2. NOTES ON RADIATIVE TRANSFER The specific intensity I ν
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1 1 2. NOTES ON RADIATIVE TRANSFER 2.1. The specific intensity I ν Let f(x, p) be the photon distribution function in phase space, summed over the two polarization states. Then fdxdp is the number of photons in volume dx with momenta in dp. The photon momentum is also often expressed in terms of the wavenumber, p = k = 2πˆp/λ. Introducing spherical coordinates in momentum space, we have dp = p 2 dpdω, where dω is an element of solid angle in the direction of photon propagation, ˆp = ˆk. Assume that the photons are propagating in a vacuum, so that p = hν/c, where h is Planck s constant. Then the density of photons in dp is fdp = fp 2 dpdω = f ( ) h 3 ν 2 dνdω, (1) c and the energy flux of these photons is ( ) h 4 fdp energy hν speed c = f ν 3 dνdω. (2) Now define the specific intensity I ν as the radiative energy flux per unit frequency, per unit solid angle i.e., the energy per time, per unit area, per unit frequency, per unit solid angle. In terms of f, this is ( ) h 4 energy flux = I ν dνdω = f c 2 ν 3 dνdω (3) ( h 4 ν 3 ) I ν = f. (4) c 2 To measure the specific intensity, consider a pixel of area da in a detector that can measure frequencies to within an accuracy dν with unit quantum efficiency. Assume that the photons are emitted by a source of angular size dω and impinge on the detector at an angle θ. Then the energy measured by the pixel in a time dt is I ν dt(da cos θ)dνdω. A key property of the specific intensity is that it is independent of the distance from the source. To see this, consider a source with a luminosity L ν, measured in erg s 1 Hz 1. If it is at a distance D, the flux per Hz is F ν = L ν /4πD 2. Let the projected area of the source be A; then it subtends a solid angle Ω = A/D 2. The specific intensity of the source is then I ν = F ν Ω = c 2 L ν 1 4πD 2 A/D 2 = L ν 4πA, (5) which is independent of the distance D. As a result, the specific intensity is often referred to as the surface brightness, and it is an intrinsic property of the source.
2 The blackbody intensity B ν The specific intensity for a blackbody is denoted B ν. Recall that a blackbody is characterized by a photon occupation number per polarization state N = For unpolarized radiation, N is related to f by 1 exp(hν/kt ) 1. (6) N = 1 2 fh3, (7) since f is summed over the two polarization states. Hence, in general ( ) 2hν I ν = λ 2 N, (8) where we have written this in an easy to remember form: I ν includes units of energy (hν) per unit area (λ 2 ), and the factor 2 is for the two polarization states of a photon. Altogether then, the intensity of a blackbody is B ν = ( ) 2hν λ 2 1 exp(hν/kt ) 1. (9) At low frequencies, the blackbody intensity approaches the Rayleigh-Jeans form The brightness temperature of a source, T b, is defined by B ν 2kT λ 2 (hν kt ). (10) B ν (T b ) = I ν. (11) At low frequencies, this simplifies to I ν = 2kT b /λ 2 ; note that T b is generally a function of frequency Moments of the specific intensity It is frequently convenient to average the specific intensity over solid angle. The most useful moments are Mean intensity J ν 1 I ν dω, (12) 4π Flux F ν I ν ˆkdΩ, (13) K ν 1 I ν ˆkˆkdΩ, (14) 4π
3 3 where ˆk is the direction of propagation. If there is a direction of symmetry n, then it is often convenient to use scalar values of the last two moments, F ν = F n = I ν µdω, (15) K ν = n K ν n = 1 I ν µ 2 dω, (16) 4π where µ cos θ n ˆk. The spectral energy density u ν and the number density of photons n ph, ν = u ν /hν are directly related to the mean intensity. Recall that the energy flux per Hz is I ν dω from equation (3). The energy density per Hz is then (I ν /c)dω. Integrating over all solid angle, we find hνn ph, ν = u ν = 4π c J ν. (17) The second moment, K ν, is proportional to the radiation pressure tensor, which represents the momentum flux. In general, the momentum flux of a beam of particles or photons impinging on a surface with normal n with speed v is momentum flux = density momentum/particle in n direction, pµ velocity in n direction, vµ. (18) For photons, the density in a beam of solid angle dω per unit frequency is (I ν /chν)dω, so the radiation pressure exerted on the surface is dp rad, ν = I νdω chν hνµ cµ = 1 c c I νµ 2 dω, (19) P rad, ν = 1 I ν µ 2 dω = 4π c c K ν. (20) For the particular cases of a beam of radiation (I = I 0 > 0 in Ω with µ 1), semi-isotropic radiation [I = I 0 H(µ), where H(x) is the step function], and isotropic radiation (I = I 0 ), we have dω = 2πdµ and the moments are: Beam Semi-isotropic Isotropic 1 Mean intensity J I 0 Ω/4π 2 I 0 I 0 Flux F I 0 Ω πi 0 0 I Radiation pressure P 0 Ω 2πI rad c = u 0 3c = 1 3 u 4πI 0 3c = 1 3 u. Note in particular that the flux from an istropically radiating surface, such as the surface of a star, is F = πi. (21) 2.2. Radiative Transfer Equation The radiative transfer equation describes how the specific intensity changes along a ray. Since I ν is proportional to the photon distribution function in phase space, Liouville s theorem states
4 4 that, in the absence of sources or sinks, di ν /ds = 0, where ds is an increment of length along a ray. We have already seen that this is the case in 2.1 above, where we showed that the specific intensity of a source is independent of distance if there is no absorption or scattering. In general, however, the specific intensity along a ray will increase because of emission and decrease because of absorption. Define the emissivity j ν as the rate of energy emission per unit volume, per unit frequency, per unit solid angle. Also, define κ ν as the opacity, such that κ ν I ν is the rate at which energy is absorbed from the beam per unit volume, per unit frequency, per unit solid angle. With this definition, the units of κ ν are inverse length; physically, the mean free path of a photon at frequency ν is 1/κ ν. Note that some authors, like Shu, define the opacity as ρκ ν, so that κ ν is the opacity per unit mass. Also note that, whereas j ν is per unit frequency, κ ν is at frequency ν. Recall that Einstein showed that in addition to normal emission, called spontaneous emission, there is also stimulated emission that is proportional to the specific intensity I ν. It thus acts as a negative absorption, and we shall include stimulated emission in the absorption coefficient. The radiative transfer equation is then di ν ds = j ν κ ν I ν. (22) This form of the radiative transfer equation treats scattering as absorption followed by re-emission. We shall usually ignore scattering in this course, however Formal solution of the equation of radiative transfer Define the source function S ν as and the optical depth by S ν j ν κ ν, (23) dτ ν κ ν ds. (24) Since the photon mean free path is κ 1 ν, it follows that the optical depth τ ν = sκ ν is the number of mean free paths in a distance s. The equation of radiative transfer then becomes which can be readily integrated to give di ν dτ ν = S ν I ν, (25) where I ν = I ν, 0 e τ ν, 0 + τ ν = τν, 0 0 observer s S ν e τ ν dτ ν, (26) κ ν ds. (27)
5 5 This has a simple interpretation: the observed intensity is the intensity at some boundary s 0 attenuated by the optical depth to s 0 plus the emission at all the intervening points (S ν dτ ν = j ν ds) attenuated by the optical depth to those points LTE and Kirchoff s Law In thermodynamic equilibrium at temperature T, particle velocity distributions are Maxwellian at T, all atomic and molecular 1 levels are populated in equilibrium, n j exp (E j/kt ), and the radiation field is B ν (T ). In thermodynamic equilibrium, I ν = B ν is independent of position, so the equation of radiative transfer implies Kirchoff s Law j ν = B ν (T ) = 2hν 1 κ ν λ 2 exp (hν/kt ) 1. (28) In Local Thermodynamic Equilibrium, LTE, the particles and the populations of the internal states are in equilibrium, but the radiation field need not be Planckian (I ν B ν ). It follows that the rate of spontaneous emission j ν is identical to that in full thermodynamic equilibrium, and Kirchoff s Law continues to apply. This is very useful in being able to infer the emissivity, as we shall see when we discuss dust grains, for example. We can now explicitly solve the radiative transfer equation for an isothermal slab in LTE. Assume that there is no radiation impinging on the slab from the rear (I ν, 0 = 0). Then, since j ν /κ ν = S ν = B ν (T ) is independent of position, we have I ν = B ν (T ) τν 0 e τν dτ ν (29) = B ν (T ) ( 1 e ) τν (30) { B ν (T )τ ν = j ν s (τ ν 1) (31) B ν (T ) (τ ν 1) Emission and Absorption Einstein Coefficients A and B Consider two levels of an atom separated by an energy E jk = hν jk. Let A kj be the rate at which an atom in the upper state k makes a spontaneous transition to the lower state j. Let B jk u ν be the rate of absorption of photons of frequency ν jk by an atom in the lower state. (Note: Sometimes B jk is defined such that B jk J ν is the rate of absorption, as in Shu, and this leads to a 1 Henceforth in this lecture, we shall refer simply to atoms, although the results apply equally well to molecules.
6 6 difference of 4π/c in some of the relations.) In 1916, Einstein introduced the idea that there is a third process, stimulated emission, that removes atoms from the upper state at a rate B kj u ν. Note that the energy levels j and k are not precise, so there is a small spread in the energy E jk of the photons that can interact with these two levels. We assume that I ν is constant over this narrow frequency range, which is termed the natural line width. In Thermodynamic Equilibrium, the ratio of the populations of the two levels are given by the Boltzmann formula, n k n j = g k g j e hν/kt, (32) where g j is the statistical weight of state j. Since the level populations are in a steady state, the rate of excitation and de-excitation must balance: [ ( g k e hν/kt Akj e hν/kt 1 ) ] g j (4π/c)(2hν/λ 2 ) + B kj [ g k cλ 2 g j 8πhν A kj (1 ) ] e hν/kt + B kj e hν/kt n k (A kj + B kj u ν = n jb jk u ν (33) n ( ) k Akj + B kj = B jk (34) n j u ν = B jk (35) = B jk. (36) This must be valid for all T, so the terms with and without the factor exp (hν/kt ) must be equal: ( λ 2 ) c B kj = A kj (37) 8πhν B jk = g k g j B kj. (38) Hence the emission and absorption properties are determined by a single quantity, A kj, that is intrinsic to the levels in the atom Emissivity: j ν For an emission line, j ν is the rate at which energy is emitted in the line per unit volume, per steradian, per unit frequency. The photons are emitted over a range of frequencies, due both to the natural line width and to the motions of the atoms. Let φ ν be the emission line profile, which describes the frequency dependence of the line emission: j ν φ ν, where the line profile is normalized to unity: φ ν dν = 1. (39) As a result, j ν = φ ν j ν dν. (40)
7 7 The total rate of emission in the line per steradian per unit volume is 1/4π times the density of atoms in the upper state, n k, times the rate of spontaneous transitions, A kj, times the energy per photon: j ν dν = 1 4π n ka kj hν jk. (41) In terms of the line profile (which is assumed to be narrow, ν ν, so that ν ν jk to high accuracy), j ν = 1 4π n ka kj hν jk φ ν (42) with the aid of equation (40) Absorption: κ ν Let σ ν be the cross section for absorption of a photon of frequency ν as measured in the rest frame of the atom. Then the rate of absorption of energy (with no correction for stimulated emission) from a radiation field with energy density du ν = (I ν /c)dω (43) is B jk du ν hν = dω I ν σ ν dν. (44) We assume that the intensity is constant over the very narrow natural line width and define s u σ ν dν, (45) which is the frequency-integrated cross section, uncorrected for stimulated emission. Equation (44) then yields hν c B jk = s u. (46) One can show that s u is related to the oscillator strength of the transition by s u = πe2 m e c f jk = f jk cm 2 Hz. (47) The absorption coefficient κ ν includes the effects of stimulated emission. Let s ν be the absorption cross section, corrected for stimulated emission; then κ ν = n j s ν. If we irradiate a gas with an intensity I ν that is independent of frequency, the net rate at which energy is absorbed in dω is (n j B jk n k B kj )du ν hν = dω κ ν I ν dν = cdu ν n j s ν dν, (48) s jk s ν dν = hν c B jk ( 1 g jn k g k n j ), (49)
8 8 where we used equation (43) in the first equation. Note that the absorption is reduced by the effects of stimulated emission represented by the second term in this expression. We assume that the absorption profile is the same as the emission profile, so that s ν = s jk φ ν. We then obtain ( κ ν = n j s jk φ ν = n j s u 1 g ) jn k φ ν. (50) g k n j Define the excitation temperature T ex by n k n j g k g j e hν/ktex. (51) In LTE, T ex = T. The expression for the absorption coefficient then simplifies to κ ν = n j s u [1 exp( hν/kt ex )] φ ν. (52) Using the results for j ν and κ ν, we obtain the generalized Kirchoff s Law j ν κ ν = B ν (T ex ). (53) REFERENCES: Aspects of this material are discussed by Spitzer (Chap 3), Shu (Vol 1, Chaps 1-4), and Osterbrock and Ferland (Appendix 1). Elements of the theory of the transfer of polarized radiation, which we shall not discuss in any detail, are given in the first chapter of Chandrasekhar s book, Radiative Transfer.
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