A A

absorption thermal emission

Iv+Mv

Iv+Mv

Figure 2.9 Passage of light through dust grains. Radiation with specific intensity Iv enters an area A^ at the left. By the time it travels a distance As, the intensity changes to Iv + AIv through absorption, scattering and thermal emission by the grains.

lattice. Photons are also lost from the beam by scattering. Here, the induced excitation quickly decays and reemits another photon in a different direction. This second photon has a frequency identical to the first in the rest frame of the grain. An external observer sees a slight Doppler shift associated with the finite grain velocity.

For present purposes, we may lump absorption and scattering together, and simply note that the total extinction must be proportional to As. In addition, the rate of photon removal should vary linearly with the incident flux, i. e., with Iv itself. Finally, for a gaseous medium with a given admixture of grains, the extinction must be proportional to the total density p. We therefore write this negative contribution to AIv as -pKvIv. Here kv is the opacity, a quantity measured in cm2 g^1 that depends on the incident frequency v, the relative number of grains, and their intrinsic physical properties.

There are also several ways that Iv can be increased in As. The lattice vibrations excited by radiation also emit it, generally at infrared wavelengths. In addition to such thermal emission, photons are added to the beam by being scattered from beams propagating in other directions. As before, we lump these processes together and define an emissivity jv, such that jv Av AQ is the energy per volume per unit time emitted into the direction n. Writing the elementary volume as AA As and recalling that Iv is defined as an energy per unit area, we see that the augmentation to Iv is simply jv As. The full change to Iv is thus

AIU = -p Kv Iv As + ju As , Dividing through by As, we obtain the equation of transfer:

Although we have motivated this result for the specific case of radiation propagating through dust grains, both equation (2.20) and the associated terminology are applicable to any continuous medium which can remove or generate photons. For example, we will use the equation of transfer in Chapter 6 to discuss the interaction of radio waves with interstellar molecules.

The quantity 1 /pKv, which has the dimensions of a length, is known as the photon mean free path. The ratio of As to this length, i. e., the product pKv As, is the optical depth, denoted Atv .4 When a photon propagates through an optically-thick medium, i. e., one for which Atv > 1, it has a high probability of extinction. Conversely, radiation can travel freely in an optically-thin environment (Atv C 1). Note that the same physical medium can be optically thin or thick, depending on the frequency of the radiation in question.

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