars or help us
Sun will then be looked at as an example of a typ- plan a probe to Mars. Radiative transfer helps us
ical star. We will then put these stellar properties understand the appearance of the Sun, the physi-
together, and describe a theoretical picture of cal conditions in interstellar clouds or the tem-
how stars work. In Part II we will develop the spe- peratures of planetary atmospheres. Tidal effects
cial and general theories of relativity, to allow us help us explain the appearance of certain galax-
to understand better the unusual states that are ies, rings around some planets and the internal
reached when stars die. We will discuss the nor- heating of Io, one of Jupiter™s moons.
mal lifetime of stars and stellar old age and death Though understanding how astronomical
in Part III. In stellar death, we will encounter a objects work is our goal, astronomy™s foundation
variety of exotic objects, including neutron stars is observation. We will see how observations
and black holes. often define a problem “ the discovery of new phe-
In Part IV, we will look at the contents of our nomena. Observations usually provide a check on
own galaxy, the Milky Way. We will start by look- theories that are developed. In this book, we will
ing at the interstellar medium and then at how therefore emphasize the interplay between obser-
stars are formed. Finally, we will look at how vation and understanding the physics. We will
stars, gas and dust are organized into a galaxy. see how some observations yield numbers with
In Part V, we will look at the overall structure great precision, while others only give order of
of the universe, including the arrangement of magnitude estimates, but both types can be
galaxies and their motions. We will start by look- equally important for deciding between theories.
ing at other galaxies. We will also study active With the current pace of astronomical discov-
galaxies, which give off much more energy than ery, there is an important caution to keep in
our own. We will follow the trail of active galax- mind. When you read an introductory text on
ies from starburst galaxies to quasars. The early classical physics, you are reading about theories
history of the universe (the big bang) will be that were worked out and tested over a century
described, and we will see can how we look for ago. No question is raised about the correctness
clues about the past and its ultimate fate. In talk- of these theories. In astronomy, new ideas or new
ing about the early universe, we will encounter observations are constantly changing the think-
one of the most fascinating recent developments, ing about various problems. Many of the topics
the merging of physics on the largest and small- discussed in this book are far from being settled.
est scales. This involves blending theories on the Sometimes, more than one explanation is pre-
ultimate structure of matter with theories of the sented for a given phenomenon. This is done
overall structure of the universe. either because we don™t know which is correct, or
1 INTRODUCTION 3



to show how one theory was eliminated in favor We next look at the Sun (Fig. 1.1b). It is 1.5
8
of another. Just because this is a “text” it doesn™t 10 km from the Earth, meaning it takes light
mean that it has the final word. If you under- over eight minutes to get here from the Sun. We
stand where the problems lie, and the reasoning call this distance the Astronomical Unit. Its mass
1033 g. This turns out to be average for a
behind the explanations, then you will be able to is 2
follow future developments as they appear in sci- star, and we even use it as a convenient measure.
The Sun™s radius is 6 105 km.
entific magazines or journals.
This, then, is the plan. As you study the material We see how far out the planets are by looking
that follows, see how far you can go with a little bit at Pluto (Fig. 1.1c). It is almost 40 astronomical
of physics and a lot of curiosity and ingenuity. units from the Sun, meaning it takes light almost
six hours to reach us from Pluto.
By the time we reach the nearest stars, they
1.2 The scale of the universe are so far away that it takes light years to reach
us. So we measure their distance in light years
The objects that we encounter in astronomy are,
for the most part, so large or distant that it is hard
to comprehend their size or distance. We will take
a brief look at the distances involved when we
study different astronomical objects. We will talk
about these sizes in more detail when we
encounter the objects in the rest of the book. In
Fig. 1.1, we show a selection of objects on the var-
ious scales.
We start by looking at the Earth and Moon
(Fig. 1.1a). Earth has a radius of about 6000 km. Its
mass is about 6 1027 grams. The Moon is about
105 km from the Earth. It takes about one
4
second for light to travel from the Moon to the
Earth.




(b)




(c)
Fig 1.1. Photographs to show different astronomical
scales. (a) The Earth and Moon from space. [NASA] (b) The
Sun. [NOAO/AURA/NSF] (c) Pluto and its moon, Charon.
[STScI/NASA]
(a)
4 1 INTRODUCTION




(d) (f)




(e)
Fig 1.1. (Continued) (d) Betelgeuse. [STScI/NASA] (e) A
globular cluster. [NOAO/AURA/NSF] (f) The Andromeda
Galaxy. [NOAO/AURA/NSF] (g) A cluster of galaxies.
[NOAO/AURA/NSF]




(almost 1013 km) or parsecs (one parsec is about (g)
three light years). Fig. 1.1(d) shows a star about as
1.1(e). These objects may contain 105 stars, and
far away as we can take direct picture of its disk.
It is the giant star Betelgeuse in the constellation have extents of tens of parsecs. Because of their
of Orion, some 500 parsecs away, meaning it took collective brightness, we can see them far away,
the light for that image about 1500 years to even on the other side of our galaxy. In fact, they
reach us. tell us that we are 8500 parsecs from our galactic
The next largest scale are groupings of stars center. That means it takes light from the galactic
called clusters, such as the globular cluster in Fig. center 25 000 years to reach us.
1 INTRODUCTION 5



In Fig. 1.1(f), we leave the Milky Way Galaxy Our final step is to a cluster of galaxies, such
and look at one of our neighbors, the as the Virgo Cluster, which is shown in Fig. 1.1(g).
Andromeda Galaxy, which we think looks a lot These clusters are groupings of thousands of
like our galaxy would look if we could view it galaxies, and are typically millions of parsecs
from outside. It is so far away that we measure its across. We detect some clusters so far away that
distance in thousands of parsecs, kiloparsecs. It their light has taken a significant fraction of the
is 700 kiloparsecs away, meaning it takes light age of the universe (which we think is about 14
109 yr) to reach us.
about 2100 years to reach us. It is about 20 kilo-
parsecs across. It has a mass equal to more than As we have said, this description is just to give
1011 Suns. When we look at larger scales, we will you a flavor of the sizes involved. The individual
see that galaxies are like the molecules of the objects will be discussed in detail throughout
universe. this book.
Part I
Properties of ordinary stars
Chapter 2




Continuous radiation from stars

that the eye has a large dynamic range; this range
2.1 Brightness of starlight is achieved at a sacrifice in our ability to discrimi-
nate small brightness differences.)
When we look at the sky, we note that some stars The next step was to make the scale continu-
appear brighter than others. At this point we are ous, so that, for example, we could accurately
not concerned with what causes these brightness describe the brightness of a star that is between
differences. (They may result from stars actually second and third magnitude. In addition we would
having different power outputs, or from stars being like to extend the scale, so that the brightnesses
at different distances.) All we know at first glance is of stars that we can see only through telescopes
that stars appear to have different brightnesses. can be included. It was found that a difference of
We would like to have some way of quantify- five magnitudes corresponds to a factor of 100 in
ing the observed brightnesses of stars. When we brightness. In setting up the magnitude scale,
speak loosely of brightness, we are really talking this relation is defined to be exact.
about the energy flux, f, which is the energy per Let b1 and b2 be the observed brightnesses of
two stars, and let m1 and m2 be the corresponding
unit area per unit time received from the star.
This can be measured with current instruments magnitudes. The statement that a five-magnitude
(as we will discuss in Chapter 4). However, the difference gives a flux ratio of 100 corresponds to
study of stellar brightness started long before
b1>b2 1001m2 m1 2>5
(2.1)
such instruments, or even telescopes, were avail-
able. Ancient astronomers made naked eye esti- We can see that this equation guarantees that
mates of brightness. Hipparchus, the Greek each time m2 m1 increases by five, b1/b2
astronomer, and later Ptolemy, a Greek living in decreases by a factor of 100. Remember, increasing
Alexandria, Egypt, around 150 BC, divided stars the brightness decreases the magnitude. This
into six classes of brightness. These classes were point sometimes confuses even professional
called magnitudes. This was an ordinal arrange- astronomers. That is why you will often hear
ment, with first-magnitude stars being the bright- astronomers talking about being so many magni-
est and sixth-magnitude stars being the faintest. tudes “brighter” or “fainter” than something else,
When quantitative measurements were made, without worrying about whether that makes m
it was found that each jump of one magnitude larger or smaller.
corresponded to a fixed flux ratio, not a flux differ- Equation (2.1) gives brightness ratios in pow-
ence. Because of this, the magnitude scale is essen- ers of 100, but we usually work in powers of ten.
To convert this we write 100 as 102, so equation
tially a logarithmic one. This is not too surprising,
since the eye is approximately logarithmic in its (2.1) becomes
response to light. This type of response allows us to
b1>b2 101m2 m1 2>2.5
(2.2)
see in very low and very high light levels. (We say
10 PART I PROPERTIES OF ORDINARY STARS




»
This equation can be used to calculate the
brightness ratio for a given magnitude difference.
If we want to calculate a magnitude difference for
a given brightness ratio, we take the logarithm
Distance
(base 10) of both sides, giving




Intensity
m2 m1 2.5 log10(b1/b2) (2.3)

To see how this works, let™s look at a few sim-
ple examples. On the original scale, the magni-
tude range for stars visible to the naked eye is 1 to
6 mag. This corresponds to a brightness ratio
10(6 1)/2.5
102
b1/b2 Fig 2.2. The wavelength is the distance between the
corresponding points of a wave in successive cycles. For
The largest ground-based telescopes extend example, it can be from peak to peak.
our range from 6 to 26 mag. This corresponds to
an additional brightness ratio
10(26 6)/2.5
108
b1/b2 different colors. We call this range of colors the
visible spectrum. These colors have different wave-
We can also find the magnitude difference,
lengths (Fig. 2.2). For example, the red light has a
m, corresponding to a factor of 106 in brightness:
wavelength around 650 nm ( 650 10 9 m
2.5 log10(106)
m (2.6)(6) mag 15 mag 6.5 10 7 m 6.5 10 5 cm). (We used to
express this in terms of angstrom units, after the
So, we have taken the original six magnitude
Swedish physicist A. J. …ngstrom, but this is not
groups and come up with a continuous scale that
part of the official metric system. The angstrom
can be extended to fainter or brighter objects.
was a convenient unit, since it is about the size of
Objects brighter than magnitude 1 can have mag-
a typical atom.) At the opposite end of the visible
nitude 0 or even negative magnitudes.
spectrum from red is violet, with a wavelength of
about 400 nm.
2.2 The electromagnetic spectrum In a vacuum, all wavelengths of light travel at
the same speed c 3.0 1010 cm/s (3.0 108 m/s,
3.0 105 km/s). At this speed light can travel a dis-
Thomas Young first demonstrated interference
tance equal to the Earth™s circumference 7.5 times
effects in light, showing that light is a wave phe-
per second. A light pulse take 1.3 s to reach the
nomenon. If we pass light through a prism (Fig.
Moon. The speed of light is so large that measur-
2.1), we can see that the light is spread out into
ing it requires the accurate measurement of time
over short intervals, or the passage of light over
long distances. Until late in the 19th century, the
large distances between astronomical objects
were used to provide reasonably long travel times.
More recently, accurate timing devices have made
Red
t
ligh laboratory measurements feasible.
Orange
hite Yellow
W
All waves have a frequency associated with
Green
Blue
them. The frequency tells us the number of oscil-
Violet
lations per second, or the number of crests that
pass per second. The product of the wavelength
Fig 2.1. The colors of visible light.When light passes
and the frequency gives the speed of the wave.
through a prism, the rays of different colors are de¬‚ected by
That is,
different amounts.The colors are listed, from top to bottom,
in order of decreasing wavelength.
c (2.4)
2 CONTINUOUS RADIATION FROM STARS 11




The electromagnetic spectrum.
Table 2.1.
Region Wavelength Frequency (Hz)

1011
Radio 1 mm 3
3 1011“4.3 1014
Infrared 700 nm“1 mm
4.3 1014 “7.5 1014
Visible 400“700 nm
7.5 1014 “3 1016
Ultraviolet 10“400 nm
3 1016 “3 1018
X-ray 0.1“10 nm
3 1018
Gamma-ray 0.1 nm



The higher the frequency, the shorter the through space, even empty space. All wavelengths
wavelength. For example, we can find the fre- are possible. The speed of these waves can be pre-
quency for light at a wavelength of 600 nm: dicted from Maxwell™s equations. The speed of
these waves in a vacuum is the same at all wave-
c>
lengths, and turns out to be numerically equal to
108 m>s
3.0 c, the speed of light. Light is just one form of elec-
9 tromagnetic wave. Other forms have wavelengths
600 10 m
that fall in different ranges.
1014 cps
5.0
The full set of electromagnetic waves is called
the electromagnetic spectrum (see Table 2.1). The vis-
For 1 cycle per second (cps), we use the unit
ible spectrum is just a small part of the electro-
1 hertz (Hz).
magnetic spectrum. At longer wavelengths are
When we talk about light waves with the above
1014 infrared and radio waves. At shorter wavelengths
frequency, what is actually varying at 5
are the ultraviolet, X-ray and gamma-ray parts of
cycles per second? This question was answered
the spectrum. Even though there is no difference
more than 100 years ago by James Clerk Maxwell,
between the waves in various parts of the spec-
who pointed out the unity between electric and
trum, we use the divisions because different tech-
magnetic fields. The behavior of these fields,
niques are used to detect electromagnetic waves
and their relationship to charged particles is
in various wavelength ranges. For example, our
described by four equations known as Maxwell™s
eyes are sensitive to wavelengths between 400 nm
equations.
and 700 nm. This is not too surprising, since this
In these equations, Maxwell was mostly sum-
is where the Sun gives off most of its energy. It
marizing the work of others, but it was he who
makes sense that we have evolved with our eyes
put the whole picture together. For example, one
able to make the best use of the illuminating
of Maxwell™s equations is Faraday™s law of induc-
light.
tion, which describes how a changing magnetic
We now know that astronomical objects give
field can produce an electric field. (This is the
off radiation in all parts of the spectrum.
basis for the production of electricity in a gener-
However, the Earth™s atmosphere limits what we
ator.) Maxwell realized that if there is a symmetry
can actually detect (Fig. 2.3). Ultraviolet and
between electric and magnetic fields, then a vary-
shorter wavelengths are blocked by the atmos-
ing electric field should be able to produce a mag-
phere. Visible light passes through the clear
netic field.
atmosphere (but is blocked by clouds). Most
This realization serves as the basis for our
infrared wavelengths are blocked by the atmos-
understanding of electromagnetic waves. An electric
phere, but some wavelengths get through. For the
field that varies sinusoidally (as a sine wave) pro-
most part, radio waves pass through the atmos-
duces a sinusoidally varying magnetic field,
phere with little absorption. We speak of visible
which in turn produces a varying electric field,
and radio windows in the atmosphere, as well as
and so on. These varying fields can propagate
12 PART I PROPERTIES OF ORDINARY STARS



100%




Atmospheric Transmission

0%
0.3 0.4 0.6 0.8 1 2 3 4 6 8 10
Wavelength (mm)

100%
Transmission




0
0.1 1 10 100 1mm 1cm 10cm 1m 10m 100m

UV Vis. IR Radio
Fig 2.3. Atmospheric transmission as a function of wave-
length.The curve shows the fraction of transmission at each
wavelength. Note the good transmission in the radio and vis-
2.3 Colors of stars
ible parts of the spectrum. Also note a few narrow ranges,
or “windows” of relatively good transmission in the infrared.
2.3.1 Quantifying color
[IRAM]
When we look at a star, we would like to know how
much energy it gives off at various wavelengths.
We sometimes refer to a graph, or some equivalent
representation, showing intensity as a function of
wavelength (or frequency) as a spectrum. It is not
some narrow windows in the infrared. A window
really proper to talk about the energy given off at
is simply a wavelength range in which the atmos-
a particular wavelength. If we can specify a wave-
phere is at least partially transparent.
length to an arbitrary number of decimal places
Until relatively recently, astronomers could
then even a small wavelength range has an infinite
only gather information in the visible part of the
number of wavelengths. If there was even a little
spectrum, because of the lack of equipment.
energy “at” each wavelength, then there would be
Much of the development of astronomy was
an infinite amount of energy.
biased by this handicap. In the middle of the 20th
Instead, we talk about the energy given off
century, the radio part of the spectrum was opened
over some wavelength (or frequency) range. For
for astronomical observations (taking advantage
example, we define the intensity function I( )
of equipment developed for radar in WW II). Even
such that I( ) d is the energy/unit time/unit sur-
more recently, other parts of the spectrum have
face area given off by an object in the wavelength
become available to us, due in part to observato-
d . Similarly, I( ) d is the energy/
range to
ries orbiting the Earth. Observing in various parts
unit time/unit surface area given off by an object
of the spectrum will be discussed throughout this
in the frequency range to d.
book.
2 CONTINUOUS RADIATION FROM STARS 13



Fig 2.4. Star cluster H and Chi
Persei. (We will talk more about
clusters of stars in Chapter 13.)
Notice the wide range of star
colors. [NOAO/AURA/NSF]




When we make a plot of I( ) vs. for a star we with a quantitative way of determining the color of
a star and therefore its temperature.
find that the graph varies smoothly over most
wavelengths. There are some wavelength ranges
2.3.2 Blackbodies
at which there is a sharp increase or decrease in
I( ) over a very narrow wavelength range. These We can understand the relationship between color
sharp increases and decreases are called spectral and temperature by considering objects called
blackbodies. A blackbody is a theoretical idea that
lines and will be discussed in the next chapter. In
closely approximates many real objects in thermo-
this chapter, we will be concerned with the smooth
dynamic equilibrium. (We say that an object is in
or continuous part of the spectrum. This is also
called the continuum. thermodynamic equilibrium with its surround-
ings when energy is freely interchanged and a
When we look at stars we see that they have
steady state is reached in which there is no net
different colors. Stars with different colors have
energy flow. That is, energy flows in and out at
different continuous spectra. In Fig. 2.4, we look
the same rate.) A blackbody is an object that
at a cluster of stars, and note a wide range of col-
absorbs all of the radiation that strikes it.
ors. If we took a continuous spectrum of various
A blackbody can also emit radiation. In fact, if
colored stars, we would find that stars that
a blackbody is to maintain a constant tempera-
appear blue have continuous spectra that peak in
ture, it must radiate energy at the same rate that
the (shorter wavelength) blue. The color of a star
it absorbs energy. If it radiates less energy than it
depends on its temperature. We know that as we
absorbs, it will heat up. If it radiates more energy
heat an object, first it glows in the red, then turns
than it absorbs, then it will cool. However, this
yellow/green, and then it turns blue as it becomes
does not mean that the spectrum of emitted radi-
even hotter.
ation must match the spectrum of absorbed radi-
We can therefore measure the temperature of a
ation. Only the total energies must balance. The
star by measuring its continuum. In fact, it is not
spectrum of emitted radiation is determined by
necessary to measure the whole spectrum in detail.
the temperature of the blackbody. As the temper-
We can measure the amounts of radiation received
ature changes, the spectrum changes. The black-
in certain wavelength ranges. These ranges are
defined by filters that let a given wavelength range body will adjust its temperature so that its emitted
spectrum contains just enough energy to balance
pass through. By comparing the intensity of radia-
the absorbed energy. When the temperature
tion received in various filters, we can come up
14 PART I PROPERTIES OF ORDINARY STARS



“3
(a) (b)




Log Intensity
Intensity
“ 4.5



Log B (5000 K)
B (5000 K)
Log B (10000 K)
B (10000 K)
Log B (15000 K)
B (15000 K)
“6
13 14 15 16
0 1 2 3
Log Frequency
(1015 Hz)
Frequency
Fig 2.5. Blackbody spectra. Note the shift of the peak
wavelength to higher frequency (shorter wavelength) at
higher temperature. Note also that, at any frequency, a hot- part of the spectrum, 550 nm. (b) The Earth has
ter blackbody gives off more radiation than a cooler one. an average temperature of about 300 K. At what
(a) Intensity as a function of frequency. Notice the big wavelength does the Earth™s blackbody spectrum
change in intensity with only a factor of three change in
peak?
temperature. For this reason, we often ¬nd it useful to make
a plot such as the set of curves in (b), which show the log of
SOLUTION
the intensity as a function of the log of the frequency.
(a) Given the wavelength, we solve equation (2.5) for
the temperature:
which allows this balance is reached, the black-
106 nm K
2.9
body is in equilibrium. T 5270 K
550 nm
Figure 2.5 shows some sample blackbody spec-
tra. If we compare these spectra to those of actual This is close to the temperature of the Sun.
stars, we see that the actual spectra are very much (b) Given the temperature, we solve equation (2.5)
like blackbody spectra. Notice that in any wave- for the wavelength:
length range, a hotter blackbody gives off more
106 nm K
2.9
energy than a cooler blackbody of the same size.
300 K
We also see that as the temperature increases the
104 nm
peak of the spectrum shifts to shorter wavelengths. 1
The relationship between the wavelength at 6
10 10 m
which the peak occurs, max, and temperature, T,
is very simple. It is given by Wien™s displacement 10 m
law:
This is in the infrared part of the spectrum.
1 6
maxT 2.90 10 cm K 2.90 10 nm K (2.5) Even though the Earth is giving off radiation, we
don™t see it glowing in the visible part of the
In this law, we must use temperature on an
spectrum. Similarly, objects around us that are at
absolute (Kelvin) scale. (The temperature on the
essentially the same temperature as the Earth
Kelvin scale is the temperature on the Celsius
give off most of their radiation in the infrared
scale plus 273.1.)
part of the spectrum, with very little visible light.
Example 2.1 Using Wien™s displacement law The visible light that we see from surrounding
(a) Find the temperature of an object whose black- objects is partially reflected sunlight or artificial
body spectrum peaks in the middle of the visible light.
2 CONTINUOUS RADIATION FROM STARS 15



quantity T4 is only the energy per second per
We could have solved (b) by scaling a known result,
such as the answer in (a): unit surface area. Therefore, to obtain the lumi-
nosity, we must multiply it by the surface area. If
TA
Earth
the star is a sphere with radius R, the surface area
TEarth
A
is (4 R2 ), so the luminosity is

a b
TA
(4 R2)( T 4)
L (2.7)
Earth A
TEarth
Example 2.2 Luminosity of the Sun
a b 1550 nm2
5270 K The surface temperature of the Sun is about 5800 K
300 K and its radius is 7 105 km (7 1010 cm). What is
the luminosity of the Sun?
103 nm
10

10 m SOLUTION
We use equation (2.7) to find the luminosity:
Scaling results can be useful because they show
how different physical parameters are related to 1010 cm)2 [5.7 5
erg/(cm2 K4 s)]
L 4 (7 10
each other. It also provides us with a way of using (5.8 103 K)4
an equation even if we don™t remember the
1033 erg/s.
4
constants.
This quantity is called the solar luminosity, L , and
Suppose we are interested in the total energy
serves as a convenient unit for expressing the
given off by a blackbody (per unit time per unit
luminosities of other stars.
surface area) over the whole electromagnetic
spectrum. We must add the contributions at all
wavelengths. This amounts to taking an integral
2.4 Planck™s law and photons
over blackbody curves, such as those in Fig. 2.5.
Since a hotter blackbody gives off more energy at
2.4.1 Planck™s law
all wavelengths than a cooler one, and is particu-
The study of blackbody radiation plays an impor-
larly dominant at shorter wavelengths, we would
tant role in the development of what we refer to
expect a hotter blackbody to give off much more
as “modern” physics (even though these develop-
energy than a cooler one. Indeed, this is the case.
ments took place early in the 20th century).
The total energy per unit time, per unit surface
When physicists tried to apply classical ideas of
area, E, given off by a blackbody is proportional to
radiation, they could not derive blackbody spec-
the fourth power of the temperature. That is
tra that agreed with the experimental results.
T4
E (2.6) The classical calculations yielded an intensity
I( , T) given by
This relationship is called the Stefan“Boltzmann
law. The constant of proportionality, , is called 2kT 2/c2
I( , T) (2.8)
the Stefan“Boltzmann constant. It has a value of
5.7 10 5 erg/(cm2 K4 s). This law was first deter- This is known as the Rayleigh“Jeans law. The con-
stant k that appears in this law is the Boltzmann con-
mined experimentally, but it can also be derived

. 1
( 28)



>>