 Solution

Use the notation $I_{\nu }(\mu ,\tau )$ throughout, with optical depth increasing inward and $0<\mu \le 1$ 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 givesIntegrating from the surface to a deeper boundary $b$ yieldsFor 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 thereforeThe 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 $I_{\nu }(\mu ,0)=a_0+a_1\mu$. 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 ${\int }_0^{\infty }{u}^j{e}^{-u}du=j!$, so the polynomial source function moments for emergent intensity giveThe 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 relationExpanding the exponentials gives the constant source transfer through a thin layer: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.