= Solution
Use the notation $I_\nu(\mu,\tau)$ throughout, with optical depth increasing inward and $0<\mu\leq1$ for an outward ray. This removes the change in argument order used in the printed surface-intensity notation. Multiplying the <radiative transfer equation> by its integrating factor gives
$$
\frac{d}{d\tau}\left(I_\nu e^{-\tau/\mu}\right)=-\frac{S_\nu(\tau)}\mu e^{-\tau/\mu}.
$$
Integrating from the surface to a deeper boundary $b$ yields
$$
I_\nu(\mu,0)=I_\nu(\mu,b)e^{-b/\mu}+\frac1\mu\int_0^b S_\nu(t)e^{-t/\mu}\,dt.
$$
For a semi-infinite atmosphere with no exponentially growing homogeneous term, the boundary term vanishes as $b\to\infty$. The <formal solution of the radiative transfer equation> is therefore
$$
I_\nu(\mu,0)=\int_0^\infty S_\nu(t)e^{-t/\mu}\frac{dt}{\mu}.
$$
The usual deep-atmosphere boundary condition is necessary; a first-order differential equation alone does not fix the emerging <specific intensity>.
For a linear <source function>, the zeroth and first exponential moments give \b[$I_\nu(\mu,0)=a_0+a_1\mu$]. Equivalently $I_\nu(\mu,0)=S_\nu(\tau=\mu)$, the <Eddington-Barbier relation>. If the source rises inward, rays near the limb sample cooler, shallower layers, producing <limb darkening>.
For the finite polynomial, put $u=t/\mu$ and integrate each term. Repeated integration by parts gives $\int_0^\infty u^je^{-u}\,du=j!$, so the <polynomial source function moments for emergent intensity> give
$$
\boxed{I_\nu(\mu,0)=\sum_{j=0}^n j!a_j\mu^j,\qquad A_j=j!a_j.}
$$
The limiting grazing-ray <specific intensity> is $a_0$. If the displayed polynomial is only a local Taylor approximation to a more general source, this formula applies to that approximation and a source-function remainder also contributes; a finite local expansion is not automatically an exact description at every optical depth.
For a constant source in an overlying layer of optical thickness $\tau$, the radial ray has the exact transfer relation
$$
I_\nu(1,0)=I_\nu(1,\tau)e^{-\tau}+S_\nu(1-e^{-\tau}).
$$
Expanding the exponentials gives the <constant source transfer through a thin layer>:
$$
\boxed{I_\nu(1,0)=I_\nu(1,\tau)+[S_\nu-I_\nu(1,\tau)]\tau+O(\tau^2).}
$$
This finite-layer result does not require the <specific intensity> below the layer to equal its source function. The sign of $S_\nu-I_\nu(1,\tau)$ already distinguishes emission from absorption.
An <absorption line> forms when a bound-bound transition increases <opacity> at selected frequencies. In a photosphere whose <temperature> decreases outward, optical depth of order one at the line frequency occurs higher than at a nearby continuum frequency. In an absorption-dominated region in <local thermodynamic equilibrium>, the source is the <Planck function>; the cooler line-forming layer then emits less <specific intensity> than the deeper continuum-forming layer. The result is a dark spectral feature. Scattering and departures from LTE modify the source, so this temperature-gradient picture is a useful mechanism rather than a universal identity for every line.
Two distinct situations can produce <emission lines>. First, a heated <chromosphere> has an outward <temperature> rise, so an optically thick line can sample gas hotter than the continuum photosphere beneath it. Secondly, a hot optically thin circumstellar envelope or <stellar wind> emits line photons through collisional excitation or recombination, with no comparably bright background along all rays; its added line flux can exceed the underlying continuum. Winds of hot luminous stars can also give a mixture of emission and absorption across a Doppler-broadened line profile. These examples explain the change in <specific intensity> through either an enhanced source function or additional emitting material.
Back to article page
