Theoretical Astrophysics Skript Heidelberg

8/19/2019 Theoretical Astrophysics Skript Heidelberg

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Theoretical Astrophysics

Matthias BartelmannInstitut f ̈ur Theoretische Astrophysik

Universität Heidelberg


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Contents

1 Macroscopic Radiation Quantities, Emission and Absorp-

tion 1

1.1 Specific Intensity . . . . . . . . . . . . . . . . . . . . 1

1.2 Relativistic Invariant . . . . . . . . . . . . . . . . . . 2

1.2.1 Lorentz Transformation of I ν . . . . . . . . . . 2

1.2.2 Example: The CMB Dipole . . . . . . . . . . 4

1.3 Einstein coefficients and the Planck spectrum . . . . . 5

1.3.1 Transition Balance . . . . . . . . . . . . . . . 5

1.3.2 Example: The CMB Spectrum . . . . . . . . . 6

1.4 Absorption and Emission . . . . . . . . . . . . . . . . 7

1.5 Radiation Transport in a Simple Case . . . . . . . . . 8

1.6 Emission and Absorption in the Continuum Case . . . 10

2 Scattering 13

2.1 Maxwell’s Equations and Units . . . . . . . . . . . . . 13

2.2 Radiation of a Moving Charge . . . . . . . . . . . . . 14

2.3 Scattering off Free Electrons . . . . . . . . . . . . . . 15

2.3.1 Polarised Thomson Cross Section . . . . . . . 15

2.3.2 Unpolarised Thomson Cross Section . . . . . . 16

2.4 Scattering off Bound Charges . . . . . . . . . . . . . . 17

2.5 Radiation Drag . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Time-Averaged Damping Force . . . . . . . . 19

2.5.2 Energy Transfer to a Radiation Field . . . . . . 20

2.6 Compton Scattering . . . . . . . . . . . . . . . . . . . 21

1


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CONTENTS 2

2.6.1 Energy-Momentum Conservation . . . . . . . 21

2.6.2 Energy Balance . . . . . . . . . . . . . . . . . 22

2.7 The Kompaneets Equation . . . . . . . . . . . . . . . 24

3 Radiation Transport and Bremsstrahlung 27

3.1 Radiation Transport Equations . . . . . . . . . . . . . 27

3.2 Local Thermodynamical Equilibrium . . . . . . . . . . 29

3.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Spectrum of a Moving Charge . . . . . . . . . 34

3.4.2 Hyperbolic Orbits . . . . . . . . . . . . . . . . 35

3.4.3 Integration over the Electron Distribution . . . 37

4 Synchrotron Radiation, Ionisation and Recombination 40

4.1 Synchrotron Radiation . . . . . . . . . . . . . . . . . 40

4.1.1 Electron Gyrating in a Magnetic Field . . . . . 40

4.1.2 Beaming and Retardation . . . . . . . . . . . . 41

4.1.3 Synchrotron Spectrum . . . . . . . . . . . . . 43

4.2 Photo-Ionisation . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Transition Amplitude . . . . . . . . . . . . . . 44

4.2.2 Transition Probability . . . . . . . . . . . . . 46

4.2.3 Transition Matrix Element . . . . . . . . . . . 48

4.2.4 Cross Section . . . . . . . . . . . . . . . . . . 50

5 Spectra 52

5.1 Natural Width of Spectral Lines . . . . . . . . . . . . 52

5.2 Cross Sections and Oscillator Strengths . . . . . . . . 52

5.2.1 Transition Probabilities . . . . . . . . . . . . . 53

5.3 Collisional Broadening of Spectral Lines . . . . . . . . 55

5.4 Velocity Broadening of Spectral Lines . . . . . . . . . 56

5.5 The Voigt Profile . . . . . . . . . . . . . . . . . . . . 57


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CONTENTS 3

5.6 Equivalent Widths and Curves-of-Growth . . . . . . . 58

6 Energy-Momentum Tensor and Equations of Motion 61

6.1 Boltzmann Equation and Energy-Momentum Tensor . 616.1.1 Boltzmann Equation . . . . . . . . . . . . . . 61

6.1.2 Moments; Continuity Equation . . . . . . . . . 62

6.1.3 Energy-Momentum Tensor . . . . . . . . . . . 64

6.2 The Tensor Virial Theorem . . . . . . . . . . . . . . . 68

6.2.1 A Corollary . . . . . . . . . . . . . . . . . . . 68

6.2.2 Second Moment of the Mass Distribution . . . 68

7 Ideal and Viscous Fluids 71

7.1 Ideal Fluids . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.1 Energy-Momentum Tensor . . . . . . . . . . . 71

7.1.2 Equations of Motion . . . . . . . . . . . . . . 73

7.1.3 Entropy . . . . . . . . . . . . . . . . . . . . . 75

7.2 Viscous Fluids . . . . . . . . . . . . . . . . . . . . . . 76

7.2.1 Stress-Energy Tensor; Viscosity and Heat Con-ductivity . . . . . . . . . . . . . . . . . . . . 76

7.2.2 Estimates for Heat Conductivity and Viscosity 78

7.2.3 Equations of Motion for Viscous Fluids . . . . 80

7.2.4 Entropy . . . . . . . . . . . . . . . . . . . . . 81

7.3 Generalisations . . . . . . . . . . . . . . . . . . . . . 82

7.3.1 Additional External Forces; Gravity . . . . . . 82

7.3.2 Example: Cloud in Pressure Equilibrium . . . 83

7.3.3 Example: Self-Gravitating Gas Sphere . . . . . 84

8 Flows of Ideal and Viscous Fluids 86

8.1 Flows of Ideal Fluids . . . . . . . . . . . . . . . . . . 86

8.1.1 Vorticity and Kelvin’s Circulation Theorem . . 86

8.1.2 Bernoulli’s Constant . . . . . . . . . . . . . . 88

8.1.3 Hydrostatic Equlibrium . . . . . . . . . . . . . 89


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CONTENTS 4

8.1.4 Curl-Free and Incompressible Flows . . . . . . 90

8.2 Flows of Viscous Fluids . . . . . . . . . . . . . . . . . 91

8.2.1 Vorticity; Incompressible Flows . . . . . . . . 91

8.2.2 The Reynolds Number . . . . . . . . . . . . . 92

8.3 Sound Waves in Ideal Fluids . . . . . . . . . . . . . . 93

8.3.1 Linear Perturbations . . . . . . . . . . . . . . 93

8.3.2 Sound Speed . . . . . . . . . . . . . . . . . . 95

8.4 Supersonic Flows . . . . . . . . . . . . . . . . . . . . 96

8.4.1 Mach’s Cone; the Laval Nozzle . . . . . . . . 96

8.4.2 Spherical Accretion . . . . . . . . . . . . . . . 97

9 Shock Waves and the Sedov Solution 102

9.1 Steepening of Sound Waves . . . . . . . . . . . . . . . 102

9.1.1 Formation of a Discontinuity . . . . . . . . . . 102

9.1.2 Specific Example . . . . . . . . . . . . . . . . 104

9.2 Shock Waves . . . . . . . . . . . . . . . . . . . . . . 107

9.2.1 The Shock Jump Conditions . . . . . . . . . . 1079.2.2 Propagation of a One-Dimensional Shock Front 108

9.2.3 The Width of a Shock . . . . . . . . . . . . . 111

9.3 The Sedov Solution . . . . . . . . . . . . . . . . . . . 112

9.3.1 Dimensional Analysis . . . . . . . . . . . . . 112

9.3.2 Similarity Solution . . . . . . . . . . . . . . . 114

10 Instabilities, Convection, Heat Conduction, Turbulence 11810.1 Rayleigh-Taylor Instability . . . . . . . . . . . . . . . 118

10.2 Kelvin-Helmholtz Instability . . . . . . . . . . . . . . 121

10.3 Thermal Instability . . . . . . . . . . . . . . . . . . . 123

10.4 Heat Conduction and Convection . . . . . . . . . . . . 127

10.4.1 Heat conduction . . . . . . . . . . . . . . . . 127

10.4.2 Convection . . . . . . . . . . . . . . . . . . . 129

10.5 Turbulence . . . . . . . . . . . . . . . . . . . . . . . 130


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CONTENTS 5

11 Collision-Less Plasmas 133

11.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . 133

11.1.1 Shielding; the Debye length . . . . . . . . . . 133

11.1.2 The plasma frequency . . . . . . . . . . . . . 134

11.2 The Dielectric Tensor . . . . . . . . . . . . . . . . . . 135

11.2.1 Polarisation and dielectric displacement . . . . 135

11.2.2 Structure of the dielectric tensor . . . . . . . . 136

11.3 Dispersion Relations . . . . . . . . . . . . . . . . . . 137

11.3.1 General form of the dispersion relations . . . . 137

11.3.2 Transversal and longitudinal waves . . . . . . 13811.4 Longitudinal Waves . . . . . . . . . . . . . . . . . . . 139

11.4.1 The longitudinal dielectricity . . . . . . . . . . 139

11.4.2 Landau Damping . . . . . . . . . . . . . . . . 141

11.5 Waves in a Thermal Plasma . . . . . . . . . . . . . . . 142

11.5.1 Longitudinal and transversal dielectricities . . 142

11.5.2 Dispersion Measure and Damping . . . . . . . 145

12 Magneto-Hydrodynamics 147

12.1 The Magneto-Hydrodynamic Equations . . . . . . . . 147

12.1.1 Assumptions . . . . . . . . . . . . . . . . . . 147

12.1.2 The induction equation . . . . . . . . . . . . . 148

12.1.3 Euler’s equation . . . . . . . . . . . . . . . . 149

12.1.4 Energy and entropy . . . . . . . . . . . . . . . 151

12.1.5 Magnetic advection and diff usion . . . . . . . 152

12.2 Generation of Magnetic Fields . . . . . . . . . . . . . 153

12.3 Ambipolar Diff usion . . . . . . . . . . . . . . . . . . 155

12.3.1 Scattering cross section . . . . . . . . . . . . . 155

12.3.2 Friction force; diff usion coefficient . . . . . . . 157

13 Waves in Magnetised Plasmas 159

13.1 Waves in magnetised cold plasmas . . . . . . . . . . . 159


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CONTENTS 6

13.1.1 The dielectric tensor . . . . . . . . . . . . . . 159

13.1.2 Contribution by ions . . . . . . . . . . . . . . 161

13.1.3 General dispersion relation . . . . . . . . . . . 162

13.1.4 Wave propagation parallel to the magnetic field 163

13.1.5 Faraday rotation . . . . . . . . . . . . . . . . 164

13.1.6 Wave propagation perpendicular to the mag-netic field . . . . . . . . . . . . . . . . . . . . 165

13.2 Hydro-Magnetic Waves . . . . . . . . . . . . . . . . . 166

13.2.1 Linearised perturbation equations . . . . . . . 166

13.2.2 Alfvén waves . . . . . . . . . . . . . . . . . . 168

13.2.3 Slow and fast hydro-magnetic waves . . . . . . 169

14 Jeans Equations and Jeans Theorem 171

14.1 Collision-less motion in a gravitational field . . . . . . 171

14.1.1 Motion in a gravitational field . . . . . . . . . 171

14.1.2 The relaxation time scale . . . . . . . . . . . . 172

14.2 The Jeans Equations . . . . . . . . . . . . . . . . . . 175

14.2.1 Moments of Boltzmann’s equation . . . . . . . 175

14.2.2 Jeans equations in cylindrical and spherical co-ordinates . . . . . . . . . . . . . . . . . . . . 177

14.2.3 Application: the mass of a galaxy . . . . . . . 178

14.3 The Virial Equations . . . . . . . . . . . . . . . . . . 179

14.3.1 The tensor of potential energy . . . . . . . . . 179

14.3.2 The tensor virial theorem . . . . . . . . . . . . 180

14.4 The Jeans Theorem . . . . . . . . . . . . . . . . . . . 182

15 Equilibrium, Stability and Disks 184

15.1 The Isothermal Sphere . . . . . . . . . . . . . . . . . 184

15.1.1 Phase-space distribution function . . . . . . . 184

15.1.2 Isothermality . . . . . . . . . . . . . . . . . . 185

15.1.3 Singular and non-singular solutions . . . . . . 186

15.2 Equilibrium and Relaxation . . . . . . . . . . . . . . . 187


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CONTENTS 7

15.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 188

15.3.1 Linear analysis and the Jeans swindle . . . . . 188

15.3.2 Jeans length and Jeans mass . . . . . . . . . . 190

15.4 The rigidly rotating disk . . . . . . . . . . . . . . . . 191

15.4.1 Equations for the two-dimensional system . . . 191

15.4.2 Analysis of perturbations . . . . . . . . . . . . 192

15.4.3 Toomre’s criterion . . . . . . . . . . . . . . . 193

16 Dynamical Friction, Fokker-Planck Approximation 195

16.1 Dynamical Friction . . . . . . . . . . . . . . . . . . . 195

16.1.1 Deflection of point masses . . . . . . . . . . . 195

16.1.2 Velocity changes . . . . . . . . . . . . . . . . 197

16.1.3 Chandrasekhar’s formula . . . . . . . . . . . . 198

16.2 Fokker-Planck Approximation . . . . . . . . . . . . . 200

16.2.1 The master equation . . . . . . . . . . . . . . 200

16.2.2 The Fokker-Planck equation . . . . . . . . . . 201


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Chapter 1

Macroscopic Radiation

Quantities, Emission and

Absorption

further reading: Shu, “ThePhysics of Astrophysics, Vol I:Radiation”, chapter 1; Rybicki,Lightman, “Radiative Processesin Astrophysics”, chapter 1; Pad-manabhan, “Theoretical Astro-physics, Vol. I: Astrophysical

Processes”, sections 6.1–6.3

1.1 Specific Intensity

• to begin with, radiation is considered as a stream of particles;energy, momentum and so on of this stream will be investigatedas well as changes of its properties;

• a screen of area d A is set up; which energy is streaming per timeinterval dt enclosing the angle θ with the direction normal to thescreen into the solid angle element dΩ and within the frequencyinterval dν?

• we begin with the occupation number: let nα p be the number den- the occupation number is thenumber density of occupiedstates per phase space element

sity of photons with momentum p and the polarisation state α(α = 1, 2);

• the energy per photon is E = hν = cp (because the photon haszero rest mass); thus

p = | p| = hνc

; (1.1)

• the volume element in momentum space is d3 p; the number of independent phase-space cells is Heisenberg’s uncertainty rela-

tion implies that points in phasespace cannot be observed; rather,observable cells in phase spacehave a finite volume

d3 p(2π)3

= p2d pdΩ

h3 =

ν2dνdΩc3

, (1.2)

where momentum has been expressed by frequency ν in the laststep;

1


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CHAPTER 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND ABSORPTION 2

• in terms of these quantities, the following amount of energy isflowing through the screen: (number of phase space cells) times(photon occupation number) times (energy per photon) times(volume filled by the photons); thus

d E = ν2dνdΩ

c3

2α=1

nα p hν d A cos θ dt (1.3)

• the energy flowing through the screen per unit time, frequencyand solid angle is

d E dt dνd AdΩ

=

2α=1

nα phν3

c2 cos θ ≡ I ν cos θ , (1.4)

where I ν is called specific intensity of the radiation;• for unpolarised light, we obviously have

I ν = 2hν3

c2 nα p ; (1.5)

1.2 Relativistic Invariant

1.2.1 Lorentz Transformation of I ν

• switching from one reference frame to another, the transformationproperties of the physical quantities is important to be known; weshall now show by Lorentz transformation that the quantity

I ν

ν3 (1.6)

is relativistically invariant;

• let us assume two observers O and O, which are moving rela-

tively to each other with velocity v in x3 direction; O is collectingphotons on a screen d A in the x1- x2 plane which move under the

angle θ with respect to the area normal into the solid angle dΩ;he finds

d N = 2 p2d pdΩ

(2π)3 n p d A

c cos θ dt (1.7)

photons on his screen;

• likewise, observer O expects the same screen to collect the photonnumber

d N = 2 p2d pdΩ

(2π)3 n p d A(c cos θ

−v) dt (1.8)

and of course the two numbers must be equal, d N = d N ;


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• the Lorentz transformation relating O and O is

Λ =

γ 0 0 βγ 0 1 0 0

0 0 1 0 βγ 0 0 γ

(1.9)

with β ≡ v/c and γ ≡ (1 − β2)−1/2;• for the screen at rest in O, d x3 = 0, thus

d x0 = cdt = γ d x0 = γ cdt , (1.10)

so that dt = γ dt , which is the usual relativistic time dilation;

• energy and momentum are combined in the four-vector

p µ = E

c, p ≡ p0, p ; p0 = | p| because | p| = E

c, (1.11)

and we obtain

p0 = γ ( p0 + β p3) ; p3 = γ ( β p0 + p3) , (1.12)

and the other components are p1 = p1, p2 = p2;

• since, from simple geometry, p3 = cos θ | p| = p0 cos θ and p3 = p0 cos θ , we then find

cos θ = p3

p0 = β p

0 + p

3

p0 + β p3 = β+

cos θ 1 + β cos θ (1.13)

for the Lorentz transformation of the angle θ ;

• this implies for the solid-angle element

d2Ω = d(cos θ )dφ = dφd

β + cos θ

1 + β cos θ

=

d2Ω

γ 2(1 + β cos θ )2 ;

(1.14)

• summarising, we find for the number of photons in the system O:

d N = 2h3γ (1 + β cos θ )

3 p2d p

= p2d p

d2Ω

γ 2(1 + β cos θ )2 =d2Ω

× n p d A =d A

c

β + cos θ

1 + β cos θ

− v

=c cos θ −v

γ dt =dt

; (1.15)

equating this to d N from (1.7) yields

n p cos θ = γ 2(1 + β cos θ )

β + cos θ

1 + β cos θ − β n p

= γ 2(1 − β2)cos θ n p = n p cos θ ; (1.16)


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• thus, the occupation number is obviously a relativistic invariant,n p = n

p , (1.17)

and I ν ∝ ν3n p implies the claimed invariance (1.6), I ν

ν3 = I

νν3

; (1.18)

the Lorentz transformation of the solid angle (1.14) will be usedlater in the discussion of synchrotron radiation

1.2.2 Example: The CMB Dipole

• this relativistic invariance of I ν/ν3 allows the dipole of the cosmicmicrowave background to be computed: a photon flying at an

angle θ relative to the x axis of the observer will be redshifted byan amount which directly follows from Lorentz transformation;

• using p µ = ( E /c, p) and p1 = | p| cos θ , the Lorentz transformationyields

p µ =

γ βγ 0 0

βγ γ 0 00 0 1 00 0 0 1

E /c

p1

p2

p3

= γ E /c + βγ | p| cos θ βγ E /c + γ

| p|cos θ

p2

p3

, (1.19)i.e. the energy in the primed system is

E = cγ

E

c+ βγ

E

ccos θ

= γ (1 + β cos θ ) E ; (1.20)

the frequency is thus increased to

ν = γ (1 + β cos θ )ν ; (1.21)

• with the occupation number n p being a relativistic invariant,

n p = 1ehν/kT − 1 = 1ehν/kT − 1 = n p , (1.22)

the temperature T must change exactly as the frequency ν, thus

T = T γ (1 + β cos θ ) ; (1.23)

• for non-relativistic velocities v c, γ ≈ 1, and thusT ≈ T

1 +

v

ccos θ

; (1.24)

the motion of the Earth relative to the microwave background thuscauses a dipolar pattern in its measured temperature; with v

∼10−3c and T ∼ 3 K, the amplitude of the dipole is of order a fewmilli-Kelvins;


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1.3 Einstein coefficients and the Planck spec-

trum

1.3.1 Transition Balance

• we consider mean transition rates in an emission- and absorptionprocess between two energy levels E 1 and E 2; the rates of absorp-tion and of stimulated emission will be proportional to the specific stimulated emission is a con-

sequence of the Bose characterof photons: if a quantum stateis occupied by photons, an in-crease in the occupation numberis more likely

intensity, absorption rate ∝ I ν B12 and stimulated emission rate ∝ I ν B21, while the rate of spontaneous emission will not depend on I ν, spontaneous emission rate ∝ A21; A and B are called the Ein-stein coefficients;

• now, let N 1 and N 2 be the mean number of states with the energies E 1 and E 2; equilibrium between transitions will require as manytransitions from E 1 to E 2 as there are from E 2 to E 1, thus

N 1 I ν B12 = N 2 [ A21 + I ν B21] , (1.25)

which can be satisfied if the specific intensity is

I ν = N 2 A21

N 1 B12 − N 2 B21 = A21

N 1 N 2

B12 − B21=

A21

B21

N 1 N 2

− 1 , (1.26)

where we have used that B12 = B21 ( E 1 and E 2 are eigenstates of

the Hamilton operator);• according to the definition of A21 and B21, we must have

[cf. Eq. (1.5)]

A21 = 2 hν3

c2 B21 ; (1.27)

• if there is thermal equilibrium between the states E 1 and E 2, wehave the Boltzmann factor between N 1 and N 2,

N 2

N 1= exp −

hν

kT , (1.28)where E 2 = E 1 + hν;

• under this condition, (1.28) implies

I ν = 2hν3

c21

ehν/kT − 1 ≡ Bν , (1.29)

which is the Planck spectrum;

• limiting cases of the Planck spectrum for high and low frequen-cies are

Bν ≈ 2hν3

c2 e−hν/kT for ν kT

h(Wien’s law) (1.30)


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and

Bν ≈ 2ν2

c2 kT for ν kT

h(Rayleigh-Jeans law) (1.31)

• the spectral energy density isdU ν =

d E dνd x3

= d E

dνd A(cdt ) =

I ν

cd2Ω , (1.32)

thus

cU ν =

I νd2Ω , (1.33)

which equals 4π I ν for isotropic radiation;

• a unit for the spectral energy density which is frequently used inastronomy is the Jansky, defined by note that 1 Jy is not the unit of

specific intensity, which wouldbe Jy/sr1 Jy = 10−26

Wm2 Hz

= 10−23 ergcm2 s Hz

; (1.34)

1.3.2 Example: The CMB Spectrum

• for example, the spectral energy density of the CMB is given by

U ν = 4π

c Bν =

4πc

2hν3

c21

ehν/kT − 1= 2.4

×10−25

erg

cm2 s Hz = 23.9 mJy (1.35)

at a frequency of ν = 30 GHz;

• the maximum of the Planck spectrum is located at

x ≡ hνkT

≈ 2.82 ; (1.36)

for the CMB, this corresponds to a frequency of

ν ≈ 1.60 × 1011 Hz = 160 GHz ; (1.37)

• inserting the Planck spectrum for I ν in (1.28) and integrating overall frequencies yields

U =

∞0

U νdν = π2

15(kT )4

(c)3 (1.38)

for the energy density of a Planckian radiation field;

• the number density of the photons is clearly

n =

∞0

U ν

hνdν =

2ζ (3)π2

kT

c

3, (1.39)

where the Riemann ζ function takes the numerical value ζ (3) ≈1.202;


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• for the cosmic microwave background, T = 2.7 K, and thus

n ≈ 400 cm−3 , U ≈ 4.0 × 10−13 ergcm3

; (1.40)

• the Rayleigh-Jeans law is often used to define a radiation temper-ature T rad by requiring

2ν2

c2 kT rad

!= I ν ; (1.41)

obviously, this agrees well with the thermodynamic temperatureif hν/kT 2.82 and I ν = Bν, but the deviation becomes consid-erable for higher frequencies;

1.4 Absorption and Emission

• the absorption coe fficient αν is defined in terms of the energy ab-sorbed per unit volume, time and frequency from the solid angled2Ω,

αν I ν =

d E

d3 xdt dνd2Ω

abs

; (1.42)

• since the stimulated emission is also proportional to I ν, an analo-gous definition applies for the “induced” emission,

αindν I ν =

d E

d3 xdt dνd2Ω

ind

; (1.43)

• for the spontaneous emission, we define the emissivity

jν =

d E

d3 xdt dνd2Ω

spn

, (1.44)

i.e. the spontaneous energy emission per unit volume, time and

frequency into the solid-angle element d2

Ω;• the eff ective net absorption is

αnetν = αν − αindν ; (1.45)

• since the unit of I ν isenergy

time × area × frequency × solid angle , (1.46)

αν must obviously have the dimension (length)−1; the “mean freepath” for a photon of frequency ν is thus approximately α−1ν ;


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• the homogeneous equation (1.53) is easily solved:d I νdl

= −αν I ν ⇒ d ln I ν = −ανdl , (1.54)

thus I ν = C 1 exp

−

ανdl

; (1.55)

• for solving the inhomogeneous equation (1.53), we assume C 1 =C 1(l) and find

d I νdl

= d

dl

C 1(l)exp

−

ανdl

(1.56)

=

C 1(l) − C 1(l)αν

exp

−

ανdl

!= jν − αν I ν = jν − ανC 1(l)exp − ανdl ;this implies

C 1(l)exp−

ανdl

= jν , (1.57)

which has the solution

C 1(l) =

dl jν exp

ανdl

+ C 2 ; (1.58)

• if αν is a constant along the light path, the integral is simply ανdl = ανl , (1.59)

and then we have

C 1(l) = jν

ανeανl , I ν(l) =

jν

αν− C 2e−ανl ; (1.60)

• for example, if the intensity satisfies the boundary condition I ν =0 at l = 0, the intensity as a function of path length becomes

I ν(l) = jναν1 − e−ανl ; (1.61)

• interesting limiting cases: let L be the entire path length throughthe medium; if

αν L 1 : I ν( L) = jναν

(αν L) = jν L ,

αν L 1 : I ν( L) = jναν

(1.62)

the former is the “optically thin”, the latter the “optically thick”

case; this amounts to comparing the mean free path α−1ν to thetotal path length L;


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• if the radiation is in thermal equilibrium with the irradiated mate-rial, we must have

I ν = Bν = jν

αν 1 − e

−αν L , (1.63)which implies that the source emits at most the intensity of thePlanck spectrum

• we consider optically thin, thermal emission of radio waves; op-tically thin implies αν L 1 and I ν = jν L, thermal equilibriumrequires I ν = Bν, and in the radio regime we have

hν

kT 1 , Bν ≈ 2ν

2

c2 kT ; (1.64)

combining these conditions, we find

I ν ≈ jν L = αν Bν L = 2ν2

c2 ανkT L =

2ν2

c2 kT b , (1.65)

where T b is the observed temperature, which is obviously relatedto the emission temperature T by

T b ≈ αν LT ; (1.66)this absurd conclusion shows indicates that the two assumptions,thermal equilibrium and optically-thin radiation, are in conflictwith each other;

1.6 Emission and Absorption in the Contin-

uum Case

• in the discrete case, the energy balance for the emitted energy was N 2 A21

transition number× hν12

energy per transition= δ E (1.67)

• the emissivity (per unit solid angle) is

jν = N 2 A21hν12

4π → N 2 A21hν

4π δD(ν − ν12) , (1.68)

with the Dirac delta function modeling a sharp line transition;

• correspondingly, we generalise this expression by a “line profilefunction” φ(ν),

jν = N 2 A21hν

4π φ(ν) , (1.69)

where φ(ν) quantifies the transition probability as a function of frequency;


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• by an analogous procedure for the absorption coefficient, we find

αν = N 1 B12

4π hν φ(ν) ; (1.70)

• we now consider an electron of energy E which emits the energyd

dνdt ≡ P(ν, E ) (1.71)

per unit time and unit frequency; let further f ( p) be the momen-tum distribution of the electrons, then the number of electronswith energies between E and E + d E is

n( E )d E = f ( p)d3 pd E

d E = 4π p2d pd E

f ( p) d E , (1.72)

if we assume the distribution to be isotropic in momentum space;since each electron emits the energy

d = P(ν, E ) dνdt , (1.73)

we obtain for the emissivity

4π jν = ∞

0n( E )P(ν, E )d E = 4π

∞0

p2 f ( p)d pd E

P(ν, E )d E

(1.74)

• by definition, we have for a continuous transition

P(ν, E 2) = hν E 2

0 A21φ(ν)d E 1 , (1.75)

i.e. electrons with the energy E 2 can emit in transitions to all pos-sible states with E 1 < E 2; thus

P(ν, E 2) = hν2hν3

c2

E 20

B21φ(ν)d E 1 ; (1.76)

• likewise, the net absorption coefficient is

αν = hν4π

d E 1

d E 2

n( E 1) B12 absorption

− n( E 2) B21 stim. emission

φ(ν) ;(1.77)

• the second term in this expression can be writtenhν

4π

d E 1

d E 2 n( E 2) B21φ(ν)

= hν

4π d E 2n( E 2)

E 2

0d E 1 B21φ(ν)

= c2

8πhν3

d E 2 n( E 2)P(ν, E 2) , (1.78)


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while the first term reads

hν

4π

d E 1

d E 2 n( E 1) B12φ(ν)

= hν4π

d E 2 n( E 2 − hν) d E 1 B12φ(ν)=

c2

8πhν3

d E 2 n( E 2 − hν)P(ν, E 2) ; (1.79)

• we thus obtain for the absorption coefficient

αν = c2

8πhν3

d E [n( E − hν) − n( E )] P(ν, E ) ; (1.80)

• in thermal equilibrium and far from the Fermi edge, the electronnumber density is

n( E ) ∝ exp− E

kT

, (1.81)

thus

n( E − hν) − n( E ) = n( E )exp

hν

kT

− 1

, (1.82)

from which we obtain

αν = c2

8πhν3ehν/kT − 1

d E n( E ) P(ν, E )

= c2

2hν3ehν/kT − 1

jν =

jν

Bν, (1.83)

just as in the discrete case;


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Chapter 2

Scattering

further reading: Rybicki, Light-

man, “Radiative Processes inAstrophysics”, chapter 7; Pad-manabhan, “Theoretical Astro-physics, Vol. I: AstrophysicalProcesses”, sections 6.4–6.7

2.1 Maxwell’s Equations and Units

• we use cgs units, i.e. the dielectric constant and the magnetic per-meability of the vacuum are both unity, 0 = 1 = µ0; Maxwell’sequations in vacuum then read

∇ · E = 4πρ , ∇ · B = 0 , ∇ × E = −1

c

∂ B

∂t , ∇ × B = 4π

c j +

1c

∂ E

∂t , (2.1)

where ρ is the charge density and j is the current density;

• the Lorentz force per unit charge is

f L = E + v

c× B ; (2.2)

• the energy density of the electromagnetic field is

U = 18π

E 2 + B2

; (2.3)

• consequently, the field components E and B have dimension erg

cm3

1/2=

g cm2

s2 cm3

1/2=

gcm s2

1/2(2.4)

• forces have the dimensiong cm

s2 ≡ dyn ; (2.5)

thus, the Lorentz force

[ F L] = [q][ E ] ⇒ g cms2 = [q]

gcm s2

1/2, (2.6)

13


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CHAPTER 2. SCATTERING 14

implies that the unit of charge must be

[q] = g1/2cm3/2

s ; (2.7)

• in these units, the elementary charge ise = 4.8033 × 10−10 cm

3/2g1/2

s ; (2.8)

• the Poynting vector, i.e. the vector of the energy current densityof the electromagnetic field, is

S = c

4π

E × B

, (2.9)

with dimension[ S ] =

cms

ergcm2s

(2.10)

which is obvious because the unit of E 2

is[ E 2] =

ergcm3

; (2.11)

2.2 Radiation of a Moving Charge

• far from its source, the electric field of an accelerated charge is,in the non-relativistic limit | β| 1 see, for example, Jackson,

Classical Electrodynamics,eq. (14.18)

E = q

cR e × e × ̇ β , (2.12)

where e is the unit vector pointing from the radiating charge tothe observer, and R is the distance;

• since B = e × E and E = B × e in vacuum, the B field is B = − q

cR

e × ̇ β

, (2.13)

and the Poynting vector is

S = c

4π

B × e

× B

=

c

4π

B2 e − ( B · e) B

=

c

4π B2 e (2.14)

because e· B = 0;

• per unit time, the energyd E dt

= S · d A (2.15)

is radiated through the area element d A; since d A is related to thesolid-angle element d Ω as d A = R2 edΩ, we find

d E dt

= c

4π B2 R2dΩ ; (2.16)

thus, the energy radiated per unit time into the solid-angle elementdΩ is

d E dt dΩ

= c4π

R B2 ; (2.17)


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2.3 Scattering off Free Electrons

2.3.1 Polarised Thomson Cross Section

• a point charge q is accelerated by an incoming electromagneticwave with the electric field component E ; the equation of motionfor the charge is

m̈ x = F L = q

E + β × B ≈ q E + O( β) , (2.18)

i.e. the last approximation employs the non-relativistic limit of the Lorentz force; thus, the acceleration is

̈ x = ċ β = q

m E ; (2.19)

• using the dipole moment d ≡ q x, we can write eq. (2.13) for themagnetic field in the form

B = − e × ̈ d

c2 R; (2.20)

• according to (2.19), the second time derivative of the dipole mo-ment is

̈ d = q2

m E , (2.21)

which, when combined with (2.20) and (2.17), implies

d E dt dΩ

= c

4π

1c2

e × ̈ d 2=

q4

4πc3m2

e × E 2 = q44πc3m2 E 2 sin2 α , (2.22)where α was introduced as the angle between the incoming elec-tric field E and the direction of the outgoing radiation, e;

• the incoming energy current density is

S = c

4π E 2 ; (2.23)

thus the diff erential scattering cross section is

dσdΩ

= 1S

d E dt dΩ

=

q2

mc2

2sin2 α ; (2.24)

• suppose the elementary charge −e is homogeneously distributedon the surface of a sphere with radius r e; then, its absolute poten-tial energy is

∼ e2r e

; (2.25)


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equating this to an electron’s rest-mass energy mec2, we can solvefor r e,

e2

r e

!= mec

2

⇒ r e =

e2

mec2 ≈

2.8×

10−13 cm ; (2.26)

this is the so-called “classical electron radius”;

• generally, the radiusr 0 ≡ q

2

mc2 (2.27)

is associated with a particle of charge q and rest-mass m; usingthis radius, the diff erential scattering cross section reads

dσ

dΩ= r 20 sin

2 α ; (2.28)

• the total cross section is

σ = r 20

sin2 αdΩ = 2πr 20

π0

sin3 αdα = 8π

3 r 20 ; (2.29)

for electrons, we obtain the Thomson cross section,

σT = 8π

3 r 2e =

8π3

e2

mec2

2≈ 6.6 × 10−25 cm2 ; (2.30)

2.3.2 Unpolarised Thomson Cross Section

• this scattering cross section is valid for one particular polarisationdirection; we now average over all incoming polarisation direc-tions; for doing so, we introduce the angle ϕ in the plane perpen-dicular to the incoming direction n; the polarisation direction isthen

e =

cos ϕsin ϕ

0

, (2.31)

if e is parallel to the x3 axis; the outgoing direction of the scat-tered radiation is

e =

sin θ

0cos θ

; (2.32)• using this, one obtains the diff erential scattering cross section

dσdΩ

= r 20 sin2 α = r 20(1 − cos2 α) = r 20

1 − ( e · e)2

; (2.33)


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using e · e = sin θ cos ϕ, averaging over ϕ yieldsdσdΩ

=

r 20

2π

2π0

dϕ

1 − sin2 θ cos2 ϕ

= r 20

1 − sin2 θ

2π

2π0

dϕ cos2 ϕ

=r 20

2 (1 + cos2 θ ) ; (2.34)

this is the unpolarised Thomson cross section;

2.4 Scattering off Bound Charges

• an accelerated charge radiates energy and thus damps the in-coming, accelerating wave; the non-relativistic Larmor formulaasserts that a non-relativistic, accelerated charge q emits the cf. Jackson, Classical Electrody-

namics, eq. (14.22)power

P = 2q2

3c3̇v2 ; (2.35)

• this is interpreted as damping with a force F D,

− F D

·v = P

⇒ v

· F D =

−2q2

3c3 ̇v

2; (2.36)

• the temporal average over a time interval T isd E dt

=

1T

T 0

dt 2q2

3c3̇v2

= 1

T

2q2

3c3

v̇vT 0 − T

0dt v̈v

; (2.37)

• the first term vanishes for bound charges and large T , thus

F D ·v

= 2q2

3c3... x · v ; (2.38)

we thus identify the expression

F D = 2q2

3c3... x (2.39)

with the time-averaged damping force;

• for bound orbits with an angular frequency of ω0, we have

̈ x = −ω20 x ⇒ ... x = −ω20̇ x ; (2.40)


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thus, the equation of motion reads

̈ x + ω20 x = q

m E 0e−iω

t − γ ̇ x (2.41)

with the damping term

γ = 2q2

3c3ω20 ; (2.42)

the first term on the right-hand side of (2.41) is the external exci-tation, the second is the damping; this equation models a driven,damped harmonic oscillator, whose solution is known to read

x = q

m

E 0e−iωt

ω20 −

ω2

−iωγ

; (2.43)

• we put this back into Larmor’s equation (2.35) and obtaind E dt

= P = 2q2

3c3̈ x2 = 2q2

3c3̈ x · ̈ x∗

= 2q4

3m2c3 E 20

ω4

(ω2 − ω20)2 + ω2γ 2 ; (2.44)

• the incoming energy current is | S | = c E 20/(4π), and thus the scat-tering cross section becomes

σ = 1

| S |d E dt

= 8π

3 r 20

ω4

(ω2 − ω20)2 + ω2γ 2 (2.45)

with the typical resonance behaviour at ω = ω0;

• interesting limiting cases are:

ω ω0 : σ ≈ 8π3 r 20 = σT ;

(binding forces are then irrelevant)

ω ω0 : σ ≈ σT ω

ω0

4;

(Rayleigh scattering)

ω ≈ ω0 : σ ≈ σTω20

4(ω − ω0)2 + γ 2

= 2π2q2

mc

γ/(2π)

(ω − ω0)2 + (γ/2)2

, (2.46)

where the term in square brackets defines the so-called Lorentz profile;


27/210


2.5 Radiation Drag

2.5.1 Time-Averaged Damping Force

• in the case of Thomson scattering, the scattering charge damps themotion which is caused by the incoming electric field accordingto the damping force (2.39)

F D = 2q2

3c3... x ; (2.47)

• an incoming electromagnetic wave exerts the Lorentz forceF L = q( E − β × B) = m̈ x (2.48)

on the scattering charge; the last equality in (2.48) assumes that F D F L, i.e. the back reaction of the radiation by the charge wasneglected;

• from (2.48), we find... x =

q

m

̇ E + ̇ β × B + β × ̇ B

=

q

m

̇ E + β × ̇ B + ̈ xc

× B= qm ̇ E + β ×

̇ B + q

mc E + β × B × B ; (2.49)in the non-relativistic limit, we can drop the terms proportional to β and find

... x =

q

m

̇ E +

q

mc( E × B)

, (2.50)

and thus F D =

2q3

3mc3

̇ E +

q

mc( E × B)

; (2.51)

• averaging over time yields

F D = 2q3

3mc3 ̇ E

=0

+ 23 q2

mc22 E × B ; (2.52)

using

E × B = E + ( e × E ) = E 2 e − ( E · e) =0

E

= 4πU e , (2.53)

we finally find

F D

=

8π

3 r 2qU e = σTU e (2.54)

for the time-averaged damping force;


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2.5.2 Energy Transfer to a Radiation Field

• we now consider a charge moving with a relative velocity vthrough a radiation field which is isotropic in its rest frame; in

the rest frame of the radiation, we have

E = 0 = B , E 2 = 4πU = B2 ; (2.55)

• in the rest frame of the charge, the Lorentz force is F L = q( E

+ β × B) = q E , (2.56)

because β = 0 in the charge’s rest frame; in addition, we have

E = E ⊥ + E

= γ E ⊥ + β ×

B + E , (2.57)where γ is the usual Lorentz factor; for Larmor’s equation, wefurther need

d E dt

= 2q2

3c3̈ x2 , ̈ x = F L

m; (2.58)

• in a first step, we computë x = q2

m2

γ ( E ⊥ + β × B) + E

2 (2.59)

= q2

m2 γ ( E − E + β × B) + E 2

= q2

m2

γ ( E + β × B) + (1 − γ ) E

2=

q2

m2

γ 2 E 2 + γ 2 β2 B2sin2 θ + (1 − γ 2) E 2

,

where we have used that

E · ( β × B) = 0 (2.60)

because the direction of β is random with respect to the direction

of E × B; thus, we obtain̈ x2 = 4πγ 2U q2m2

1 +

2 β2

3 +

1 − γ 23γ 2

= 4πγ 2U q2

m2

1 +

β2

3

; (2.61)

• with that, we find the resultd E dt

= 4πγ 2U 2q4

3m2c3 1 + β2

3 = σTUcγ 2

1 + β2

3 (2.62)for the radiation which is on average radiated by the charge;


29/210


• according to the radiation damping, the energy which is on aver-age absorbed by the charge is

d E

dt abs = σTUc , (2.63)

and thus the total energy change of the radiation field per unittime is

d E dt

= σTUc

γ 21 +

β2

3

− 1

=

43

σTUcγ 2 β2 ; (2.64)

this amount of energy is added to the radiation field per unit timeby a single charge;

• the number of collisions between the charge and photons per unittime is

d N cdt

= σTc U hν

; (2.65)

combining this with (2.64), we find the energy gain of the radia-tion field by scattering of the charge per scattering process,

∆ E γ

=

d E cdt

d N cdt

−1=

43

hνγ 2 β2 = 43

γ 2 β2 E γ ; (2.66)

2.6 Compton Scattering

2.6.1 Energy-Momentum Conservation

• we now consider electromagnetic radiation as being composed of photons; if an ensemble of charges is embedded into a radiationfield, energy is transfered by scattering from the photons to thecharges and back; if the radiation temperature is higher than thetemperature of the charge ensemble, energy flows from the radi-ation to the charges; this process is called Compton scattering;in astrophysics, the inverse Compton scattering is typically more

important, during which energy is transfered from the charges tothe radiation field;

• an incoming photon with momentum hν e/c hits an electron withmomentum p; after scattering, the photon and the electron havemomenta hν e/c and p;

• conservation of momentum and energy implyhν e + c p = hν e + c p , hν + E = hν + E , (2.67)

where

E 2

= c2

p2

+ m2

c4

(2.68)according to the relativistic energy-momentum relation;


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• solving the energy equation for E 2 and inserting (2.68) yields

c2 p2 = c2 p2 + h2(ν − ν)2 + 2 Eh(ν − ν) , (2.69)

while the momentum equation impliesc2 p2 = c2 p2 + h2(ν e − ν e)2 + 2h(ν e − ν e)c p ; (2.70)

• subtracting (2.69) from (2.70) and cancelling suitable terms gives

hνν(1 − cos θ ) = E (ν − ν) − c p(ν e − ν e) , (2.71)

where θ is the angle between e and e;

• if the electron is originally at rest, p = 0 and E = mc2, and (2.71)

simplifies toν − ν = h

mc2νν(1 − cos θ ) , (2.72)

and in the limit of very low photon energy, hν mc2, we find forthe relative energy change of the Compton-scattered photon

∆ E γ

E γ =

ν − νν

= − E γ mc2

(1 − cos θ ) , (2.73)

and if hν mc2, quantum electrodynamics must be used anyway;

• averaging (2.73) over all scattering angles θ , we find the meanenergy loss per Compton scattering,

E γ = − E 2γ

mc2

1−1

(1 − cos θ )d(cos θ ) = − E 2γ

mc2 ; (2.74)

2.6.2 Energy Balance

• the total energy transfer to the radiation field due to the motion of a single charge is given by the diff erence between the energy gain

(2.66) per scattering and the energy loss per Compton scattering(2.74),

∆ E γ

=

43

γ 2 β2 − E γ mc2

E γ ; (2.75)

• for photons with E γ mc2 in the relativistic limit, β ≈ 1, and

∆ E γ ≈ 4

3γ 2 E γ , (2.76)

which can become a very large number; in that way, for example,

CMB photons can be converted to X-ray photons;


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• in the thermal limit of (2.75), we can approximate v c, thusγ ≈ 1, and mv2 = 3kT e; then

∆ E γ ≈ 4v2

3c2 − E γ

mc2 E γ = 4kT e − E γ E γ

mc2

; (2.77)

thus, the photons gain energy (on average), if

4kT e > E γ (2.78)

(inverse Compton scattering), and lose energy otherwise (Comp-ton scattering)

• Compton scattering causes fast charges to lose energy; typicaltime scales are, according to (2.64)

t c ≡ E d E /dt = γ mc2

43 σTUcγ

2 β2 =

34

mc2

γβ2σTU ; (2.79)

for non-relativistic, thermal electrons, E = 3kT e/2 and γ ≈ 1, and

t c =

32 kT ec

2

43 σTUcv

2 =

98

mc

σTU ; (2.80)

• after N s scatterings, the total energy transfer from thermal elec-trons to the photons is

E

E =

1 +

4kT emc2

N s≈ exp

4kT e N s

mc2

≡ e4y , (2.81)

where the Compton parameter

y ≡ 4kT e N smc2

(2.82)

was introduced;

• if the electron number density is n

e, the number of scatterings per

path length dl is

d N s = neσTdl ⇒ N s =

σTnedl , (2.83)

and thus the Compton-y parameter becomes

y = kT e

mc2

σTnedl ; (2.84)


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2.7 The Kompaneets Equation

• we need an additional equation which specifies how the photonspectrum is changed due to the scatterings; for deriving it, we

assume that a homogeneous, thermal distribution of electrons islocated in a homogeneous sea of radiation, such as, for example,a galaxy cluster in the microwave background; the collisions withthe electrons change the photon energy, but not their number, andthus their spectrum cannot remain a Planck spectrum;

• let n(ν) be the occupation number of photon states with frequencyν; then, the Boltzmann equation requires

∂n(ν)∂t

= d3 p dΩ

dσdΩ c (2.85)× n(ν) [1 + n(ν)] N ( E ) − n(ν) 1 + n(ν) N ( E ) ;

this equation has the following meaning: the occupation numberat the frequency ν changes due to scattering from ν to ν, and fromν to ν; the term

n(ν)1 + n(ν)

N ( E ) (2.86)

quantifies how many photons there are at frequency ν, correctedby the factor for stimulated emission from ν to ν, and multiplieswith the number of collision partners N ( E ) at energy E ; in other

words, it quantifies the number of collisions away from frequencyν; analogously, the term

n(ν) [1 + n(ν)] N ( E ) (2.87)

quantifies the opposite scattering, i.e. scattering processes in-creasing the occupation number at frequency ν; of course, theenergy diff erence between photon frequencies ν and ν must bebalanced by the diff erence between the energies E and E ; the in-tegral over d3 p integrates over the electron distribution, and thefactor

dσdΩ dΩ (2.88)

specifies the probability for scattering photons from frequency νto frequency ν or backward;

• we assume thermal photon and electron distributions, and restrictourselves to the limit of Thomson scattering, which applies if

hν mc2 ; (2.89)

moreover, we assume small changes in the photon frequency,

hence δν ≡ ν − ν ν ; (2.90)


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moreover, the electron energy distribution is

N ( E ) ∝ exp− E

kT e

, (2.91)

and energy conservation requires E = hν − hδν ; (2.92)

• now, both n(ν) and N ( E ) can be expanded in Taylor series up tosecond order,

n(ν) = n(ν) + ∂n

∂νδν +

12

∂2n

∂ν2δν2 + O(δν3) , (2.93)

N ( E ) = N ( E ) − ∂ N ∂ E

hδν + 12

∂2 N

∂ E 2h2δν2 + O(δν3) ,

where (2.91) allows us to use

∂ N

∂ E = − N ( E )

kT e,

∂2 N

∂ E 2 =

N ( E )(kT e)2

; (2.94)

• for simplification, we now define the dimension-less photon en-ergy, scaled by the thermal electron energy

x ≡ hνkT e

(2.95)

and find

n( x) ≈

n( x) + ∂n

∂ xδ x +

1

2

∂2n

∂ x2δ x2 ,

N ( E ) ≈ N ( E )1 + δ x +

δ x2

2

; (2.96)

• with these approximations, we return to the original equation(2.86) for n(ν) and obtain

∂n

∂t =

∂n

∂ x+ n(n + 1)

I 1

+ 1

2 ∂2n

∂ x2 + 2(1 + n)

∂n

∂ x+ n(n + 1) I 2 , (2.97)

with the abbreviations

I i ≡

d3 p

dΩdσdΩ

cδ xi N ( E ) (2.98)

• the energy change of a photon scattering off a moving electronfollows from (2.71), adopting the non-relativistic limit

E = mc2 + p2

2m(2.99)

and using (2.89) and (2.90); this yields

hδν = − hνmc

( e − e) · p ⇒ δ x = − xmc

( e − e) · p ; (2.100)


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• using this result, the integrals I i can be carried out straightfor-wardly; with the unpolarised Thomson cross section (2.34), wefirst find

I 2 = 2σTneckT e x

2

mc2

; (2.101)

• for evaluating I 1, we note that I 1 is the mean rate of relative en-ergy transfer, quantified by δ x from the electrons to the photons,and therefore the mean energy transfer rate, divided by kT e; from(2.77), we know that this is

∆ E γ

=

x(kT e)2

mec2 (4 − x) (2.102)

per scattering, and multiplying with the collision rate neσTc gives

I 1 = kT e

mec2 neσTc x(4 − x) ; (2.103)

• with these two expression for I i, we find the time derivative of nto be

mec2

kT e

1neσTc

∂n

∂t =

1 x2

∂

∂ x

x4

∂n

∂ x+ n + n2

; (2.104)

• we finally transform the time t to the Compton parameter, using

dy = kT emec2

neσTc dt (2.105)

to find the Kompaneets equation

∂n

∂y =

1 x2

∂

∂ x

x4

∂n

∂ x+ n + n2

; (2.106)

• the hot gas in galaxy clusters is much hotter than the cosmic back-ground radiation; then, we can approximate the right-hand side of (2.106) to lowest order in x,

∂n

∂y ≈ x2 ∂

2n

∂ x2 + 4 x

∂n

∂ x; (2.107)

• inserting here the occupation number in thermal equilibrium, n ≈(e x − 1)−1, we find

δn

n= δy

x2e x(1 + e x)

(e x − 1)2 − 4 xe x

e x − 1

(2.108)

for the relative change of the occupation number, where x is now

hν/kT and no longer hν/kT e!


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Chapter 3

Radiation Transport and

Bremsstrahlung

further reading: Shu, “ThePhysics of Astrophysics, Vol I:Radiation”, chapters 2, 3, and15; Rybicki, Lightman, “Radia-tive Processes in Astrophysics”,chapter 5; Padmanabhan, “Theo-retical Astrophysics, Vol. I: As-trophysical Processes”, sections6.8–6.9

3.1 Radiation Transport Equations

• we start with the collision-less Boltzmann equation for describingthe temporal change of the photon distribution function in phasespace,

∂n

∂t + ∇ · ∂n

∂ x+ ̇ p · ∂n

∂ p= 0 , (3.1)

which is valid in absence of collisions;• for photons, we have v = c e, where e is the unit vector in the

direction of light propagation; moreover, ̇ p = 0 in absence of systematic external forces (such as gravitational lensing); sincethe intensity I ν is proportional to n, the Boltzmann equation forphotons can also be written as

1c

∂ I ν

∂t + e · ∂ I ν

∂ x= 0 ; (3.2)

• we now define the following quantities: F ν ≡

dΩ e · I ν , Pν,i j ≡ 1

c

dΩ eie j I ν (3.3)

and recall the spectral energy density

U ν = 1c

dΩ I ν ; (3.4)

• integrating the Boltzmann equation (3.2) first over dΩ, we obtain

the equation ∂U ν∂t

+ ∇ · F ν = 0 ; (3.5)

27


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CHAPTER 3. RADIATION TRANSPORT AND BREMSSTRAHLUNG 28

which has the form of a continuity equation and identifies F ν asthe spectral radiation current density (spectral because it retainsthe dependence on frequency ν); this equation expresses energyconservation in the radiation field;

• if we multiply (3.2) with ei first before integrating over dΩ, wefind

1c

dΩ ei

∂ I ν

∂t +

dΩ eie j

∂ I ν

∂ x j= 0 , (3.6)

and hence1c

∂F ν,i

∂t + c

∂Pν,i j

∂ x j= 0 ; (3.7)

• this equation describes the change of the momentum current den-sity, because U ν

c

(c e) (3.8)

is the momentum density of the radiation field, and thus

1c

∂ F

∂t =

1c2

∂

∂t

dΩ I ν e (3.9)

is c times the temporal change of the momentum current density;Eq. (3.7) expresses momentum conservation;

• in presence of emission, stimulated emission and absorption, weknow from the first chapter that the energy equation must be aug-

mented by source and sink terms on its right-hand side; we had

d I νdl

= jν − αν I ν = 1c

d I νdt

; (3.10)

integrating over dΩ, and assuming that jν and αν are isotropic, wefind

dU νdt

= 4π jν − ανU νc = 4π jν − ρκ νcU ν ; (3.11)we now re-define the emissivity,

4π jν →

ρ jν ≡

ρ jν , (3.12)

i.e. we refer it to the mass density, and write

dU νdt

= ρ( jν − κ νcU ν) ; (3.13)

• likewise, the momentum-conservation equation1c

d I νdt

= jν − αν I ν (3.14)

becomes after multiplication with e and integration over dΩ

1c

ddt

dΩ e · I ν =

dΩ jν e −

dΩ αν I ν e , (3.15)


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and thus1c

d F dt

= −αν F ν = − ρκ ν F ν , (3.16)where we have assumed again that jν and κ ν are isotropic

• including the emission and absorption terms, the transport equa-tions are modified to read

∂U ν

∂t + ∇ · F ν = ρ( jν − κ νcU ν)

1c

∂F ν,i

∂t + c

∂Pν,i j

∂ x j= − ρκ νF ν,i ; (3.17)

these equations do not contain scattering terms yet!

• since the change in the momentum current density corresponds toa force density, and this force is caused by the interaction betweenradiation and matter, an oppositely directed and equally strongforce must act on the matter as radiation pressure force; thus

f rad = ρ

c

∞0

κ ν F ν dν (3.18)

is the density of the radiation pressure force;

3.2 Local Thermodynamical Equilibrium

• the moment equations for U ν and F ν are by no means easier tohandle than the Boltzmann equation whose moments they are;we obviously need an additional approximation, or condition, inorder to “close” the moment equations; the “closure” means thatthey can then be solved without progressing indefinitely to higherorders of moments;

• often, the mean free path of the photons is much smaller thanthe dimensions of the system under consideration; then, we canassume that thermodynamical equilibrium is locally establishedbetween the radiation field and the matter; under this condition,

I ν ≈ Bν(T ) , (3.19)

i.e. the specific intensity of the radiation field is the Planckianintensity of a black body, and

U ν = 1c

dΩ I ν =

4πc

Bν(T ) ; (3.20)


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• under such circumstances, there is obviously no radiation flux anymore because the radiation field is isotropic; in order to estimatethe flux nonetheless, we study the orders of magnitude of the dif-ferent terms in the moment equations;

• time derivatives can typically be neglected because temporalchanges of the quantities U ν, F ν and Pν,i j occur on an evolutionarytime scale, while the other terms change according to the stream-ing of the photons, thus approximately on time scales of order(mean free path)/c;

• if we first ignore ∂ F ν/∂t , we obtain

F ν,i ≈ − c ρκ ν

∂Pν,i j

∂ x j(3.21)

in the approximation of Local Thermodynamical Equilibrium(LTE), we further have

Pν,i j ≈ Pνδi j = U ν3 δi j (3.22)

because of the (local) isotropy of the radiation field, and thus


∂U ν

∂ xi≈ − c

ρκ ν

U ν

R, (3.23)

where R is a typical dimension of the system; the mean free pathλν is determined by

λνnσν = λν ρκ ν ≈ 1 , (3.24)

and (3.23) can thus be approximated by

|F ν,i| ≈ cU ν

λν

R

, (3.25)

which is smaller by a factor λν/ R compared to the transparentcase (in which κ

→ 0 and λ

ν → R;

• using this estimate for F ν, we return to the Eq. (3.17) for the par-tial time derivative of U ν; as before, we ignore the time derivative,such that the only term remaining on the left-hand side is

∇ · F ν ≈ cU ν R

λν

R; (3.26)

the second term on the right-hand side is

ρκ νcU ν ≈ c

λν U ν ≈ Rλν 2

∇ ·

F ν

∇ ·

F ν ; (3.27)


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thus, because of the assumption of local thermodynamical equi-librium, the divergence of F ν is negligibly small; consequently,we must require

ρ jν ≈ ρκ νcU ν ⇒ U ν ≈ ρκ νcαν

= 4πc Bν(T ) , (3.28)

as anticipated;

• accordingly, if λν R and t evol λν/c, the solutions of themoment equations are


∂Pν,i j

∂ x j, U ν ≈ 4π

c Bν(T ) ; (3.29)

• because of (local) isotropy, we hadPν,i j ≈ U ν3 δi j ≈

4π3c

Bν(T ) δi j , (3.30)

and thus

F ν,i ≈ − 4π3 ρκ ν

∂ Bν

∂T

∂T

∂ xi, (3.31)

i.e. the flux will become proportional to the temperature, which ischaracteristic for diff usion processes;

• for convenience, we now introduce the Rosseland mean opacity,

κ −1R ≡ ∞

0 dν

κ −1ν

∂ Bν(T )∂T

∞

0 dν

∂ Bν(T )

∂T

; (3.32)here, we can use the fact that ∞

0dν

∂ Bν(T )∂T

=

∂

∂T

∞0

dν Bν(T ) = ∂

∂T

caT 4

4π

, (3.33)

where

a ≡ π2

15 k 4

(c)3 = 7.6 × 10−15 ergK4cm3 (3.34)is the so-called Stefan-Boltzmann constant;

• using this, we obtain the expression

F =

∞0

dν F ν = −4π3 ρ∂T

∂ x

∞0

dνκ ν

∂ Bν

∂T

= − 4π3 ρκ R

∇T ddT

caT 4

4π

= − c

3 ρκ R ∇aT 4

(3.35)

for the radiative energy flux;


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• the energy which is streaming away interacts with the absorbingmatter and thus exerts a force on it, which is determined by theright-hand side of the momentum-conservation equation, as de-scribed above:

f rad = ρ

c

∞0

dν κ ν F ν = − ρc

4π3 ρ

∞0

dν ∂ Bν

∂T

∂T

∂ x

= −13

∇(aT 4) = − ∇P , (3.36)

which equals just the negative pressure gradient;

• a remark on units: the unit of U ν is

[U ν] = ergcm3 Hz

, (3.37)

the unit of κ ν is

[κ ν] = cm2

g , (3.38)

and thus the unit of F ν is

[ F ν] = [c][U ν] = ergcm2 t Hz

, (3.39)

and the unit of f rad is

[ f rad] = g

cm3s

cm

cm2

g

erg

cm2 s HzHz = erg

cm4 (3.40)

= g cm2

s41

cm4 =

g cms2

1cm3

= dyncm3

= forcevolume

,

as it should be;

3.3 Scattering

• so far, we have only considered emission and absorption, but ne-glected scattering; scattering changes the distribution function of the photons by exchanging photons with diff erent momenta; if weassume for simplicity that the scattering process changes the pho-ton’s momentum, but not its energy, we can write the scatteringcross section in the form

dσ( e → e)dΩ

= σφ( e, e) , (3.41)

where e and e are unit vectors in the propagation directions of the incoming and the outgoing photon; the function φ( e, e) is

normalised, symmetric in its arguments and dimension-less anddescribes the directional distribution of the scattered photons;


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• scattering increases the distribution function n( e) according todn( e)

dt

+

=

dΩ

N ecσφ( e, e)

# of scatterings e → en( e) [1 + n( e)] ,

(3.42)where the factor [1 + n( e)] is included for describing stimulatedemission of photons with momentum direction e;

• analogously, losses due to scattering are given bydn( e)

dt

−

=

dΩ

N ecσφ( e, e

)

n( e) [1 + n( e)] , (3.43)

and thus the total change of n( e) due to scatterings becomes

dn( e)

dt = dΩ N ecσφ( e, e)× n( e) [1 + n( e)] − n( e) [1 + n( e)]=

dΩ

N ecσφ( e, e)

n( e) − n( e) , (3.44)

in which the terms from stimulated emission cancel exactly;

• since the integral over the solid angle only concerns the directionof e, we obtain from (3.44)

n( e)dt

= − N ecσn( e) + N ecσ

dΩ φ( e, e)n( e) , (3.45)

and thus1c

dn( e)dt

= − ρκ scaν n( e) + ρκ scaν

dΩ φ( e, e)n( e) , (3.46)

where we have introduced the scattering opacity through κ scaν = N eσ;

• since the intensity at fixed frequency is proportional to the occu-pation number, the same equation (3.46) also holds for I ν; there-fore, the transport equation for the specific intensity is changed inpresence of scattering to

1c

d I νdt

= 1c

∂ I ν

∂t + e · ∂ I ν

∂ x(3.47)

= ρ jν

4π − ρκ absν I ν − ρκ scaν

I ν −

dΩφ( e, e) I ν( e)

;

• again, we now take the moments of the transport equation in orderto see how the moment equations are changed by scattering; thefirst moment is obtained by integrating (3.48) over dΩ,

∂U ν

∂t + ∇ · F ν = ρ jν − ρκ absν cU ν − ρκ scaν cU ν + ρκ scaν cU ν (3.48)

due to the normalisation of φ( e, e); therefore, the scattering termscancel, and the equation for U ν remains unchanged ;


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• the next moment equation simplifies if we further assume that dΩφ( e, e) e = 0 =

dΩφ( e, e) e , (3.49)

which holds for many scattering processes (e.g. Thomson scatter-ing); then, the second moment equation reads

1c

∂F ν,i

∂t + c

∂Pν,i j

∂ x j= − ρκ absν F ν,i − ρκ scaν F ν,i= − ρ(κ absν + κ scaν )F ν,i , (3.50)

i.e. the scattering opacity is simply added to the absorption opac-ity here; with a suitable modification of the Rosseland mean opac-ity, the diff usion approximation remains valid which we have ob-tained above;

3.4 Bremsstrahlung

3.4.1 Spectrum of a Moving Charge

• a radiation process which is very important in astrophysics is dueto electrons which are scattered off ions and radiate due to theacceleration they experience; in order to describe it, we start again

from Larmor’s equation, which saysd E dt

= 2e2

3c3̈ x2 , (3.51)

where e is the charge of the accelerated particle (the electron, inmost cases), ̈ x is its acceleration, and d E /dt is the power radiatedaway;

• as a function of frequency, this equation can be written as follows:

d E = 2e2

3c3 ̈ x2 ⇒ E =

2e2

3c3

∞

−∞dt ̈ x2 ; (3.52)

if we Fourier-transform the particle’s trajectory,

̂ x(ω) = ∞

−∞dt x(t )eiωt , x(t ) =

∞−∞

dω2π

̂ x(ω)e−iωt , (3.53)

we first have

̈ x =

∞−∞

dω2π

−ω2̂ x(ω)

e−iωt ⇒ ˆ̈ x = −ω2̂ x(ω) , (3.54)

and we can use Parseval’s equation, ∞−∞

dt f 2(t ) = ∞

−∞dω2π

ˆ f (ω)2 ; (3.55)


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combining these results yields

E = 2e2

3c3

∞−∞

dω2π

ˆ̈ x(ω)

2

= 4e2

3c3

∞0

dω2π

ω4

̂ x(ω)

2

, (3.56)

and, by diff erentiation,

d E dω

= 2e2

3πc3ω4̂ x(ω)2 ; (3.57)

this is a general expression valid for all radiation processes; inorder to make progress, we need the Fourier transform of the spe-cific particle trajectory;

3.4.2 Hyperbolic Orbits• classically, the electron follows a hyperbola around the ion in the

orbital plane perpendicular to the (conserved) angular momen-tum; the focal point of the hyperbola is the centre of mass, whichwe assume to coincide with the centre of the scattering ion, i.e. weneglect the mass of the electron; in polar coordinates, the trajec-tory is given by

r (ϕ) = p

1 + cos ϕ , (3.58)

with the parameters

p = L2 z

αm= a( 2 − 1) and ≡

1 +

2 L2 z E

α2m

1/2, (3.59)

where L z = bmv∞ is the angular momentum in z direction, v∞ isthe initial velocity at infinity, E = mv2∞/2 is the kinetic energy atinfinity and thus the total energy, and α quantifies the couplingstrength; for electrons orbiting nuclei at rest with charge Ze,

α = Z e2 (3.60)

• as for solving the Kepler problem, we introduce a parameter ψ(the eccentric anomaly), of which we require that it satisfy

r = a( cosh ψ − 1) ; (3.61)

we find the relation between ϕ and ψ by inserting (3.61) into(3.58), using (3.59)

a( 2 − 1)1 + cos ϕ

= a( cosh ψ − 1) ⇒ cos ϕ = − cosh ψ cosh ψ − 1 ,

(3.62)if we want ψ = 0 where ϕ = 0;


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• energy conservation implies that the time t when the electronreaches the distance r from the scattering ion is

t = r

r 0

d x 2m

E + α x − L2 z2mx2 1/2

= r

r 0

xd x 2m

E x2 + α x − L2 z2m1/2 ,

(3.63)where

L2 z = mαa( 2 − 1) (3.64)

was used from (3.59); furthermore, we have

a = p

2 − 1 = L2 z

αm

α2m

2 L2 z E =

α

2 E ⇒ E = α

2a; (3.65)

combining, we first find

t =

m

2α

r r 0

xd x x2

2a + x − a2 ( 2 − 1)1/2 , (3.66)

which can be transformed with (3.61) to obtain

t =

ma3

α

ψ0

( cosh ψ − 1)dψ =

ma3

α ( sinh ψ − ψ) ; (3.67)

• the coordinates x and y in the orbital plane satisfy

x = r cos ϕ = a( cosh ψ − 1) − cosh ψ cosh ψ − 1 = a( − cosh ψ)

y = r sin ϕ = a√

2 − 1sinh ψ , (3.68)

where we have used (3.61) and (3.62); with these expressions, wereturn to the Fourier transform of x and y

• sinceˆ̇ x = −iω ˆ x(ω) , (3.69)

we haveˆ x(ω) =

iω

ˆ̇ x(ω) = iω

∞−∞

dt ˙ x(t )eiωt

= i

ω

∞−∞

dt d xdψ

dψdt

eiωt (ψ)

= − iaω

∞−∞

dψ sinh ψeiωt (ψ) ; (3.70)

• using (3.67), we can write

eiωt (ψ) = expiω ma3α ( sinh ψ − ψ) ; (3.71)


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from (3.65), we finda =

α

mv2∞, (3.72)

moreover,

ω

ma3

α = ω

mα2

m3v6∞=

αω

mv3∞≡ µ , (3.73)

and thuseiωt (ψ) = ei µ( sinh ψ−ψ) (3.74)

• putting this result into (3.70), we write

ˆ x(ω) = − iaω

∞−∞

dψ sinh ψei µ( sinh ψ−ψ)

ŷ(ω) = ia √ 2 − 1ω

∞−∞

dψ cosh ψei µ( sinh ψ−ψ) ; (3.75)

these expressions can be analytically integrated and lead to first-order Hankel functions, which will not be discussed in detail here;

• forming|̂ x(ω)|2 = ˆ x(ω) ˆ x∗(ω) + ŷ(ω)ŷ∗(ω) (3.76)

and inserting the result into (3.57) yields the desiredbremsstrahlung spectrum;

3.4.3 Integration over the Electron Distribution

• having obtained the spectrum d E /dω for a single charge, we nowhave to integrate over a distribution of charges; we do this by inte-grating over all impact parameters b from 0 to ∞ after multiplyingd E /dω with

nine v · 2πbdb , (3.77)which is the number of scatterings per unit volume and unit timebetween ions and electrons with the number densities n

i and n

e,

respectively; for a fully ionised pure hydrogen gas, ni = ne ≡ n;• preparing the integration, we note from (3.59) and (3.72) that

2 = 1 +b2m2v4∞

α2 = 1 +

b2

a2 , (3.78)

such that the integration over b can be transformed into an inte-gration over ,

d = bdb

a2

= bdb

a2

⇒ bdb = a2 d , (3.79)

where ∈ [1, ∞) while b ∈ [0, ∞);


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• inserting suitable approximations for the first-order Hankel func-tions, we find after carrying out the b integration and inserting(3.60)

d E dV dt dω ≈ 16 Z

2e

6n

2

3m2c3v ln

2γ

mv3

Ze2ω ω mv3 Ze2 π√ 3

ω mv3

Ze2

, (3.80)where γ ≈ 1.78 is Euler’s constant;

• we write this result asd E

dV dt dω =

16π Z 2n2e6

3√

3m2c31v

gff (v, ω) , (3.81)

introducing the so-called gaunt factor gff , which usually dependsat most weakly on v;

• in a dilute thermal plasma, the electrons have a Maxwellian ve-locity distribution, but for emitting a photon of energy ω, anelectron needs at least an energy of

m

2 v2min = ω ⇒ vmin =

2ω

m; (3.82)

the thermal average of the inverse velocity is then

1v = m

2πkT

3/2

∞

vmin

4πv2dv 1

v

e−mv2/2kT

=

2mπkT

e−ω/kT ; (3.83)

• replacing 1/v in (3.81) by the average (3.83) finally yields thespecific emissivity of a thermal plasma due to bremsstrahlung,

d E dV dt dω

≡ j(ω) = 16π Z 2n2e6

3√

3m2c3

2mπkT

e−ω/kT gff (v, ω) ; (3.84)

• the volume emssivity is the integral of j(ω) over frequency ω, j =

d E dV dt

= 16 Z 2n2e6

3m2c3

2mkT

π gff (v, ω) ∝ n2

√ T ; (3.85)

• numerically, the volume emissivity in cgs units is

j = Z 2 gff

n

cm−3

2

6.69 × 10−20

kT

erg

1/22.68 × 10−24

kT

keV1/2

7.86 × 10−28 T K1/2

(3.86)


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• as an example, we consider the X-ray emission of a massivegalaxy cluster with kT = 10 keV; typical clusters reach electrondensities of n ≈ 10−3 cm−3 in their cores; let us assume for sim-plicity that the X-ray emitting gas with that electron density fills

a volume of 1 Mpc3;

• assuming fully ionised pure hydrogen, we put Z = 1 and gff = 1for simplicity; then, (3.86) yields

LX ≈ V j ≈ (3.1 × 1024)3 · 10−6 · 2.68 × 10−24 ·√

10

≈ 2.5 × 1044 ergs−1 , (3.87)

which makes galaxy clusters the most luminous X-ray sources onthe sky;

• with an average energy of ∼ 10keV = 1.6e − 8erg per photon,this luminosity corresponds to

N X ≈ 1.6 × 1052 s−1 (3.88)

photons emitted by the cluster per second; if the cluster is at adistance of, say, 100 Mpc ≈ 3.1 × 1026 cm, these photons are dis-tributed over an area of ≈ 1.2 × 1054 cm2, such that an X-ray de-tector with a typical collecting area of a few hundred cm2 sees

≈ 100 1.6

×1052

1.2 × 1054 s−1

≈ 1 s−1

(3.89)

i.e. this enormous X-ray luminosity produces a flux of approxi-mately one photon per second in a typical X-ray detector;


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Chapter 4

Synchrotron Radiation,

Ionisation and Recombination

further reading: Shu, “ThePhysics of Astrophysics, Vol I:Radiation”, chapters 18, 19, 21–23; Rybicki, Lightman, “Radia-tive Processes in Astrophysics”,chapters 6 and 10; Padmanab-han, “Theoretical Astrophysics,Vol. I: Astrophysical Processes”,sections 6.10–6.12

4.1 Synchrotron Radiation

4.1.1 Electron Gyrating in a Magnetic Field

• a further very important radiation process is the emission of radi-ation by electrons moving in a magnetic field B; in such a field,electrons spiral around field lines, with their angular frequency

given byωB =

ceB

E =

eB

γ mc, (4.1)

where E is the electron energy, and γ is the usual Lorentz factor;numerically, we have

ωB ≈ 17.6γ −1 MHz

B

1 G

, (4.2)

i.e. synchrotron radiation is typically emitted at radio frequencies;

• the radius of the projection of the spiral orbit perpendicular to themagnetic field isr B =

v

ωB=

γ mcv

eB, (4.3)

and the complete motion is the circular motion around B, super-posed by a drift along B;

• we employ Larmor’s equationd E dt

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Theoretical Astrophysics Skript Heidelberg

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