Emission, absorption, and source functions

For positive extinction define $S_\nu=j_\nu/\alpha_\nu$, which has the same units as specific intensity. Dividing the transfer equation by extinction gives a derivative with respect to ray optical depth equal to source function minus intensity. If S is constant, an integrating factor gives an attenuated boundary contribution plus an emitted contribution. The two coefficients sum to one, so a uniform slab produces a weighted interpolation between its incoming intensity and S. This statement concerns a prescribed uniform source function, not every inhomogeneous atmosphere.

In the thin limit the emitted increment is approximately j times path length and the incident beam is only slightly attenuated. In the thick limit the incident contribution is exponentially suppressed and the emerging intensity approaches S. Neither limit identifies S with a blackbody unless the emission and absorption processes justify that identification. A cloud in front of a brighter background can therefore appear in absorption even while emitting photons locally. Whether its line is bright or dark depends on both its source function and the background intensity.

Separate true absorption alpha a from scattering extinction alpha s. For coherent isotropic scattering at the same frequency, the radiation added to a ray is alpha s times the mean intensity J. If the true thermal process has emissivity alpha a times the Planck function B, the total source function is a weighted mean of B and J. The effective thermal weight is $\varepsilon=\alpha_a/(\alpha_a+\alpha_s)$. This simplified expression assumes isotropic redistribution and no frequency shift; Compton scattering requires a frequency-coupled treatment instead.

A scattering-dominated medium need not establish the gas's thermal spectrum after only one extinction mean free path. Scattering changes directions without necessarily exchanging enough energy to equilibrate photons with matter. Moreover, J is generally determined by the entire transfer problem and its boundaries. Inserting an externally prescribed J into the local formula is a useful algebraic illustration, but does not solve the coupled angular radiation field self-consistently. This nonlocal dependence is why source functions cannot generally be tabulated from local density and temperature alone.

For a narrow isolated transition, normalize its profile $\phi_\nu$ to unit frequency integral. Spontaneous line emissivity is $j_\nu=(h\nu_0/4\pi)n_uA_{ul}\phi_\nu$ for isotropic emission. Net line absorption includes stimulated emission: $\alpha_\nu=(h\nu_0/4\pi)(n_lB_{lu}-n_uB_{ul})\phi_\nu$. Here the Einstein B coefficients are defined to multiply profile-averaged mean intensity, not spectral energy density; that choice fixes the four-pi factors and gives $A_{ul}/B_{ul}=2h\nu_0^3/c^2$. Densities are per cubic metre and the profile has inverse-hertz units.

Using the statistical-weight relation between the two B coefficients gives $S_\nu=(2h\nu_0^3/c^2)/[(g_un_l)/(g_ln_u)-1]$ under the common-profile approximation. In LTE the population ratio reduces this to the Planck function, but outside LTE it need not. A population inversion makes net absorption negative and permits amplification. The positive-optical-depth slab interpretations above deliberately exclude that maser regime. Even with positive absorption, a measured source function constrains level populations and radiative transfer together rather than automatically measuring the gas kinetic temperature.