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–1– 2. NOTES ON RADIATIVE TRANSFER 2.1. The speciﬁc intensity Iν Let f (x, p) be the photon distribution function in phase space, summed over the two polariza- tion states. Then f dxdp 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 = 2πˆ/λ. Introducing p spherical coordinates in momentum space, we have dp = p 2 dpdΩ, where dΩ is an element of solid ˆ ˆ angle in the direction of photon propagation, p = k. 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 3 h f dp = f p2 dpdΩ = f ν 2 dνdΩ, (1) c and the energy ﬂux of these photons is h4 f dp × energy hν × speed c = f ν 3 dνdΩ. (2) c2 Now deﬁne the speciﬁc intensity Iν 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 h4 energy ﬂux = Iν dνdΩ = f ν 3 dνdΩ (3) c2 h4 ν 3 ⇒ Iν = f. (4) c2 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 Iν 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 Lν , measured in erg s−1 Hz−1 . If it is at a distance D, the ﬂux per Hz is Fν = Lν /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 Fν Lν 1 Lν Iν = = 2 A/D 2 = , (5) ∆Ω 4πD 4πA 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– 2.1.1. The blackbody intensity Bν The speciﬁc intensity for a blackbody is denoted Bν . Recall that a blackbody is characterized by a photon occupation number per polarization state 1 N = . (6) exp(hν/kT ) − 1 For unpolarized radiation, N is related to f by 1 N = f h3 , (7) 2 since f is summed over the two polarization states. Hence, in general 2hν Iν = N, (8) λ2 where we have written this in an easy to remember form: Iν includes units of energy (hν) per unit area (λ2 ), and the factor 2 is for the two polarization states of a photon. Altogether then, the intensity of a blackbody is 2hν 1 Bν = 2 . (9) λ exp(hν/kT ) − 1 At low frequencies, the blackbody intensity approaches the Rayleigh-Jeans form 2kT Bν → (hν kT ). (10) λ2 The brightness temperature of a source, Tb , is deﬁned by Bν (Tb ) = Iν . (11) At low frequencies, this simpliﬁes to Iν = 2kTb /λ2 ; note that Tb is generally a function of frequency. 2.1.2. Moments of the speciﬁc intensity It is frequently convenient to average the speciﬁc intensity over solid angle. The most useful moments are 1 Mean intensity Jν ≡ Iν dΩ, (12) 4π Flux Fν ≡ ˆ Iν kdΩ, (13) 1 ˆˆ Kν ≡ Iν kkdΩ, (14) 4π –3– ˆ where k 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, Fν = F ·n= Iν µdΩ, (15) 1 Kν = n · Kν · n = Iν µ2 dΩ, (16) 4π ˆ where µ ≡ cos θ ≡ n · k. The spectral energy density uν and the number density of photons nph, ν = uν /hν are directly related to the mean intensity. Recall that the energy ﬂux per Hz is Iν dΩ from equation (3). The energy density per Hz is then (Iν /c)dΩ. Integrating over all solid angle, we ﬁnd 4π hνnph, ν = uν = Jν . (17) c The second moment, Kν , 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 (Iν /chν)dΩ, so the radiation pressure exerted on the surface is Iν dΩ hνµ 1 dPrad, ν = × × cµ = Iν µ2 dΩ, (19) chν c c 1 4π ⇒ Prad, ν = Iν µ2 dΩ = Kν . (20) c c For the particular cases of a beam of radiation (I = I0 > 0 in ∆Ω with µ 1), semi-isotropic radiation [I = I0 H(µ), where H(x) is the step function], and isotropic radiation (I = I0 ), we have dΩ = 2πdµ and the moments are: Beam Semi-isotropic Isotropic 1 Mean intensity J I0 ∆Ω/4π 2 I0 I0 (21) Flux F I0 ∆Ω πI0 0 I0 ∆Ω 2πI0 1 4πI0 1 Radiation pressure Prad c =u 3c = 3 u 3c = 3 u. Note in particular that the ﬂux from an istropically radiating surface, such as the surface of a star, is F = πI. 2.2. Radiative Transfer Equation The radiative transfer equation describes how the speciﬁc intensity changes along a ray. Since Iν is proportional to the photon distribution function in phase space, Liouville’s theorem states –4– that, in the absence of sources or sinks, dIν /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 jν as the rate of energy emission per unit volume, per unit frequency, per unit solid angle. Also, deﬁne κν as the opacity, such that κν Iν 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 κν are inverse length; physically, the mean free path of a photon at frequency ν is 1/κν . Note that some authors, like Shu, deﬁne the opacity as ρκν , so that κν is the opacity per unit mass. Also note that, whereas jν is per unit frequency, κν 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 dIν = j ν − κν Iν . (22) ds This form of the radiative transfer equation treats scattering as absorption followed by re-emission. We shall usually ignore scattering in this course, however. 2.2.1. Formal solution of the equation of radiative transfer Deﬁne the source function Sν as jν Sν ≡ , (23) κν and the optical depth by dτν ≡ κν ds. (24) Since the photon mean free path is κ−1 , it follows that the optical depth τν = sκν is the number ν of mean free paths in a distance s. The equation of radiative transfer then becomes dIν = Sν − Iν , (25) dτν which can be readily integrated to give τν, 0 Iν = Iν, 0 e−τν, 0 + Sν e−τν dτν , (26) 0 where observer τν = κν ds . (27) s –5– This has a simple interpretation: the observed intensity is the intensity at some boundary s 0 attenuated by the optical depth to s0 plus the emission at all the intervening points (Sν dτν = jν ds) attenuated by the optical depth to those points. 2.2.2. LTE and Kirchoﬀ ’s Law In thermodynamic equilibrium at temperature T , particle velocity distributions are Maxwellian at T , all atomic and molecular1 levels are populated in equilibrium, n∗ ∝ exp −(Ej /kT ), and the j radiation ﬁeld is Bν (T ). In thermodynamic equilibrium, Iν = Bν is independent of position, so the equation of radiative transfer implies Kirchoﬀ ’s Law jν 2hν 1 = Bν (T ) = 2 . (28) κν λ exp −(hν/kT ) − 1 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 (Iν = Bν ). It follows that the rate of spontaneous emission jν 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 jν /κν = Sν = Bν (T ) is independent of position, we have τν Iν = Bν (T ) e−τν dτν (29) 0 = Bν (T ) 1 − e−τν (30) Bν (T )τν = jν s (τν 1) → (31) Bν (T ) (τν 1). 2.3. Emission and Absorption 2.3.1. Einstein Coeﬃcients A and B Consider two levels of an atom separated by an energy Ejk = hνjk . Let Akj be the rate at which an atom in the upper state k makes a spontaneous transition to the lower state j. Let Bjk uν be the rate of absorption of photons of frequency νjk by an atom in the lower state. (Note: Sometimes Bjk is deﬁned such that Bjk Jν 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– 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 uν . 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 Iν 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 gk −hν/kT nj∗ = g e , (32) j where gj 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: n∗ (Akj + Bkj uν ) = n∗ Bjk uν k j (33) n∗ k Akj + Bkj = Bjk (34) n∗ j uν gk e−hν/kT Akj ehν/kT −1 + Bkj = Bjk (35) gj (4π/c)(2hν/λ2 ) gk cλ2 Akj 1 − e−hν/kT + Bkj e−hν/kT = Bjk . (36) gj 8πhν This must be valid for all T , so the terms with and without the factor exp −(hν/kT ) must be equal: λ2 c Bkj = Akj (37) 8πhν gk Bjk = Bkj . (38) gj Hence the emission and absorption properties are determined by a single quantity, A kj , that is intrinsic to the levels in the atom. 2.3.2. Emissivity: jν For an emission line, jν 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 φν be the emission line proﬁle, which describes the frequency dependence of the line emission: jν ∝ φν , where the line proﬁle is normalized to unity: φν dν = 1. (39) As a result, jν = φ ν jν dν. (40) –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, nk , times the rate of spontaneous transitions, Akj , times the energy per photon: 1 jν dν = nk Akj hνjk . (41) 4π In terms of the line proﬁle (which is assumed to be narrow, ∆ν ν, so that ν ν jk to high accuracy), 1 jν = nk Akj hνjk φν (42) 4π with the aid of equation (40). 2.3.3. Absorption: κν Let σν 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 duν = (Iν /c)dΩ (43) is Bjk duν hν = dΩ Iν σν dν . (44) We assume that the intensity is constant over the very narrow natural line width and deﬁne su ≡ σν dν , (45) which is the frequency-integrated cross section, uncorrected for stimulated emission. Equation (44) then yields hν Bjk = su . (46) c One can show that su is related to the oscillator strength of the transition by πe2 su = fjk = 0.0265fjk cm2 Hz. (47) me c The absorption coeﬃcient κν includes the eﬀects of stimulated emission. Let sν be the absorp- tion cross section, corrected for stimulated emission; then κν = nj sν . If we irradiate a gas with an intensity Iν that is independent of frequency, the net rate at which energy is absorbed in dΩ is (nj Bjk − nk Bkj )duν hν = dΩ κν Iν dν = cduν nj sν dν, (48) hν g j nk ⇒ sjk ≡ sν dν = Bjk 1 − , (49) c g k nj –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 sν = sjk φν . We then obtain g j nk κν = nj sjk φν = nj su 1 − φν . (50) g k nj Deﬁne the excitation temperature Tex by nk gk −hν/kTex ≡ e . (51) nj gj In LTE, Tex = T . The expression for the absorption coeﬃcient then simpliﬁes to κν = nj su [1 − exp(−hν/kTex )] φν . (52) Using the results for jν and κν , we obtain the “generalized Kirchoﬀ’s Law” jν = Bν (Tex ). (53) κν 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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