Theoretical Astrophysics. Matthias Bartelmann Institut für Theoretische Astrophysik Universität Heidelberg

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1 Theoretical Astrophysics Matthias Bartelmann Institut für Theoretische Astrophysik Universität Heidelberg

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3 Inhaltsverzeichnis 1 Macroscopic Radiation Quantities, Emission and Absorption Specific Intensity Relativistic Invariant Lorentz Transformation of I ν Example: The CMB Dipole Einstein coefficients and the Planck spectrum Transition Balance Example: The CMB Spectrum Absorption and Emission Radiation Transport in a Simple Case Emission and Absorption in the Continuum Case Scattering 13.1 Maxwell s Equations and Units Radiation of a Moving Charge Scattering off Free Electrons Polarised Thomson Cross Section Unpolarised Thomson Cross Section Scattering off Bound Charges Radiation Drag Time-Averaged Damping Force Energy Transfer to a Radiation Field Compton Scattering

4 4 INHALTSVERZEICHNIS.6.1 Energy-Momentum Conservation Energy Balance The Kompaneets Equation Radiation Transport and Bremsstrahlung Radiation Transport Equations Local Thermodynamical Equilibrium Scattering Bremsstrahlung Spectrum of a Moving Charge Hyperbolic Orbits Integration over the Electron Distribution Synchrotron Radiation, Ionisation and Recombination Synchrotron Radiation Electron Gyrating in a Magnetic Field Beaming and Retardation Synchrotron Spectrum Photo-Ionisation Transition Amplitude Transition Probability Transition Matrix Element Cross Section Spectra Natural Width of Spectral Lines Cross Sections and Oscillator Strengths Transition Probabilities Collisional Broadening of Spectral Lines Velocity Broadening of Spectral Lines The Voigt Profile

5 INHALTSVERZEICHNIS Equivalent Widths and Curves-of-Growth Energy-Momentum Tensor and Equations of Motion Boltzmann Equation and Energy-Momentum Tensor Boltzmann Equation Moments; Continuity Equation Energy-Momentum Tensor The Tensor Virial Theorem A Corollary Second Moment of the Mass Distribution Ideal and Viscous Fluids Ideal Fluids Energy-Momentum Tensor Equations of Motion Entropy Viscous Fluids Stress-Energy Tensor; Viscosity and Heat Conductivity Estimates for Heat Conductivity and Viscosity Equations of Motion for Viscous Fluids Entropy Generalisations Additional External Forces; Gravity Example: Cloud in Pressure Equilibrium Example: Self-Gravitating Gas Sphere Flows of Ideal and Viscous Fluids Flows of Ideal Fluids Vorticity and Kelvin s Circulation Theorem Bernoulli s Constant Hydrostatic Equlibrium

6 6 INHALTSVERZEICHNIS Curl-Free and Incompressible Flows Flows of Viscous Fluids Vorticity; Incompressible Flows The Reynolds Number Sound Waves in Ideal Fluids Linear Perturbations Sound Speed Supersonic Flows Mach s Cone; the Laval Nozzle Spherical Accretion Shock Waves and the Sedov Solution Steepening of Sound Waves Formation of a Discontinuity Specific Example Shock Waves The Shock Jump Conditions Propagation of a One-Dimensional Shock Front The Width of a Shock The Sedov Solution Dimensional Analysis Similarity Solution Instabilities, Convection, Heat Conduction, Turbulence Rayleigh-Taylor Instability Kelvin-Helmholtz Instability Thermal Instability Heat Conduction and Convection Heat conduction Convection Turbulence

7 INHALTSVERZEICHNIS 7 11 Collision-Less Plasmas Basic Concepts Shielding; the Debye length The plasma frequency The Dielectric Tensor Polarisation and dielectric displacement Structure of the dielectric tensor Dispersion Relations General form of the dispersion relations Transversal and longitudinal waves Longitudinal Waves The longitudinal dielectricity Landau Damping Waves in a Thermal Plasma Longitudinal and transversal dielectricities Dispersion Measure and Damping Magneto-Hydrodynamics The Magneto-Hydrodynamic Equations Assumptions The induction equation Euler s equation Energy and entropy Magnetic advection and diffusion Generation of Magnetic Fields Ambipolar Diffusion Scattering cross section Friction force; diffusion coefficient Waves in Magnetised Plasmas Waves in magnetised cold plasmas

8 8 INHALTSVERZEICHNIS The dielectric tensor Contribution by ions General dispersion relation Wave propagation parallel to the magnetic field Faraday rotation Wave propagation perpendicular to the magnetic field Hydro-Magnetic Waves Linearised perturbation equations Alfvén waves Slow and fast hydro-magnetic waves Jeans Equations and Jeans Theorem Collision-less motion in a gravitational field Motion in a gravitational field The relaxation time scale The Jeans Equations Moments of Boltzmann s equation Jeans equations in cylindrical and spherical coordinates Application: the mass of a galaxy The Virial Equations The tensor of potential energy The tensor virial theorem The Jeans Theorem Equilibrium, Stability and Disks The Isothermal Sphere Phase-space distribution function Isothermality Singular and non-singular solutions Equilibrium and Relaxation

9 INHALTSVERZEICHNIS Stability Linear analysis and the Jeans swindle Jeans length and Jeans mass The rigidly rotating disk Equations for the two-dimensional system Analysis of perturbations Toomre s criterion Dynamical Friction, Fokker-Planck Approximation Dynamical Friction Deflection of point masses Velocity changes Chandrasekhar s formula Fokker-Planck Approximation The master equation The Fokker-Planck equation

10 10 INHALTSVERZEICHNIS

11 Kapitel 1 Macroscopic Radiation Quantities, Emission and Absorption 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 investigated as well as changes of its properties; a screen of area da is set up; which energy is streaming per time interval dt enclosing the angle θ with the direction normal to the screen into the solid angle element dω and within the frequency interval dν? we begin with the occupation number: let n α p be the number density of photons with momentum p and the polarisation state α (α = 1, ); further reading: Shu, The Physics of Astrophysics, Vol I: Radiation, chapter 1; Rybicki, Lightman, Radiative Processes in Astrophysics, chapter 1; Padmanabhan, Theoretical Astrophysics, Vol. I: Astrophysical Processes, sections the occupation number is the number density of occupied states per phase space element the energy per photon is E = hν = cp (because the photon has zero rest mass); thus p = p = hν c ; (1.1) the volume element in momentum space is d 3 p; the number of independent phase-space cells is d 3 p (π ) = p dpdω 3 h 3 = ν dνdω c 3, (1.) where momentum has been expressed by frequency ν in the last step; Heisenberg s uncertainty relation implies that points in phase space cannot be observed; rather, observable cells in phase space have a finite volume 1

12 KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A in terms of these quantities, the following amount of energy is flowing through the screen: (number of phase space cells) times (photon occupation number) times (energy per photon) times (volume filled by the photons); thus de = ν dνdω c 3 n α p hν da cos θ dt (1.3) α=1 the energy flowing through the screen per unit time, frequency and solid angle is de dtdνdadω = α=1 n α p hν 3 c 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 = hν3 c n α p ; (1.5) 1. Relativistic Invariant 1..1 Lorentz Transformation of I ν switching from one reference frame to another, the transformation properties of the physical quantities is important to be known; we shall now show by Lorentz transformation that the quantity is relativistically invariant; I ν ν 3 (1.6) let us assume two observers O and O, which are moving relatively to each other with velocity v in x 3 direction; O is collecting photons on a screen da in the x 1 -x plane which move under the angle θ with respect to the area normal into the solid angle dω ; he finds dn = p dp dω n (π ) 3 p da c cos θ dt (1.7) photons on his screen; likewise, observer O expects the same screen to collect the photon number dn = p dpdω (π ) 3 n p da(c cos θ v) dt (1.8) and of course the two numbers must be equal, dn = dn;

13 1.. RELATIVISTIC INVARIANT 3 the Lorentz transformation relating O and O is γ 0 0 βγ Λ = βγ 0 0 γ (1.9) with β v/c and γ (1 β ) 1/ ; for the screen at rest in O, dx 3 = 0, thus dx 0 = cdt = γ dx 0 = 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 ( E ) p µ = c, p ( p 0, p ) ; p 0 = p because p = E c, (1.11) and we obtain p 0 = γ(p 0 + βp 3 ) ; p 3 = γ(βp 0 + p 3 ), (1.1) and the other components are p 1 = p 1, p = p ; since, from simple geometry, p 3 p 3 = p 0 cos θ, we then find = cos θ p = p 0 cos θ and cos θ = p3 p = βp 0 + p 3 β + cos θ = (1.13) 0 p 0 + βp β cos θ for the Lorentz transformation of the angle θ; this implies for the solid-angle element [ ] β + cos θ d Ω = d(cos θ)dφ = dφ d = 1 + β cos θ d Ω γ (1 + β cos θ ) ; (1.14) summarising, we find for the number of photons in the system O: dn = [ γ(1 + β cos θ ) ]3 p dp h 3 } {{ } =p dp ( ) β + cos θ [ n p }{{} da =da d Ω γ (1 + β cos θ ) } {{ } ] v c 1 + β cos θ } {{ } =c cos θ v =d Ω γdt }{{} ; (1.15) =dt equating this to dn from (1.7) yields [ ( )] β + cos θ n p cos θ = γ (1 + β cos θ ) 1 + β cos θ β = γ (1 β ) cos θ n p = n p cos θ ; (1.16) n p

14 4KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A thus, the occupation number is obviously a relativistic invariant, n p = n p, (1.17) and I ν ν 3 n p implies the claimed invariance (1.6), I ν I ν = 3 ; (1.18) ν ν 3 the Lorentz transformation of the solid angle (1.14) will be used later in the discussion of synchrotron radiation 1.. Example: The CMB Dipole this relativistic invariance of I ν /ν 3 allows the dipole of the cosmic microwave background to be computed: a photon flying at an angle θ relative to the x axis of the observer will be redshifted by an amount which directly follows from Lorentz transformation; using p µ = (E/c, p) and p 1 = p cos θ, the Lorentz transformation yields γ βγ 0 0 E/c p µ βγ γ 0 0 p = p = γe/c + βγ p cos θ βγe/c + γ p cos θ p p 3 p 3, (1.19) i.e. the energy in the primed system is E = c (γ E c + βγ E ) c cos θ = γ(1 + β cos θ)e ; (1.0) the frequency is thus increased to ν = γ(1 + β cos θ)ν ; (1.1) with the occupation number n p being a relativistic invariant, 1 n p = e hν/kt 1 = 1 e hν /kt 1 = n p, (1.) the temperature T must change exactly as the frequency ν, thus T = Tγ(1 + β cos θ) ; (1.3) for non-relativistic velocities v c, γ 1, and thus T T (1 + v ) c cos θ ; (1.4) the motion of the Earth relative to the microwave background thus causes a dipolar pattern in its measured temperature; with v 10 3 c and T 3 K, the amplitude of the dipole is of order a few milli-kelvins;

15 1.3. EINSTEIN COEFFICIENTS AND THE PLANCK SPECTRUM5 1.3 Einstein coefficients and the Planck spectrum Transition Balance we consider mean transition rates in an emission- and absorption process between two energy levels E 1 and E ; the rates of absorption and of stimulated emission will be proportional to the specific intensity, absorption rate I ν B 1 and stimulated emission rate I ν B 1, while the rate of spontaneous emission will not depend on I ν, spontaneous emission rate A 1 ; A and B are called the Einstein coefficients; now, let N 1 and N be the mean number of states with the energies E 1 and E ; equilibrium between transitions will require as many transitions from E 1 to E as there are from E to E 1, thus stimulated emission is a consequence of the Bose character of photons: if a quantum state is occupied by photons, an increase in the occupation number is more likely N 1 I ν B 1 = N [A 1 + I ν B 1 ], (1.5) which can be satisfied if the specific intensity is I ν = N A 1 N 1 B 1 N B 1 = A 1 N 1 = N B 1 B 1 A 1 B 1 ( N1 N 1 ), (1.6) where we have used that B 1 = B 1 (E 1 and E are eigenstates of the Hamilton operator); according to the definition of A 1 and B 1, we must have [cf. Eq. (1.5)] A 1 = hν3 c B 1 ; (1.7) if there is thermal equilibrium between the states E 1 and E, we have the Boltzmann factor between N 1 and N, ( N = exp hν ), (1.8) N 1 kt where E = E 1 + hν; under this condition, (1.8) implies which is the Planck spectrum; I ν = hν3 c 1 e hν/kt 1 B ν, (1.9) limiting cases of the Planck spectrum for high and low frequencies are B ν hν3 c e hν/kt for ν kt h (Wien s law) (1.30)

16 6KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A and B ν ν c kt for ν kt h (Rayleigh-Jeans law) (1.31) the spectral energy density is thus du ν = de dνdx 3 = cu ν = which equals 4πI ν for isotropic radiation; note that 1 Jy is not the unit of specific intensity, which would be Jy/sr 1 Jy = 10 6 W de dνda(cdt) = I ν c d Ω, (1.3) I ν d Ω, (1.33) a unit for the spectral energy density which is frequently used in astronomy is the Jansky, defined by m Hz = Example: The CMB Spectrum erg cm s Hz ; (1.34) for example, the spectral energy density of the CMB is given by U ν = 4π c B ν = 4π hν 3 1 c c e hν/kt 1 = erg = 3.9 mjy (1.35) cm s Hz at a frequency of ν = 30 GHz; the maximum of the Planck spectrum is located at x hν kt for the CMB, this corresponds to a frequency of.8 ; (1.36) ν Hz = 160 GHz ; (1.37) inserting the Planck spectrum for I ν in (1.8) and integrating over all frequencies yields U = 0 U ν dν = π (kt) 4 (1.38) 15 ( c) 3 for the energy density of a Planckian radiation field; the number density of the photons is clearly n = 0 U ν ζ(3) dν = hν π ( kt c ) 3, (1.39) where the Riemann ζ function takes the numerical value ζ(3) 1.0;

17 1.4. ABSORPTION AND EMISSION 7 for the cosmic microwave background, T =.7 K, and thus n 400 cm 3, 13 erg U cm ; (1.40) 3 the Rayleigh-Jeans law is often used to define a radiation temperature T rad by requiring ν c kt rad! = I ν ; (1.41) obviously, this agrees well with the thermodynamic temperature if hν/kt.8 and I ν = B ν, but the deviation becomes considerable for higher frequencies; 1.4 Absorption and Emission the absorption coefficient α ν is defined in terms of the energy absorbed per unit volume, time and frequency from the solid angle d Ω, ( ) de α ν I ν = ; (1.4) d 3 xdtdνd Ω since the stimulated emission is also proportional to I ν, an analogous definition applies for the induced emission, ( ) α ind de ν I ν = ; (1.43) d 3 xdtdνd Ω for the spontaneous emission, we define the emissivity ( ) de j ν =, (1.44) d 3 xdtdνd Ω i.e. the spontaneous energy emission per unit volume, time and frequency into the solid-angle element d Ω; the effective net absorption is α net ν abs spn ind = α ν α ind ν ; (1.45) since the unit of I ν is energy time area frequency solid angle, (1.46) α ν must obviously have the dimension (length) 1 ; the mean free path for a photon of frequency ν is thus approximately α 1 ν ;

18 8KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A let σ ν be the cross section of an atom, molecule or other particle for the absorption of light of frequency ν, then α ν = nσ ν ρκ, (1.47) where n is the number density of absorbing systems and ρ is their mass density; κ is called opacity, whose physical meaning is the absorption cross section per unit mass, [κ] = cm g ; (1.48) if matter is in equilibrium with a radiation field, the emitted and absorbed amounts of energy must equal, hence j ν + α ind ν I ν = α ν I ν j ν = α net ν I ν ; (1.49) using (1.8) and (1.9), we then find I ν = j ν α net ν = hν3 c ( ) 1 N1 1 ; (1.50) N i.e. if the occupation numbers are known, the emission and absorption properties in equilibrium can be calculated, and vice versa; in particular, in thermal equilibrium with matter, we have I ν = B ν α net ν = j ν B ν ; (1.51) 1.5 Radiation Transport in a Simple Case we consider an emitting and absorbing medium which does not scatter for now and is being irradiated by a light bundle; let the medium be characterised by an emissivity j ν and a net absorption coefficient α net ν ; per unit of the traversed distance, the intensity of the light bundle changes according to di ν = j ν dl }{{}} α ν {{ I ν dl }, (1.5) emission absorption from which we obtain the equation of radiation transport in its simplest case, di ν = j ν α ν I ν. (1.53) dl

19 1.5. RADIATION TRANSPORT IN A SIMPLE CASE 9 the homogeneous equation (1.53) is easily solved: thus di ν dl = α ν I ν d ln I ν = α ν dl, (1.54) ( I ν = C 1 exp ) α ν dl ; (1.55) for solving the inhomogeneous equation (1.53), we assume C 1 = C 1 (l) and find [ ( )] C 1 (l) exp α ν dl (1.56) di ν dl = d dl = [ C 1 (l) C ] 1(l)α ν exp (! = j ν α ν I ν = j ν α ν C 1 (l) exp ) α ν dl ( ) α ν dl ; this implies C 1 ( (l) exp ) α ν dl = j ν, (1.57) which has the solution C 1 (l) = [ ( dl j ν exp )] α ν dl + C ; (1.58) if α ν is a constant along the light path, the integral is simply α ν dl = α ν l, (1.59) and then we have C 1 (l) = j ν α ν e α νl, I ν (l) = j ν α ν C e α νl ; (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) = j ν α ν ( 1 e α ν l ) ; (1.61) interesting limiting cases: let L be the entire path length through the medium; if α ν L 1 : I ν (L) = j ν α ν (α ν L) = j ν L, α ν L 1 : I ν (L) = j ν α ν (1.6) the former is the optically thin, the latter the optically thick case; this amounts to comparing the mean free path α 1 ν to the total path length L;

20 10KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A if the radiation is in thermal equilibrium with the irradiated material, we must have I ν = B ν = j ν α ν ( 1 e α ν L ), (1.63) which implies that the source emits at most the intensity of the Planck spectrum we consider optically thin, thermal emission of radio waves; optically thin implies α ν L 1 and I ν = j ν L, thermal equilibrium requires I ν = B ν, and in the radio regime we have hν kt 1, combining these conditions, we find B ν ν kt ; (1.64) c I ν j ν L = α ν B ν L = ν c α νkt L = ν c kt b, (1.65) where T b is the observed temperature, which is obviously related to the emission temperature T by T b α ν LT ; (1.66) this absurd conclusion shows indicates that the two assumptions, thermal equilibrium and optically-thin radiation, are in conflict with each other; 1.6 Emission and Absorption in the Continuum Case in the discrete case, the energy balance for the emitted energy was }{{} N A 1 }{{} hν 1 = δe (1.67) transition number energy per transition the emissivity (per unit solid angle) is j ν = N A 1 hν 1 4π N A 1 hν δ D (ν ν 1 ), (1.68) 4π with the Dirac delta function modeling a sharp line transition; correspondingly, we generalise this expression by a line profile function φ(ν), j ν = N A 1 hν φ(ν), (1.69) 4π where φ(ν) quantifies the transition probability as a function of frequency;

21 1.6. EMISSION AND ABSORPTION IN THE CONTINUUM CASE11 by an analogous procedure for the absorption coefficient, we find α ν = N 1B 1 4π hν φ(ν) ; (1.70) we now consider an electron of energy E which emits the energy dɛ dνdt P(ν, E) (1.71) per unit time and unit frequency; let further f ( p) be the momentum distribution of the electrons, then the number of electrons with energies between E and E + de is n(e)de = f ( p) d3 p dp de = 4πp de de f ( p) de, (1.7) if we assume the distribution to be isotropic in momentum space; since each electron emits the energy we obtain for the emissivity 4π j ν = 0 dɛ = P(ν, E) dνdt, (1.73) n(e)p(ν, E)dE = 4π by definition, we have for a continuous transition P(ν, E ) = hν E 0 0 p f (p) dp P(ν, E)dE de (1.74) A 1 φ(ν)de 1, (1.75) i.e. electrons with the energy E can emit in transitions to all possible states with E 1 < E ; thus P(ν, E ) = hν hν3 c E 0 B 1 φ(ν)de 1 ; (1.76) likewise, the net absorption coefficient is α ν = hν de 1 de 4π n(e } {{ 1)B 1 n(e } )B } {{ 1 } φ(ν) ; absorption stim. emission (1.77) the second term in this expression can be written hν de 1 de n(e )B 1 φ(ν) 4π = hν E de n(e ) de 1 B 1 φ(ν) 4π 0 c = de 8πhν 3 n(e )P(ν, E ), (1.78)

22 1KAPITEL 1. MACROSCOPIC RADIATION QUANTITIES, EMISSION AND A while the first term reads hν de 1 de n(e 1 )B 1 φ(ν) 4π = hν de n(e hν) de 1 B 1 φ(ν) 4π c = de 8πhν 3 n(e hν)p(ν, E ) ; (1.79) we thus obtain for the absorption coefficient α ν = c de [n(e hν) n(e)] P(ν, E) ; (1.80) 8πhν 3 in thermal equilibrium and far from the Fermi edge, the electron number density is thus ( n(e) exp E ) kt [ ( ) ] hν n(e hν) n(e) = n(e) exp 1 kt from which we obtain c ( α ν = e hν/kt 1 ) 8πhν 3 = just as in the discrete case;, (1.81) de n(e) P(ν, E), (1.8) c hν 3 ( e hν/kt 1 ) j ν = j ν B ν, (1.83)

23 Kapitel Scattering.1 Maxwell s Equations and Units we use cgs units, i.e. the dielectric constant and the magnetic permeability of the vacuum are both unity, ɛ 0 = 1 = µ 0 ; Maxwell s equations in vacuum then read further reading: Rybicki, Lightman, Radiative Processes in Astrophysics, chapter 7; Padmanabhan, Theoretical Astrophysics, Vol. I: Astrophysical Processes, sections E = 4πρ, B = 0, E = 1 c B t, B = 4π c j + 1 c E t, (.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 ; (.) the energy density of the electromagnetic field is U = 1 ( E + B ) ; (.3) 8π consequently, the field components E and B have dimension ( erg ) 1/ ( ) g cm 1/ ( g ) 1/ = = cm 3 s cm 3 cm s (.4) forces have the dimension g cm s dyn ; (.5) thus, the Lorentz force [ F L ] = [q][ E] g cm s ( g ) 1/ = [q], (.6) cm s 13

24 14 KAPITEL. SCATTERING implies that the unit of charge must be [q] = g1/ cm 3/ s in these units, the elementary charge is e = cm3/ g 1/ ; (.7) s ; (.8) the Poynting vector, i.e. the vector of the energy current density of the electromagnetic field, is S = c ( E B ), (.9) 4π with dimension [ S ] = cm erg (.10) s cm s which is obvious because the unit of E is [ E ] = erg cm ; (.11) 3. Radiation of a Moving Charge see, for example, Jackson, Classical Electrodynamics, eq. (14.18) far from its source, the electric field of an accelerated charge is, in the non-relativistic limit β 1 E = q [ ( e e β )], (.1) cr where e is the unit vector pointing from the radiating charge to the observer, and R is the distance; since B = e E and E = B e in vacuum, the B field is B = q ( e β ), (.13) cr and the Poynting vector is S = c [( B e ) B ] = c [ B e ( B e) B ] = c B 4π 4π 4π e (.14) because e B = 0; per unit time, the energy de dt = S d A (.15) is radiated through the area element d A; since d A is related to the solid-angle element d Ω as d A = R edω, we find de = c B dt 4π R dω ; (.16) thus, the energy radiated per unit time into the solid-angle element dω is de dtdω = c 4π R B ; (.17)

25 .3. SCATTERING OFF FREE ELECTRONS 15.3 Scattering off Free Electrons.3.1 Polarised Thomson Cross Section a point charge q is accelerated by an incoming electromagnetic wave with the electric field component E ; the equation of motion for the charge is m x = F L = q ( E + β B ) q E + O(β), (.18) i.e. the last approximation employs the non-relativistic limit of the Lorentz force; thus, the acceleration is x = c β = q m E ; (.19) using the dipole moment d q x, we can write eq. (.13) for the magnetic field in the form e d B = c R ; (.0) according to (.19), the second time derivative of the dipole moment is d = q E m, (.1) which, when combined with (.0) and (.17), implies de dtdω = c ( 1 e d ) 4π c q 4 = e E q 4 = E 4πc 3 m 4πc 3 m sin α, (.) where α was introduced as the angle between the incoming electric field E and the direction of the outgoing radiation, e; the incoming energy current density is S = c 4π E ; (.3) thus the differential scattering cross section is dσ dω = 1 ( ) de q S dtdω = sin α ; (.4) mc suppose the elementary charge e is homogeneously distributed on the surface of a sphere with radius r e ; then, its absolute potential energy is e r e ; (.5)

26 16 KAPITEL. SCATTERING equating this to an electron s rest-mass energy m e c, we can solve for r e, e r e! = me c r e = e m e c cm ; (.6) this is the so-called classical electron radius ; generally, the radius r 0 q mc (.7) is associated with a particle of charge q and rest-mass m; using this radius, the differential scattering cross section reads the total cross section is σ = r 0 sin αdω = πr 0 dσ dω = r 0 sin α ; (.8) π 0 sin 3 αdα = 8π 3 r 0 ; (.9) for electrons, we obtain the Thomson cross section, ( e ) cm ; (.30) σ T = 8π 3 r e = 8π 3 m e c.3. Unpolarised Thomson Cross Section this scattering cross section is valid for one particular polarisation direction; we now average over all incoming polarisation directions; for doing so, we introduce the angle ϕ in the plane perpendicular to the incoming direction n ; the polarisation direction is then cos ϕ e = sin ϕ, (.31) 0 if e is parallel to the x 3 axis; the outgoing direction of the scattered radiation is sin θ e = 0 ; (.3) cos θ using this, one obtains the differential scattering cross section dσ dω = r 0 sin α = r 0 (1 [ cos α) = r 0 1 ( e e ) ] ; (.33)

27 .4. SCATTERING OFF BOUND CHARGES 17 using e e = sin θ cos ϕ, averaging over ϕ yields dσ dω π = r 0 dϕ ( 1 sin θ cos ϕ ) π 0 [ = r 0 1 sin θ π ] dϕ cos ϕ π 0 = r 0 (1 + cos θ) ; (.34) this is the unpolarised Thomson cross section;.4 Scattering off Bound Charges an accelerated charge radiates energy and thus damps the incoming, accelerating wave; the non-relativistic Larmor formula asserts that a non-relativistic, accelerated charge q emits the power cf. Jackson, Classical Electrodynamics, eq. (14.) P = q 3c 3 v ; (.35) this is interpreted as damping with a force F D, F D v = P v F D = q 3c 3 v ; (.36) the temporal average over a time interval T is de dt = 1 T = 1 T T 0 q 3c 3 dt q v 3c 3 [ v v T T 0 0 ] dt v v ; (.37) the first term vanishes for bound charges and large T, thus we thus identify the expression with the time-averaged damping force; F D v = q 3c 3... x v ; (.38) F D = q 3c 3... x (.39) for bound orbits with an angular frequency of ω 0, we have... x = ω 0 x x = ω x 0 ; (.40)

28 18 KAPITEL. SCATTERING thus, the equation of motion reads x + ω 0 x = q m E 0 e iωt γ x (.41) with the damping term γ = q 3c 3 ω 0 ; (.4) the first term on the right-hand side of (.41) is the external excitation, the second is the damping; this equation models a driven, damped harmonic oscillator, whose solution is known to read x = q E 0 e iωt m ω 0 ω iωγ ; (.43) we put this back into Larmor s equation (.35) and obtain de dt = P = q 3c 3 x = q 3c 3 x x = q 4 E 3m c ω (ω ω 0 ) + ω γ ; (.44) the incoming energy current is S = c E 0 /(4π), and thus the scattering cross section becomes σ = 1 de S dt ω 4 = 8π 3 r 0 (.45) (ω ω 0 ) + ω γ with the typical resonance behaviour at ω = ω 0 ; interesting limiting cases are: ω ω 0 : σ 8π 3 r 0 = σ T ; (binding forces are then irrelevant) ( ) 4 ω ω ω 0 : σ σ T ; ω 0 (Rayleigh scattering) ω 0 ω ω 0 : σ σ T 4(ω ω 0 ) + γ [ ] = π q γ/(π) mc (ω ω 0 ) + (γ/), (.46) where the term in square brackets defines the so-called Lorentz profile;

29 .5. RADIATION DRAG 19.5 Radiation Drag.5.1 Time-Averaged Damping Force in the case of Thomson scattering, the scattering charge damps the motion which is caused by the incoming electric field according to the damping force (.39) F D = q 3c 3... x ; (.47) an incoming electromagnetic wave exerts the Lorentz force F L = q( E β B) = m x (.48) on the scattering charge; the last equality in (.48) assumes that F D F L, i.e. the back reaction of the radiation by the charge was neglected; from (.48), we find... x = q ( E + β B + β B) m = q E + β B + x m c B = q [ E + β B + q ( E + β B ) B] m mc ; (.49) in the non-relativistic limit, we can drop the terms proportional to β and find... x = q [ E + q m mc ( E B)], (.50) and thus averaging over time yields [ F D = q3 E + q ] 3mc 3 mc ( E B) ; (.51) using F D = q3 3mc 3 E + ( ) q E B ; (.5) }{{} 3 mc =0 E B = E + ( e E) = E e ( E e) E }{{} =0 = 4πU e, (.53) we finally find F D = 8π 3 r qu e = σ T U e (.54) for the time-averaged damping force;

30 0 KAPITEL. SCATTERING.5. Energy Transfer to a Radiation Field we now consider a charge moving with a relative velocity v through a radiation field which is isotropic in its rest frame; in the rest frame of the radiation, we have E = 0 = B, E = 4πU = B ; (.55) in the rest frame of the charge, the Lorentz force is F L = q( E + β B ) = q E, (.56) because β = 0 in the charge s rest frame; in addition, we have E = E + E = γ ( E + β B ) + E, (.57) where γ is the usual Lorentz factor; for Larmor s equation, we further need de dt in a first step, we compute x = q = q 3c 3 x, m [ γ( E + β B) + E ] = q m [ γ( E E + β B) + E ] = q m [ γ( E + β B) + (1 γ) E ] x = F L m ; (.58) (.59) = q m [ γ E + γ β B sin θ + (1 γ ) E ], where we have used that E ( β B) = 0 (.60) because the direction of β is random with respect to the direction of E B; thus, we obtain x = 4πγ U (1 q + β m ) γ 3γ ) = 4πγ U (1 q + β ; (.61) m 3 with that, we find the result de dt = 4πγ U q4 3m c 3 (1 + β 3 ) ) = σ T Ucγ (1 + β 3 (.6) for the radiation which is on average radiated by the charge;

31 .6. COMPTON SCATTERING 1 according to the radiation damping, the energy which is on average absorbed by the charge is ( ) de = σ T Uc, (.63) dt abs and thus the total energy change of the radiation field per unit time is ) ] de = σ T Uc [γ (1 + β 1 = 4 dt 3 3 σ TUcγ β ; (.64) this amount of energy is added to the radiation field per unit time by a single charge; the number of collisions between the charge and photons per unit time is dn c = σ T c U dt hν ; (.65) combining this with (.64), we find the energy gain of the radiation field by scattering of the charge per scattering process, Eγ = de c dt ( dnc dt ) 1 = 4 3 hνγ β = 4 3 γ β E γ ; (.66).6 Compton Scattering.6.1 Energy-Momentum Conservation we now consider electromagnetic radiation as being composed of photons; if an ensemble of charges is embedded into a radiation field, energy is transfered by scattering from the photons to the charges and back; if the radiation temperature is higher than the temperature of the charge ensemble, energy flows from the radiation 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 to the radiation field; an incoming photon with momentum hν e/c hits an electron with momentum p; after scattering, the photon and the electron have momenta hν e /c and p ; conservation of momentum and energy imply where hν e + c p = hν e + c p, hν + E = hν + E, (.67) E = c p + m c 4 (.68) according to the relativistic energy-momentum relation;

32 KAPITEL. SCATTERING solving the energy equation for E and inserting (.68) yields c p = c p + h (ν ν ) + Eh(ν ν ), (.69) while the momentum equation implies c p = c p + h (ν e ν e ) + h(ν e ν e )c p ; (.70) subtracting (.69) from (.70) and cancelling suitable terms gives hνν (1 cos θ) = E(ν ν ) c p(ν e ν e ), (.71) where θ is the angle between e and e ; if the electron is originally at rest, p = 0 and E = mc, and (.71) simplifies to ν ν = h mc νν (1 cos θ), (.7) and in the limit of very low photon energy, hν mc, we find for the relative energy change of the Compton-scattered photon E γ E γ = ν ν ν = E γ (1 cos θ), (.73) mc and if hν mc, quantum electrodynamics must be used anyway; averaging (.73) over all scattering angles θ, we find the mean energy loss per Compton scattering, E γ = E γ mc 1 1 (1 cos θ)d(cos θ) = E γ mc ; (.74).6. Energy Balance the total energy transfer to the radiation field due to the motion of a single charge is given by the difference between the energy gain (.66) per scattering and the energy loss per Compton scattering (.74), ( 4 Eγ = 3 γ β E ) γ E mc γ ; (.75) for photons with E γ mc in the relativistic limit, β 1, and Eγ 4 3 γ E γ, (.76) which can become a very large number; in that way, for example, CMB photons can be converted to X-ray photons;

33 .6. COMPTON SCATTERING 3 in the thermal limit of (.75), we can approximate v c, thus γ 1, and mv = 3kT e ; then ( 4v Eγ 3c E ) γ mc thus, the photons gain energy (on average), if E γ = ( 4kT e E γ ) E γ mc ; (.77) 4kT e > E γ (.78) (inverse Compton scattering), and lose energy otherwise (Compton scattering) Compton scattering causes fast charges to lose energy; typical time scales are, according to (.64) t c E de/dt = γmc 4 σ 3 TUcγ β = 3 4 mc γβ σ T U ; (.79) for non-relativistic, thermal electrons, E = 3kT e / and γ 1, and t c = 3 kt ec 4 σ 3 TUcv = 9 8 mc σ T U ; (.80) after N s scatterings, the total energy transfer from thermal electrons to the photons is ( E E = 1 + 4kT ) Ns ( ) e 4kTe N s exp e 4y, (.81) mc mc where the Compton parameter was introduced; y 4kT en s mc (.8) if the electron number density is n e, the number of scatterings per path length dl is dn s = n e σ T dl N s = σ T n e dl, (.83) and thus the Compton-y parameter becomes y = kt e mc σ T n e dl ; (.84)

34 4 KAPITEL. SCATTERING.7 The Kompaneets Equation we need an additional equation which specifies how the photon spectrum is changed due to the scatterings; for deriving it, we assume that a homogeneous, thermal distribution of electrons is located in a homogeneous sea of radiation, such as, for example, a galaxy cluster in the microwave background; the collisions with the electrons change the photon energy, but not their number, and thus their spectrum cannot remain a Planck spectrum; let n(ν) be the occupation number of photon states with frequency ν; then, the Boltzmann equation requires ( ) n(ν) dσ = d 3 p dω c (.85) t dω { n(ν ) [1 + n(ν)] N(E ) n(ν) [ 1 + n(ν ) ] N(E) } ; this equation has the following meaning: the occupation number at the frequency ν changes due to scattering from ν to ν, and from ν to ν; the term n(ν) [ 1 + n(ν ) ] N(E) (.86) quantifies how many photons there are at frequency ν, corrected by the factor for stimulated emission from ν to ν, and multiplies with the number of collision partners N(E) at energy E; in other words, it quantifies the number of collisions away from frequency ν; analogously, the term n(ν ) [1 + n(ν)] N(E ) (.87) quantifies the opposite scattering, i.e. scattering processes increasing the occupation number at frequency ν; of course, the energy difference between photon frequencies ν and ν must be balanced by the difference between the energies E and E ; the integral over d 3 p integrates over the electron distribution, and the factor dσ dω (.88) dω specifies the probability for scattering photons from frequency ν to frequency ν or backward; we assume thermal photon and electron distributions, and restrict ourselves to the limit of Thomson scattering, which applies if hν mc ; (.89) moreover, we assume small changes in the photon frequency, hence δν ν ν ν ; (.90)

35 .7. THE KOMPANEETS EQUATION 5 moreover, the electron energy distribution is ( N(E) exp E ), (.91) kt e and energy conservation requires E = hν hδν ; (.9) now, both n(ν) and N(E) can be expanded in Taylor series up to second order, n(ν ) = n(ν) + n ν δν + 1 n ν δν + O(δν 3 ), (.93) N(E ) = N(E) N E hδν + 1 N E h δν + O(δν 3 ), where (.91) allows us to use N E = N(E) kt e, N E = N(E) (kt e ) ; (.94) for simplification, we now define the dimension-less photon energy, scaled by the thermal electron energy and find x hν kt e (.95) n(x ) n(x) + n x δx + 1 n x δx, ] N(E ) N(E) [1 + δx + δx ; (.96) with these approximations, we return to the original equation (.86) for n(ν) and obtain [ ] n n = + n(n + 1) I 1 t x + 1 [ ] n + (1 + n) n + n(n + 1) I x, (.97) x with the abbreviations I i d 3 p dω dσ dω cδxi N(E) (.98) the energy change of a photon scattering off a moving electron follows from (.71), adopting the non-relativistic limit E = mc + p m and using (.89) and (.90); this yields (.99) hδν = hν mc ( e e ) p δx = x mc ( e e ) p ; (.100)

36 6 KAPITEL. SCATTERING using this result, the integrals I i can be carried out straightforwardly; with the unpolarised Thomson cross section (.34), we first find I = σ T n e c kt ex mc ; (.101) for evaluating I 1, we note that I 1 is the mean rate of relative energy transfer, quantified by δx from the electrons to the photons, and therefore the mean energy transfer rate, divided by kt e ; from (.77), we know that this is Eγ = x(kt e ) m e c (4 x) (.10) per scattering, and multiplying with the collision rate n e σ T c gives I 1 = kt e m e c n eσ T c x(4 x) ; (.103) with these two expression for I i, we find the time derivative of n to be m e c 1 n kt e n e σ T c t = 1 [ ( )] n x 4 x x x + n + n ; (.104) we finally transform the time t to the Compton parameter, using dy = kt e m e c n eσ T c dt (.105) to find the Kompaneets equation n y = 1 x x [ x 4 ( n x + n + n )] ; (.106) the hot gas in galaxy clusters is much hotter than the cosmic background radiation; then, we can approximate the right-hand side of (.106) to lowest order in x, n y n x x + 4x n x ; (.107) inserting here the occupation number in thermal equilibrium, n (e x 1) 1, we find ( ) δn x n = δy e x (1 + e x ) 4xex (.108) (e x 1) e x 1 for the relative change of the occupation number, where x is now hν/kt and no longer hν/kt e!

37 Kapitel 3 Radiation Transport and Bremsstrahlung 3.1 Radiation Transport Equations we start with the collision-less Boltzmann equation for describing the temporal change of the photon distribution function in phase space, n t + n x + p n p = 0, (3.1) which is valid in absence of collisions; further reading: Shu, The Physics of Astrophysics, Vol I: Radiation, chapters, 3, and 15; Rybicki, Lightman, Radiative Processes in Astrophysics, chapter 5; Padmanabhan, Theoretical Astrophysics, Vol. I: Astrophysical Processes, sections 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); since the intensity I ν is proportional to n, the Boltzmann equation for photons can also be written as 1 I ν c t + e I ν x = 0 ; (3.) we now define the following quantities: F ν dω e I ν, P ν,i j 1 c dω e i e j I ν (3.3) and recall the spectral energy density U ν = 1 dω I ν ; (3.4) c integrating the Boltzmann equation (3.) first over dω, we obtain the equation U ν + F ν = 0 ; (3.5) t 7

38 8KAPITEL 3. RADIATION TRANSPORT AND BREMSSTRAHLUNG which has the form of a continuity equation and identifies F ν as the spectral radiation current density (spectral because it retains the dependence on frequency ν); this equation expresses energy conservation in the radiation field; if we multiply (3.) with e i first before integrating over dω, we find 1 I ν dω e i c t + I ν dω e i e j = 0, (3.6) x j and hence 1 F ν,i + c P ν,i j = 0 ; (3.7) c t x j this equation describes the change of the momentum current density, because ( Uν ) (c e) (3.8) c is the momentum density of the radiation field, and thus 1 F = 1 dω I c t c ν e (3.9) t 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, we know from the first chapter that the energy equation must be augmented by source and sink terms on its right-hand side; we had di ν dl = j ν α ν I ν = 1 c di ν dt ; (3.10) integrating over dω, and assuming that j ν and α ν are isotropic, we find du ν = 4π j ν α ν U ν c = 4π j ν ρκ ν cu ν ; (3.11) dt we now re-define the emissivity, 4π j ν ρ j ν ρ j ν, (3.1) i.e. we refer it to the mass density, and write du ν dt = ρ( j ν κ ν cu ν ) ; (3.13) likewise, the momentum-conservation equation 1 di ν c dt = j ν α ν I ν (3.14) becomes after multiplication with e and integration over dω 1 d dω e I ν = dω j ν e dω α ν I ν e, (3.15) c dt

39 3.. LOCAL THERMODYNAMICAL EQUILIBRIUM 9 and thus 1 d F = α ν F ν = ρκ ν F ν, (3.16) c dt where we have assumed again that j ν and κ ν are isotropic including the emission and absorption terms, the transport equations are modified to read 1 c U ν + F ν = ρ( j ν κ ν cu ν ) t F ν,i t + c P ν,i j x j = ρκ ν F ν,i ; (3.17) these equations do not contain scattering terms yet! since the change in the momentum current density corresponds to a force density, and this force is caused by the interaction between radiation and matter, an oppositely directed and equally strong force must act on the matter as radiation pressure force; thus f rad = ρ c is the density of the radiation pressure force; 0 κ ν F ν dν (3.18) 3. Local Thermodynamical Equilibrium the moment equations for U ν and F ν are by no means easier to handle than the Boltzmann equation whose moments they are; we obviously need an additional approximation, or condition, in order to close the moment equations; the closure means that they can then be solved without progressing indefinitely to higher orders of moments; often, the mean free path of the photons is much smaller than the dimensions of the system under consideration; then, we can assume that thermodynamical equilibrium is locally established between the radiation field and the matter; under this condition, I ν B ν (T), (3.19) i.e. the specific intensity of the radiation field is the Planckian intensity of a black body, and U ν = 1 dω I ν = 4π c c B ν(t) ; (3.0) under such circumstances, there is obviously no radiation flux any more because the radiation field is isotropic; in order to estimate the flux nonetheless, we study the orders of magnitude of the different terms in the moment equations;

40 30KAPITEL 3. RADIATION TRANSPORT AND BREMSSTRAHLUNG time derivatives can typically be neglected because temporal changes of the quantities U ν, F ν and P ν,i j occur on an evolutionary time scale, while the other terms change according to the streaming 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 P ν,i j x j (3.1) in the approximation of Local Thermodynamical Equilibrium (LTE), we further have P ν,i j P ν δ i j = U ν 3 δ i j (3.) because of the (local) isotropy of the radiation field, and thus F ν,i c ρκ ν U ν x i c ρκ ν U ν R, (3.3) where R is a typical dimension of the system; the mean free path λ ν is determined by λ ν nσ ν = λ ν ρκ ν 1, (3.4) and (3.3) can thus be approximated by ( λν ) F ν,i cu ν, (3.5) R which is smaller by a factor λ ν /R compared to the transparent case (in which κ 0 and λ ν R; using this estimate for F ν, we return to the Eq. (3.17) for the partial 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 λ ν R ; (3.6) the second term on the right-hand side is ρκ ν cu ν c ( ) R U ν F ν F ν ; (3.7) λ ν λ ν thus, because of the assumption of local thermodynamical equilibrium, the divergence of F ν is negligibly small; consequently, we must require ρ j ν ρκ ν cu ν U ν ρκ ν cα ν = 4π c B ν(t), (3.8) as anticipated;

41 3.. LOCAL THERMODYNAMICAL EQUILIBRIUM 31 accordingly, if λ ν R and t evol λ ν /c, the solutions of the moment equations are F ν,i c ρκ ν P ν,i j x j, U ν 4π c B ν(t) ; (3.9) because of (local) isotropy, we had and thus P ν,i j U ν 3 δ i j 4π 3c B ν(t) δ i j, (3.30) F ν,i 4π 3ρκ ν ( ) Bν T, (3.31) T x i i.e. the flux will become proportional to the temperature, which is characteristic for diffusion processes; for convenience, we now introduce the Rosseland mean opacity, κ 1 R dν ( ) κ 1 B ν (T) 0 ν T dν ( ) ; (3.3) B ν (T) T here, we can use the fact that ( ) Bν (T) dν = T T dν B ν (T) = T ( ) cat 4, (3.33) 4π where a π k 4 erg = (3.34) 15 ( c) 3 K 4 cm 3 is the so-called Stefan-Boltzmann constant; using this, we obtain the expression F = dν F ν = 4π T 0 3ρ x = 4π T d ( ) cat 4 3ρκ R dt 4π for the radiative energy flux; 0 dν B ν κ ν T = c 3ρκ R ( at 4) (3.35) the energy which is streaming away interacts with the absorbing matter and thus exerts a force on it, which is determined by the right-hand side of the momentum-conservation equation, as described above: f rad = ρ c 0 dν κ ν F ν = ρ c 4π 3ρ 0 dν B ν T T x = 1 3 (at 4 ) = P, (3.36) which equals just the negative pressure gradient;

42 3KAPITEL 3. RADIATION TRANSPORT AND BREMSSTRAHLUNG a remark on units: the unit of U ν is [U ν ] = erg cm 3 Hz, (3.37) the unit of κ ν is [κ ν ] = cm g, (3.38) and thus the unit of F ν is and the unit of f rad is [ f rad ] = as it should be; g cm 3 [ F ν ] = [c][u ν ] = s cm cm g = g cm s 4 1 erg cm t Hz, (3.39) erg erg Hz = (3.40) cm s Hz cm 4 cm = g cm 1 4 s cm = dyn 3 cm = force 3 volume, 3.3 Scattering so far, we have only considered emission and absorption, but neglected scattering; scattering changes the distribution function of the photons by exchanging photons with different momenta; if we assume for simplicity that the scattering process changes the photon s momentum, but not its energy, we can write the scattering cross 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 and describes the directional distribution of the scattered photons; scattering increases the distribution function n( e) according to [ ] dn( e) dt + = dω [ Ne cσφ( e, e ) ] n( e } {{ } ) [1 + n( e)], # of scatterings e e (3.4) where the factor [1 + n( e)] is included for describing stimulated emission of photons with momentum direction e;

43 3.3. SCATTERING 33 analogously, losses due to scattering are given by [ ] dn( e) = dω [ N e cσφ( e, e ) ] n( e) [1 + n( e )], (3.43) dt and thus the total change of n( e) due to scatterings becomes dn( e) = dω [ N e cσφ( e, e ) ] dt { n( e ) [1 + n( e)] n( e) [1 + n( e )] } = dω [ N e cσφ( 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 direction of e, we obtain from (3.44) n( e) = N e cσn( e) + N e cσ dω φ( e, e )n( e ), (3.45) dt and thus 1 dn( e) c dt = ρκ sca ν n( e) + ρκ sca ν dω φ( e, e )n( e ), (3.46) where we have introduced the scattering opacity through κ sca ν = N e σ; since the intensity at fixed frequency is proportional to the occupation number, the same equation (3.46) also holds for I ν ; therefore, the transport equation for the specific intensity is changed in presence of scattering to 1 di ν c dt = 1 c I ν t + e I ν x = ρ j ν 4π ρκabs ν I ν ρκ sca ν [ I ν (3.47) ] dω φ( e, e )I ν ( e ) ; again, we now take the moments of the transport equation in order to see how the moment equations are changed by scattering; the first moment is obtained by integrating (3.48) over dω, U ν t + F ν = ρ j ν ρκ abs ν cu ν ρκ sca cu ν + ρκν sca cu ν (3.48) due to the normalisation of φ( e, e ); therefore, the scattering terms cancel, and the equation for U ν remains unchanged; the next moment equation simplifies if we further assume that dωφ( e, e ) e = 0 = dω φ( e, e ) e, (3.49) ν