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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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