Astrophysics Decoding the Cosmos
Judith A. Irwin Queen’s University, Kingston, Canada
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Judith A. Irwin Queen’s University, Kingston, Canada Copyright ß 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) 1243 779777
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Irwin, Judith A., 1954- Astrophysics : decoding the cosmos / Judith A. Irwin. p. cm. ISBN 978-0-470-01305-2 (cloth) 1. Astrophysics. I. Title. QB461.I79 2007 523.01–dc22 2007009980
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Contents
Preface xiii Acknowledgments xv Introduction xvii Appendix: dimensions, units and equations xxii
PART I THE SIGNAL OBSERVED
1 Defining the signal 3 1.1 The power of light – luminosity and spectral power 3 1.2 Light through a surface – flux and flux density 7 1.3 The brightness of light – intensity and specific intensity 9 1.4 Light from all angles – energy density and mean intensity 13 1.5 How light pushes – radiation pressure 16 1.6 The human perception of light – magnitudes 19 1.6.1 Apparent magnitude 19 1.6.2 Absolute magnitude 22 1.6.3 The colour index, bolometric correction, and HR diagram 23 1.6.4 Magnitudes beyond stars 24 1.7 Light aligned – polarization 25 Problems 25
2 Measuring the signal 29 2.1 Spectral filters and the panchromatic universe 29 2.2 Catching the signal – the telescope 32 2.2.1 Collecting and focussing the signal 34 2.2.2 Detecting the signal 36 2.2.3 Field of view and pixel resolution 37 2.2.4 Diffraction and diffraction-limited resolution 38 viii CONTENTS
2.3 The Corrupted signal – the atmosphere 41 2.3.1 Atmospheric refraction 42 2.3.2 Seeing 43 2.3.3 Adaptive optics 45 2.3.4 Scintillation 48 2.3.5 Atmospheric reddening 48 2.4 Processing the signal 49 2.4.1 Correcting the signal 49 2.4.2 Calibrating the signal 50 2.5 Analysing the signal 50 2.6 Visualizing the signal 52 Problems 55 Appendix: refraction in the Earth’s atmosphere 58
PART II MATTER AND RADIATION ESSENTIALS
3 Matter essentials 65 3.1 The Big Bang 65 3.2 Dark and light matter 66 3.3 Abundances of the elements 70 3.3.1 Primordial abundance 70 3.3.2 Stellar evolution and ISM enrichment 70 3.3.3 Supernovae and explosive nucleosynthesis 75 3.3.4 Abundances in the Milky Way, its star formation history and the IMF 77 3.4 The gaseous universe 82 3.4.1 Kinetic temperature and the Maxwell–Boltzmann velocity distribution 84 3.4.2 The ideal gas 88 3.4.3 The mean free path and collision rate 89 3.4.4 Statistical equilibrium, thermodynamic equilibrium, and LTE 93 3.4.5 Excitation and the Boltzmann Equation 96 3.4.6 Ionization and the Saha Equation 100 3.4.7 Probing the gas 101 3.5 The dusty Universe 104 3.5.1 Observational effects of dust 105 3.5.2 Structure and composition of dust 109 3.5.3 The origin of dust 111 3.6 Cosmic rays 112 3.6.1 Cosmic ray composition 112 3.6.2 The cosmic ray energy spectrum 113 3.6.3 The origin of cosmic rays 117 Problems 118 Appendix: the electron/proton ratio in cosmic rays 121
4 Radiation essentials 123 4.1 Black body radiation 123 4.1.1 The brightness temperature 127 4.1.2 The Rayleigh–Jeans Law and Wien’s Law 129 4.1.3 Wien’s Displacement Law and stellar colour 130 CONTENTS ix
4.1.4 The Stefan–Boltzmann Law, stellar luminosity and the HR diagram 132 4.1.5 Energy density and pressure in stars 134 4.2 Grey bodies and planetary temperatures 134 4.2.1 The equilibrium temperature of a grey body 136 4.2.2 Direct detection of extrasolar planets 140 Problems 143 Appendix: derivation of the Planck function 146 4.A.1 The statistical weight 146 4.A.2 The mean energy per state 147 4.A.3 The specific energy density and specific intensity 148
PART III THE SIGNAL PERTURBED
5 The interaction of light with matter 153 5.1 The photon redirected – scattering 154 5.1.1 Elastic scattering 157 5.1.2 Inelastic scattering 165 5.2 The photon lost – absorption 168 5.2.1 Particle kinetic energy – heating 168 5.2.2 Change of state – ionization and the Stro¨mgren sphere 169 5.3 The wavefront redirected – refraction 172 5.4 Quantifying opacity and transparency 175 5.4.1 Total opacity and the optical depth 175 5.4.2 Dynamics of opacity – pulsation and stellar winds 178 5.5 The opacity of dust – extinction 182 Problems 183
6 The signal transferred 187 6.1 Types of energy transfer 187 6.2 The equation of transfer 189 6.3 Solutions to the equation of transfer 191 6.3.1 Case A: no cloud 191 6.3.2 Case B: absorbing, but not emitting cloud 192 6.3.3 Case C: emitting, but not absorbing cloud 192 6.3.4 Case D: cloud in thermodynamic equilibrium (TE) 193 6.3.5 Case E: emitting and absorbing cloud 193 6.3.6 Case F: emitting and absorbing cloud in LTE 194 6.4 Implications of the LTE solution 195 6.4.1 Implications for temperature 195 6.4.2 Observability of emission and absorption lines 196 6.4.3 Determining temperature and optical depth of HI clouds 200 Problems 204 7 The interaction of light with space 207 7.1 Space and time 207 7.2 Redshifts and blueshifts 210 7.2.1 The Doppler shift – deciphering dynamics 210 7.2.2 The expansion redshift 217 7.2.3 The gravitational redshift 219 x CONTENTS
7.3 Gravitational refraction 220 7.3.1 Geometry and mass of a gravitational lens 220 7.3.2 Microlensing – MACHOs and planets 225 7.3.3 Cosmological distances with gravitational lenses 226 7.4 Time variability and source size 227 Problems 228
PART IV THE SIGNAL EMITTED
8 Continuum emission 235 8.1 Characteristics of continuum emission – thermal and non-thermal 236 8.2 Bremsstrahlung (free–free) emission 237 8.2.1 The thermal Bremsstrahlung spectrum 237 8.2.2 Radio emission from HII and other ionized regions 242 8.2.3 X-ray emission from hot diffuse gas 245 8.3 Free–bound (recombination) emission 251 8.4 Two-photon emission 253 8.5 Synchrotron (and cyclotron) radiation 255 8.5.1 Cyclotron radiation – planets to pulsars 258 8.5.2 The synchrotron spectrum 262 8.5.3 Determining synchrotron source properties 268 8.5.4 Synchrotron sources – spurs, bubbles, jets, lobes and relics 270 8.6 Inverse Compton radiation 273 Problems 276 9 Line emission 279 9.1 The richness of the spectrum – radio waves to gamma rays 280 9.1.1 Electronic transitions – optical and UV lines 280 9.1.2 Rotational and vibrational transitions – molecules, IR and mm-wave spectra 281 9.1.3 Nuclear transitions – g-rays and high energy events 286 9.2 The line strengths, thermalization, and the critical gas density 288 9.3 Line broadening 290 9.3.1 Doppler broadening and temperature diagnostics 291 9.3.2 Pressure broadening 295 9.4 Probing physical conditions via electronic transitions 296 9.4.1 Radio recombination lines 297 9.4.2 Optical recombination lines 302 9.4.3 The 21 cm line of hydrogen 307 9.5 Probing physical conditions via molecular transitions 311 9.5.1 The CO molecule 311 Problems 313
PART V THE SIGNAL DECODED
10 Forensic astronomy 319 10.1 Complex spectra 319 CONTENTS xi
10.1.1 Isolating the signal 319 10.1.2 Modelling the signal 321 10.2 Case studies – the active, the young, and the old 325 10.2.1 Case study 1: the Galactic Centre 326 10.2.2 Case study 2: the Cygnus star forming complex 329 10.2.3 Case study 3: the globular cluster, NGC 6397 331 10.3 The messenger and the message 335 Problems 336 Appendix A: Mathematical and geometrical relations 339 A.1 Taylor series 339 A.2 Binomial expansion 339 A.3 Exponential expansion 340 A.4 Convolution 340 A.5 Properties of the ellipse 340
Appendix B: Astronomical geometry 343 B.1 One-dimensional and two-dimensional angles 343 B.2 Solid angle and the spherical coordinate system 344
Appendix C: The hydrogen atom 347 C.1 The hydrogen spectrum and principal quantum number 347 C.2 Quantum numbers, degeneracy, and statistical weight 352 C.3 Fine structure and the Zeeman effect 353 C.4 The l 21 cm line of neutral hydrogen 354
Appendix D: Scattering processes 357 D.1 Elastic, or coherent scattering 358 D.1.1 Scattering from free electrons – Thomson scattering 358 D.1.2 Scattering from bound electrons I: the oscillator model 359 D.1.3 Scattering from bound electrons II: quantum mechanics 362 D.1.4 Scattering from bound electrons III: resonance scattering and the natural line shape 363 D.1.5 Scattering from bound electrons IV: Rayleigh scattering 366 D.2 Inelastic scattering – Compton scattering from free electrons 367 D.3 Scattering by dust 369 Appendix E: Plasmas, the plasma frequency, and plasma waves 373
Appendix F: The Hubble relation and the expanding Universe 377 F.1 Kinematics of the Universe 377 F.2 Dynamics of the Universe 383 F.3 Kinematics, dynamics and high redshifts 386
Appendix G: Tables and figures 389 References 401 Index 407
Preface
Like many textbooks, this one originated from lectures delivered over a number of years to undergraduate students at my home institution – Queen’s University in Kingston, Canada. These students had already taken a first year (or two) of physics and one introductory astronomy course. Thus, this book is aimed at an intermediate level and is meant to be a stepping stone to more sophisticated and focussed courses, such as stellar structure, physics of the interstellar medium, cosmology, or others. The text may also be of some help to beginning graduate students with little background in astronomy or those who would like to see how physics is applied, in a practical way, to astronomical objects. The astronomy prerequisite is helpful, but perhaps not required for students at a more senior level, since I make few assumptions as to prior knowledge of astronomy. I do assume that students have some familiarity with celestial coordinate systems (e.g. Right Ascension and Declination or others), although it is not necessary to know the details of such systems to understand the material in this text. I also do not provide any explanation as to how astronomical distances are obtained. Distances are simply assumed to be known or not known, as the case may be. I provide some figures that are meant to help with ‘astronomical geography’, but a basic knowledge of astronom- ical scales would also be an asset, such as understanding that the Solar System is tiny in comparison to the Galaxy and rotates about the Galactic centre. As for approach, I had several goals in mind while organizing this material. First of all, I did not want to make the book too ‘object-oriented’. That is, I did not want to write a great deal of descriptive material about specific astronomical objects. For one thing, astronomy is such a fast-paced field that these descriptions could easily and quickly become out of date. And for another, in the age of the internet, it is very easy for students to quickly download any number of descriptions of various astronomical objects at their leisure. What is more difficult is finding the thread of physics that links these objects, and it is this that I wanted to address. xiv PREFACE
Another goal was to keep the book practical, focussing on how we obtain informa- tion about our Universe from the signal that we actually detect. In the process, many equations are presented. While this might be a little intimidating to some students, the point should be made that the equations are our ‘tools of the trade’. Without these tools, we would be quite helpless, but with them, we have access to the secrets that astronomical signals bring to us. With the increasing availability of computer algebra or other software, there is no longer any need to be encumbered by mathematics. Nevertheless, I have kept problems that require computer-based solutions to a mini- mum in this text, and have tried to include problems over a range of difficulty. Astrophysics – Decoding the Cosmos will maintain a website at http://decoding.phy.queensu.ca. A Solutions manual to the problems is also available. I invite readers to visit the website and submit some problems of their own so that these can be shared with others. It is my sincere desire that this book will be a useful stepping-stone for students of astrophysics and, more importantly, that it may play a small part in illuminating this most remarkable and marvelous Universe that we live in.
Judith A. Irwin Kingston, Ontario October 2006 Acknowledgments
There are some important people that I feel honoured to thank for their help, patience, and critical assistance with this book. First of all, to the many people who generously allowed me to use their images and diagrams, I am very grateful. Astronomy is a visual science and the impact of these images cannot be overstated. Thanks to the students, past and present, of Physics 315 for their questions and suggestions. I have more than once had to make corrections as a result of these queries and appreciate the keen and lively intelligence that these students have shown. Thanks to Jeff Ross for his assistance with some of the ‘nuts and bolts’ of the references. And special thanks to Kris Marble and Aimee Burrows for their many contributions, working steadfastly through problems, and offering scientific expertise. With much gratitude, I wish to acknowledge those individuals who have read sections or chapters of this book and offered constructive criticism – Terry Bridges, Diego Casadei, Mark Chen, Roland Diehl, Martin Duncan, David Gray, Jim Peebles, Ernie Seaquist, David Turner and Larry Widrow. To my dear children, Alex and Irene, thank you for your understanding and good cheer when mom was working behind a closed door yet again, and also thanks to the encouragement of friends – Joanne, Wendy, and Carolyn. My tenderest thanks to my husband, Richard Henriksen, who not only suggested the title to this book, but also read through and critiqued the entire manuscript. For patience, endurance, and gentle encouragement, I thank you. It would not have been accomplished without you.
J.A.I.
Introduction
Knowledge of our Universe has grown explosively over the past few decades, with discoveries of cometary objects in the far reaches of our Solar System, new-found planets around other stars, detections of powerful gamma ray bursts, galaxies in the process of formation in the infant Universe, and evidence of a mysterious force that appears to be accelerating the expansion of the Universe. From exotic black holes to the microwave background, the modern understanding of our larger cosmological home could barely be imagined just a generation ago. Headlines exclaim astonishing proper- ties for astronomical objects – stars with densities equivalent to the mass of the sun compressed to the size of a city, million degree gas temperatures, energy sources of incredible power, and luminosities as great as an entire galaxy from a single dying star. How could we possibly have reached these conclusions? How can we dare to describe objects so inconceivably distant that the only influence they have on our lives is through our very astonishment at their existence? With the exception of a few footprints on the Moon, no human being has ever ventured beyond the confines of the Earth. No spaceprobe has ever approached a star other than the Sun, let alone returned with material evidence. Yet we continue to amass information about our Universe and, indeed, to believe it. How? In contrast to our attempts to reach into the Universe with space probes and radio beacons, the natural Universe itself has been continuously and quite effectively reach- ing us. The Earth has been bombarded with astronomical information in its various forms and in its own language. The forms that we think we understand are those of matter and radiation. Our challenge, in the absence of an ability to travel amongst the stars, is to find the best ways to detect and decipher such communications. What astronomical matter is reaching us? The high energy subatomic particles and nuclei which make up cosmic rays that continuously bombard the earth (Sect. 3.6) as well as an influx of meteoritic dust, occasional meteorites and those rare objects that xviii INTRODUCTION
Figure I.1 (a) Early photograph of cosmic ray tracks in a cloud chamber. (Ref. [30]) (b) The 1.8 km diameter Lonar meteorite crater in India are large enough to create impact craters. Such incoming matter provides us with information on a variety of astronomical sources, including our Sun, our Solar System, supernovae in the Galaxy1, and other more mysterious sources of the highest energy cosmic rays. The striking contrast in size and effect between such particulate matter is illustrated in Figure I.1. Radiation refers to electromagnetic radiation (or just ‘light’) over all wavebands from the radio to the gamma-ray part of the spectrum (Table G.6). Electromagnetic radiation can be described as a wave and identified by its wavelength, l or its frequency, n. However, it can also be thought of as a massless particle called a photon which has a particular energy, Eph. This energy can be expressed in terms of wavelength or frequency, Eph ¼ hc=l ¼ h n, where h is Planck’s constant. The wavelengths, frequencies and photon energies of various wavebands are given in Table G.6. The wave–particle duality of light is a deep issue in physics and related via the concept of probability. To quote Max Born, ‘the wave and corpuscular descriptions are only to be regarded as complementary ways of viewing one and the same objective process, a process which only in definite limiting cases admits of complete pictorial interpretation’ (Ref. [23]). Although, in principle, it may be possible to understand a physical process involving light from both points of view, there are some problems that are more easily addressed by one approach rather than another. For example, it is often more straightforward to consider waves when dealing with an interaction between light and an object that is small in comparison to the wavelength, and to consider photons when dealing with an interaction between light and an object that is large in comparison to the equivalent photon wavelength, l ¼ hc=Eph. Given this wave-particle duality, in this text we will apply whatever form is most useful for the task at hand. Some helpful expressions relating various properties of electromagnetic radiation are provided in Table I.1 and a diagram illustrating the wave nature of light is shown in Figure I.2.
1When ‘Galaxy’ is written with a capital G, it refers to our own Milky Way galaxy. Table I.1. Useful expressions relevant to light, matter, and fieldsa
Meaning Equation
Relation between wavelength and frequency c ¼ ln pffiffiffiffiffiffiffiffiffi1 Lorentz factor g ¼ 2 1 v c2 hc Energy of a photon E ¼ h n ¼ l Equivalent mass of a photon m ¼ E=c2 Equivalent wavelength of a mass (de Broglie wavelength) l ¼ h=ðm vÞ Momentum of a photon p ¼ E=c Momentum of a particle p ¼ g m v b Snell’s law of refraction n1 sin u1 ¼ n2 sin u2 c Index of refraction n ¼ c=v Dl l l 1 þ vr 1=2 Doppler shiftd ¼ obs 0 ¼ c 1 l l 1 vr 0 0 c Dn n n 1 vr 1=2 obs 0 c 1 ¼ ¼ vr n0 n0 1 þ c Dl v Dn v r ; r ; ðv cÞ l0 c n0 c Electric field vector ~E ¼ÀÁ~F=q Electric field vector of a wavee ~E ¼ E~ cos½2 p z n t þ Df 0 ÀÁl Magnetic field vector of a wavee ~B ¼ ~B cos½2 p z n t þ Df 0 ÀÁl f ~ c ~ ~ Poynting flux S ¼ 4 p E B c Time averaged Poynting fluxf hSi¼ E B 8 p 0 0 g B2 Energy density of a magnetic field uB ¼ 8 p h ~ ~ ~v ~ Lorentz force F ¼ q ðE þ c BÞ Electric field magnitude in a parallel-plate capacitori E ¼ð4 p NeÞ=A Electric dipole momentj ~p ¼ q~r 2 dE 2 q 2 Larmor’s formula for powerk P ¼ ¼ j~r€j dt 3 c3 h Heisenberg Uncertainty Principlel D x D p ¼ x 2 p h D E D t ¼ 2 p m Mm Universal law of gravitation FG ¼ G r2 m v2 Centripetal forcen F ¼ c r a See Table A.3 for a list of symbols if they are not defined here. b If an incoming ray is travelling from medium 1 with index of refraction, n1, into medium 2 with index of refraction, n2,thenu1 is the angle between the incoming ray and the normal to the surface dividing the two media and u2 is the angle between the outgoing ray and the normal to the surface. c v is the speed of light in the medium (the phase velocity) and c is the speed of light in a vacuum. Note that the index of refraction may also be expressed as a complex number whose real part is given by this equation and whose imaginery part corresponds to an absorbed component of light. See Appendix D.3 for an example. d l0 is the wavelength of the light in the source’s reference frame (the ‘true’ wavelength), lobs is the wavelength in the observer’s reference frame (the measured wavelength), and vr is the relative radial velocity between the source and the observer. vr is taken to be positive if the source and observer are receding with respect to each other and negative if the source and observer are approaching each other. e The wave is propagating in the z direction and D f is an arbitrary phase shift. The magnetic field strength, H,isgivenbyB ¼ H m where m is the permeability of the substance through which the wave is travelling (unitless in the cgs system). For em radiation in a vacuum (assumed here and throughout), this becomes B ¼ H since the permeability of free space takes the value, 1, in cgs units. Thus B is often stated as the magnetic field strength, rather than the magnetic flux density and is commonly expressed in units of Gauss. In cgs units, E ðdyn esu 1Þ¼B (Gauss). f Energy flux carried by the wave in the direction of propagation. The cgs units are erg s 1 cm 2. The time-averaged value is over one cycle. g 3 2 uB has cgs units of erg cm or dyn cm (see Table A.2.). h Force on a charge, q with velocity, ~v, by an electric field, ~E and magnetic field, ~B. i Here Neis the charge on a plate and A is its area. In SI units, this equation would be E ¼ s=e0, where s is the charge per unit area on a plate and e0 is the permittivity of free space (where we assume that free space is between the plates). In cgs units, 4 pe0 ¼ 1. j ~r is the separation between the two charges of the dipole and q is the strength of one of the charges. The direction is negative to positive. k Power emitted by a non-relativistic particle of charge, q, that is accelerating at a rate, ~r€. l One cannot know the position and momentum (x; p) or the energy and time (E; t) of a particle or photon to arbitrary accuracy. mForce between two masses, M and m, a distance, r, apart. n Force on an object of mass, m, moving at speed, v, in a circular path of radius, r. xx INTRODUCTION
E z λ
E0
x S
y B0 B
Figure I.2 Illustration of an electromagnetic wave showing the electric field and magnetic field perpendicular to each other and perpendicular to the direction of wave propagation which is in the x direction. The wavelength is denoted by l.
If we now ask which of these information bearers provides us with most of our current knowledge of the universe, the answer is undoubtedly electromagnetic radia- tion, with cosmic rays a distant second. The world’s astronomical volumes would be empty indeed were it not for an understanding of radiation and its interaction with matter. The radiation may come directly from the object of interest, as when sunlight travels straight to us, or it may be indirect, such as when we infer the presence of a black hole by the X-rays emitted from a surrounding accretion disk. Even when we send out exploratory astronomical probes, we still rely on man-made radiation to transfer the images and data back to earth. This volume is thus largely devoted to understanding radiative processes and how such an understanding informs us about our Universe and the astronomical objects that inhabit it. It is interesting that, in order to understand the largest and grandest objects in the Universe, we must very often appeal to microscopic physics, for it is on such scales that the radiation is actually being generated and it is on such scales that matter interacts with it. We also focus on the ‘how’ of astronomy. How do we know the temperature of that asteroid? How do we find the speed of that star? How do we know the density of that interstellar cloud? How can we find the energy of that distant quasar? The answers are hidden in the radiation that they emit and, earthbound, we have at least a few keys to unlock their secrets. We can truly think of the detected signal as a coded message. To understand the message requires careful decoding. In the future, there may be new, exotic and perhaps unexpected ways in which we can gather information about our Universe. Already, we are seeing a trend towards such diversification. The decades-old ‘Solar neutrino problem’ has finally been solved from the vantage point of an underground mine in the Canadian shield (Figure I.3.a). Creative experiments are underway to detect the elusive dark matter that is believed to make up most of the mass of the Universe but whose nature is unknown (Sect. 3.2). Enormous international efforts are also underway to detect gravitational waves, weak perturbations in space-time predicted by Einstein’s General Theory of Relativity INTRODUCTION xxi
Figure I.3 (a) The acrylic tank of the Sudbury Neutrino Observatory (SNO) looks like a coiled snakeskin in this fish-eye view from the bottom of the tank before the bottom-most panel of photomultiplier tubes was installed. The Canadian-led SNO project determined that neutrinos change ‘flavour’ en route from the Sun’s interior, thus answering why previous experiments had detected fewer solar neutrinos than theory predicted. (Photo credit: Ernest Orlando, Lawrence Berkeley National Laboratory. Courtesy A. Mc Donald, SNO) (b) Aerial photo of the ’L’ shaped Laser Interferometer Gravitational-Wave Observatory (LIGO) in Livingston, Louisiana, of length 4 km on each arm. Together with its sister observatory in Hanford, Washington, these interferometers may detect gravitational waves, tiny distortions in space–time produced by accelerating masses. (Reproduced courtesy of LIGO, Livingston, Lousiana)
(Figure I.3.b). Our astronomical observatories are no longer restricted to lonely mountain tops. They are deep underground, in space, and in the laboratory. Thus, ‘decoding the cosmos’ is an on-going and evolving process. Here are a few steps along the path.
Problems
0.1 Calculate the repulsive force of an electron on another electron a distance, 1 m, away in cgs units using Eq. 0.A.1 and in SI units using the equation, q q F ¼ k 1 2 ðI:1Þ r2 where the charge on an electron, in SI units, is 1:6 10 19 Coulombs (C) and the constant, k ¼ 8:988 109 Nm2 C 2. Verify that the result is the same in the two systems.
0.2 Verify that the following equations have matching units on both sides of the equals sign: Eq. D.2 (the expression that includes the mass of the electron, me), Eq. 9.8 and Eq. F.19. xxii INTRODUCTION
0.3 Verify that the equation for the Ideal Gas Law (Table 0.A.2) is equivalent to P V ¼NRT, where N is the number of moles and R is the molar gas constant (see Table G.2).
Appendix
Dimensions, Units and Equations
When [chemists] had unscrambled the difficulties caused by the fact that chemists and engineers use different units ... they found that their strength predictions were not only frequently a thousandfold in error but bore no consistent relationship with experiment at all. After this they were inclined to give the whole thing up and to claim that the subject was of no interest or importance anyway. –The New Science of Strong Materials, by J. E. Gordon
The centimetre-gram-second (cgs) system of units is widely used by astronomers internationally and is the system adopted in this text. A summary of the units is given in Table 0.A.1 as well as corresponding conversions to Syste`me Internationale (SI). If an equation is given without units, the cgs system is understood. The same symbols are generally used in both systems (Table 0.A.3) and SI prefixes (Table 0.A.4) are also equally applied to the cgs system (note that the base unit, cm, already has a prefix).
Table 0.A.1. Selected cgs – SI conversionsa
Dimension cgs unit (abbrev.) Factor SI unitb (abbrev.) Length centimetre (cm) 10 2 metre (m) Mass gram (g) 10 3 kilogram (kg) Time second (s) 1 second (s) Energy erg (erg) 10 7 joule (J) Power erg second 1 (erg s 1)10 7 watt (W) Temperature kelvin (K) 1 kelvin (K) Force dyne (dyn) 10 5 newton (N) Pressure dyne centimetre 2 0.1 newton metre 2 (N m 2) (dyn cm 2) barye ðbaÞc 0.1 pascal (Pa) Magnetic flux density (field)d gauss (G) 10 4 tesla (T) Angle radian (rad) 1 radian (rad) Solid angle steradian (sr) 1 steradian (sr) a Value in cgs units times factor equals value in SI units. b Syste`me International d’Unite´s. c This unit is rarely used in astronomy in favour of dyn cm 2. d See notes to Table I.1. APPENDIX xxiii
Table 0.A.2. Examples of equivalent units
Equationa Name Units 1 erg P ¼ nkT Ideal Gas Law dyn cm 2 ¼ ðÞK cm3 K ¼ erg cm 3
F ¼ m a Newton’s Second Law dyn ¼ gcms 2 W ¼ F s Work Equation erg ¼ dyn cm ¼ gcm2 s 2 1 2 2 2 Ek ¼ 2 m v Kinetic Energy Equation erg ¼ gcm s 2 2 EPG ¼ m g h Gravitational Potential Energy Equation erg ¼ gcm s q q E ¼ 1 2 Electrostatic Potential Energy Equation erg ¼ esu2 cm 1 Pe r See Table 0.A.3 for the meaning of the symbols and Table G.2 for the meaning of the constants. Table 0.A.3. List of symbols
Symbol Meaning a radius, acceleration A atomic weight A area, albedo, total extinction B magnetic flux density, magnetic field strengtha BðTÞ intensity of a black body (or specific intensity if subscripted with n or l) Dp degree of polarization e charge of the electron E energy, selective extinction E electric field strength EM emission measure fi;j oscillator strength between levels, i and j f correction factor F; S; f flux (or flux density, if subscripted with n or l) F force gn statistical weight of level, n g Gaunt factor I intensity (or specific intensity, if subscripted with n or l) jn emission coefficient J mean intensity (or mean specific intensity, if subscripted with n or l) JðEÞ cosmic ray flux density per unit solid angle l mean free path L luminosity (or spectral luminosity, if subscripted with n or l) m, M apparent magnitude & absolute magnitude, respectively M; m mass n index of refraction, principal quantum number n, N number density and number of (object), respectively N map noise N number of moles, column density p momentum, electric dipole moment P power (or spectral power, if subscripted with n or l), probability P pressure q charge
(Continued) xxiv INTRODUCTION
Table 0.A.3. (Continued)
Symbol Meaning Q efficiency factor r, d, D, s, x, y, z, l,h,R Position, separation, or distance R Rydberg constant R collision rate S source function t time T kinetic energy T temperature T period u energy density (or spectral energy density, if subscripted with n or l) U, V, B, etc. apparent magnitudes U excitation parameter v velocity, speed V volume X; Y; Z mass fraction of hydrogen, helium and heavier elements, respectively z redshift, zenith angle Z atomic number a synchrotron spectral index, fine structure constant an absorption coefficient ar recombination coefficient g Lorentz factor gcoll collision rate coefficient G spectral index of cosmic ray power law dist’n, damping constant e permittivity, cosmological energy density u, f one-dimensional angle kn mass absorption coefficient l wavelength m permeability, mean molecular weight n frequency, collision rate per unit volume r mass density s cross-sectional area, Stefan-Boltzmann constant, Gaussian dispersion tn, t optical depth & timescale, respectively F line shape function x ionization potential v angular frequency V solid angle, cosmological mass or energy density a See notes to Table I.1.
Almost all equations used in this text look identical in the two systems and one need only ensure that the constants and input parameters are all consistently used in the adopted system. There are, however, a few cases in which the equation itself changes between cgs and SI. An example is the Coulomb (electrostatic) force,
q q F ¼ 1 2 ð0:A:1Þ r2 APPENDIX xxv
Table 0.A.4. SI prefixes
Name Prefix Factor yocto y 10 24 zepto z 10 21 atto a 10 18 femto f 10 15 pico p 10 12 nano n 10 9 micro m 10 6 milli m 10 3 centi c 10 2 deci d 10 1 kilo k 103 mega M 106 giga G 109 tera T 1012 peta P 1015 exa E 1018 zeta Z 1021 yota Y 1024
With the two charges, q1 and q2 expressed in electrostatic units (esu, see Table G.2), and the separation, r, in cm, the answer will be in dynes. Note that there is no constant of proportionality in this equation, unlike the SI equivalent (Prob. 0.1). Equations in the cgs system show the most difference with their SI equivalents when electric and magnetic quantities are used. For example, in the cgs system, the permittivity and permeability of free space, e0 and m0, respectively, are both unitless and equal to 1 (see also notes to Table I.1). Astronomers also have a number of units specific to the discipline. There are a variety of reasons for this, but the most common involves a process of normalisation. The value of some parameter is expressed in comparison to another known, or at least more familiar value. Some examples (see Table G.3) are the astronomical unit (AU) which is the distance between the Earth and the Sun. It is much easier to visualise the distance to Pluto as 40 AU than as 6 1014 cm. Expressing the masses of stars and galaxies in Solar masses (M ) or an object’s luminosity in Solar luminosities (L )is also very common. When such units are used, the parameter is often written as M=M , or L=L . Some examples can be seen in Eqs. 8.15 through 8.18 and in many other equations in this book. A very valuable tool for checking the answer to a problem, or to help understand an equation, is that of dimensional analysis. The dimensions of an equation (e.g. time, velocity, distance) must agree and therefore their units (s, cm s 1, cm, respectively) must also agree. Two quantities can be added or subtracted only if they have the same units, and logarithms and exponentials are unitless. In this process, it is helpful to recall some equivalent units which are revealed by writing xxvi INTRODUCTION down some simple well-known equations in physics. A few examples are provided in Table 0.A.2. The example of the Ideal Gas Law in this table also shows the process of dimensional analysis, which involves writing down the units to every term and then cancelling where possible. A more complex example of dimensional analysis is given in Example A.1.
Example 0.A.1
For a gas in thermal equilibrium at some uniform temperature, T, and uniform density, n, the number density of particles with speeds2 between v and v þ dv is given by the Maxwell–Boltzmann (or simply ‘Maxwellian’) velocity distribution, m 3=2 m v2 nðvÞ dv ¼ n exp 4 p v2 dv ð0:A:2Þ 2 p k T 2 k T where nðvÞ is the gas density per unit velocity interval, m is the mass of a gas particle (taken here to be the same for all particles), and v is the particle speed. A check of the units gives, ! 3=2 cm 2 2 1 cm 1 g g ð s Þ cm cm 3 cm ¼ 3 erg exp erg ð0:A:3Þ cm s s cm K K K K s s
Simplifying yields, ! 1 1 g 3=2 g ðcmÞ2 ¼ exp s ð0:A:4Þ cm3 s3 erg erg
Using the equivalent units for energy (Table A.2), the exponential is unitless, as required, and we find, 1 1 ¼ ð0:A:5Þ cm3 cm3
Figure 3.12 shows a plot of this function. If Eq. (0.A.2) is integrated over all velocities, either on the left hand side (LHS) or the right hand side (RHS), the total density should result. Since an integration over all velocities is equivalent to a sum over the individual 1 infinitesimal velocity intervals, the total density will also have the required units of cm3. This dimensional analysis helps to clarify the fact that, since the total density, n, appears on the RHS of Eq. (0.A.2) and is a constant, an integration over all terms, except n, on the RHS must be unitless and equal 1. (In fact these remaining terms represent a probability distribution function, see Sect. 3.4.1 for a description.) Also, since the number density is just the number of particles, N, divided by a constant (the volume), we could have substituted NðvÞ dv on the LHS and N on the RHS for the density terms. Similarly, since
2Speed and velocity are taken to be equivalent in this text unless otherwise indicated. APPENDIX xxvii the density and temperature are constant, we could have multiplied Eq. (0.A.2) by k Tto turn it into an equation for the particle pressure in a velocity interval PðvÞ dv with the total pressure, P on the RHS (Table 0.A.2). Thus, while dimensional analysis says nothing about the origin or fundamentals of an equation, it can go a long way in revealing the meaning of one and how it might be manipulated. An important comment is that one should be very careful of simplifying units without thinking about their meaning. A good example is the unit, erg s 1 Hz 1 which is a representation of a luminosity or power (see Sect. 1.1) per unit frequency (Hz) in some waveband. Since Hz can be represented as s 1, the above could be written erg s 1 s1 ¼ erg which is simply an energy and does not really express the intended meaning of the term. Similar difficulties can arise when a term is expressed as ‘per cm of waveband’. Units of frequency or wavelength should not be combined with units of time or distance, respectively.
PART I The Signal Observed
The radiation from an astronomical source can be thought of as a signal that provides us with information about it. In order to relate the signal received at the Earth to the physical conditions within an astronomical source, however, we first need ways to describe and measure light. This requires setting out the basic definitions for quantities involving light and the relationships between them. The definitions range from those associated with values measured at the Earth to those that are intrinsic to the source itself. We do this without regard (yet) for the processes that actually generate the light, an approach similar to what is often followed in Mechanics. For example, first one studies Kinematics which relates distances, velocities, and accelerations and the relations between them. Later, one considers Dynamics which deals with the forces that produce these motions. Measuring light involves a deep understanding of how the measurement process itself affects the signal and also how the Earth’s atmosphere interferes. Our instrumentation imposes its own signature on an astronomical signal and it is important to account for this imposition. These steps are fundamental and lay the groundwork for turning the measurement of a weak glimmer of light into an understanding of what drives the most powerful objects in the Universe.
1 Defining the Signal
...the distance of the invisible background [is] so immense that no ray from it has yet been able to reach us at all. –Edgar Allan Poe in Eureka, 1848
1.1 The power of light – luminosity and spectral power
The luminosity, L, of an object is the rate at which the object radiates away its energy (cgs units of erg s 1 or SI units of watts),
dE ¼ Ldt ð1:1Þ
This quantity has the same units as power and is simply the radiative power output from the object. It is an intrinsic quantity for a given object and does not depend on the observer’s distance or viewing angle. If a star’s luminosity is L at its surface, then at a distance r away, its luminosity is still L . Any object that radiates, be it spherical or irregularly shaped, can be described by 33 1 its luminosity. The Sun, for example, has a luminosity of L ¼ 3:85 10 erg s (Table G.3), most of which is lost to space and not intercepted by the Earth (Example 1.1).
Example 1.1
Determine the fraction of the Sun’s luminosity that is intercepted by the Earth. What luminosity does this correspond to?
At the distance of the Earth, the Sun’s luminosity, L , is passing through the imaginary surface of a sphere of radius, r ¼ 1 AU. The Earth will be intercepting photons over only
Astrophysics: Decoding the Cosmos Judith A. Irwin # 2007 John Wiley & Sons, Inc. ISBN: 978-0-470-01305-2(HB) 978-0-470-01306-9(PB)
4 CH1 DEFINING THE SIGNAL
m
k
8
0
1
x
5
.
1
;
s
iu
d
a R
Venus Sun Mercury
Dyson Sphere
Infrared Radiation
Figure 1.1. Illustration of a Dyson Sphere that could capture the entire luminous ouput from the Sun. Some have suggested that advanced civilizations, if they exist, would have discovered ways to build such spheres to harness all of the energy of their parent stars. the cross-sectional area that is facing the Sun. This is because the Sun is so far away that incoming light rays are parallel. Thus, the fraction will be
2 R f ¼ 2 ð1:2Þ 4 r where R is the radius of the Earth. Using the values of Table G.3, the fraction is 10 f ¼ 4:5 10 and the intercepted luminosity is therefore Lint ¼ f L ¼ 1:73 1024 erg s 1. A hypothetical shell around a star that would allow a civilization to intercept all of its luminosity is called a Dyson Sphere (Figure 1.1).
When one refers to the luminosity of an object, it is the bolometric luminosity that is understood, i.e. the luminosity over all wavebands. However, it is not possible to determine this quantity easily since observations at different wavelengths require different techniques, different kinds of telescopes and, in some wavebands, the 1.1 THE POWER OF LIGHT – LUMINOSITY AND SPECTRAL POWER 5
Figure 1.2. The supernova remnant, Cas A, at a distance of 3:4 kpc and with a linear diameter of 4 pc, was produced when a massive star exploded in the year AD 1680. It is currently expanding at a rate of 4000 km s 1 (Ref. [181]) and the proper motion (angular motion in the plane of the sky, see Sect. 7.2.1.1) of individual filaments have been observed. One side of the bipolar jet, emanating from the central object, can be seen at approximately 10 o’clock. (a) Radio image at 20 cm shown in false colour (see Sect. 2.6) from Ref. [7]. Image courtesy of NRAO/AUI/NSF. (b) X-ray emission, with red, green and blue colours showing, respectively, the intensity of low, medium and high energy X-ray emission. (Reproduced courtesy of NASA/CXC/SAO) (see colour plate) necessity of making measurements above the obscuring atmosphere of the Earth. Thus, it is common to specify the luminosity of an object for a given waveband (see Table G.6). For example, the supernova remnant, Cas A (Figure 1.2), has a radio 7 10 35 1 luminosity (from 1 ¼ 2 10 Hz to 2 ¼ 2 10 Hz) of Lradio ¼ 3 10 erg s 37 1 (Ref. [6]) and an X-ray luminosity (from 0:3 to 10 keV) of LX-ray ¼ 3 10 erg s (Ref. [37]). Its bolometric luminosity is the sum of these values plus the luminosities from all other bands over which it emits. It can be seen that the radio luminosity might justifiably be neglected when computing the total power output of Cas A. Clearly, the source spectrum (the emission as a function of wavelength) is of some importance in understanding which wavebands, and which processes, are most important in terms of energy output. The spectrum may be represented mostly by continuum emission as implied here for Cas A (that is, emission that is continuous over some spectral region), or may include spectral lines (emission at discrete wavelengths, see Chapter 3, 5, or 9). Even very weak lines and weak continuum emission, however, can provide important clues about the processes that are occurring within an astronomical object, and must not be neglected if a full understanding of the source is to be achieved. In the optical region of the spectrum, various passbands have been defined (Figure 1.3). The Sun’s luminosity in V-band, for example, represents 93 per cent of its bolometric luminosity. The spectral luminosity or spectral power is the luminosity per unit bandwidth and 1 1 can be specified per unit wavelength, L (cgs units of erg s cm ) or per unit 6 CH1 DEFINING THE SIGNAL
1 (a) 1 (b)
UB V R I
0.8 0.8 JH K LL*
0.6 0.6
Filter Response 0.4 0.4
0.2 0.2
0 0 300 400 500 600 700 800 900 1000 1500 2000 2500 3000 3500 4000 Wavelength (nm) Wavelength (nm)
Figure 1.3. Filter bandpass responses for (a) the UBVRI bands (Ref. [17]) and (b) the JHKLL bands (Ref. [19]). (The U and B bands correspond to UX and BX of Ref. [17].) Corresponding data can be found in Table 1.1