Propagation of radiation in a medium, also called radiative transfer, is one of the basic problems of astrophysics. The subject is too complicated to be discussed here in any detail. The fundamental equation of radiative transfer is, however, easily derived.

Assume we have a small cylinder, the bottom of which has an area d A and the length of which is dr. Let Iv be the intensity of radiation perpendicular to the bottom surface going into a solid angle dw ([Iv] = Wm—2 Hz-1 sterad-*). If the intensity changes by an amount dIv in the distance dr, the energy changes d Eem = jv dr d A dv dm dt. The equation d E = —d Eabs + d Eem gives then d Iv = —av Iv dr + jv dr

avdr av

We shall denote the ratio of the emission coefficient jv to the absorption coefficient or opacity av by Sv :

Sv is called the source function. Because av dr = drv, where tv is the optical thickness at frequency v, (5.37) can be written as dTv = -iv+Sv dTv

Equation (5.39) is the basic equation of radiative transfer. Without solving the equation, we see that if Iv < Sv, then dIv/drv > 0, and the intensity tends to increase in the direction of propagation. And, if Iv > Sv, then dIv/drv < 0, and Iv will decrease. In an equilibrium the emitted and absorbed energies are equal, in which case we find from (5.35) and (5.36)

Substituting this into (5.39), we see that dIv/drv = 0. In thermodynamic equilibrium the radiation of the medium is blackbody radiation, and the source function is given by Planck's law:

2hv3

Even if the system is not in thermodynamic equilibrium, it may be possible to find an excitation temperature Texc such that Bv(Texc) = Sv. This temperature may depend on frequency.

A formal solution of (5.39) is

Here Iv(0) is the intensity of the background radiation, coming through the medium (e. g. an interstellar cloud) and decaying exponentially in the medium. The second term gives the emission in the medium. The solution is only formal, since in general, the source function Sv is unknown and must be solved simultaneously with the intensity. If Sv(tv) is constant in the cloud and the background radiation is ignored, we get

ement dz is related to that along the light ray, dr, according to dz = dr cos 0 .

With these notational conventions, (5.39) now yields

This is the form of the equation of radiative transfer usually encountered in the study of stellar and planetary atmospheres.

A formal expression for the intensity emerging from an atmosphere can be obtained by integrating (5.44) from tv = (we assume that the bottom of the atmosphere is at infinite optical depth) to tv = 0 (corresponding to the top of the atmosphere). This yields

sec 0 dTv

Iv(Tv) = Sv I e—(Tv —t)dt = Sv(1 — e—Tv). (5.42)

If the cloud is optically thick (tv ^ 1), we have

i. e. the intensity equals the source function, and the emission and absorption processes are in equilibrium.

An important field of application of the theory of radiative transfer is in the study of planetary and stellar atmospheres. In this case, to a good approximation, the properties of the medium only vary in one direction, say along the z axis. The intensity will then depend only on z and 0, where 0 is the angle between the z axis and the direction of propagation of the radiation.

In applications to atmospheres it is customary to define the optical depth tv in the vertical direction as dTv = —av dz .

Conventionally z increases upwards and the optical depth inwards in the atmosphere. The vertical line el-

This expression will be used later in Chap. 8 on the interpretation of stellar spectra. Telescopes Mastery

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