Formal solution of the radiative transfer equation

For an outward ray through a nonscattering atmosphere,

Along a radial ray through a layer with constant source function, the incident specific intensity is attenuated by $e^{-\tau }$ and the layer adds $S(1-e^{-\tau })$. At small optical depth, $I_{\mathrm{out}}=I_{\mathrm{in}}+(S-I_{\mathrm{in}})\tau +O({\tau }^2)$. Whether the source exceeds the incident specific intensity determines the first-order emission or absorption contrast.

For a semi-infinite plane-parallel atmosphere with polynomial source function $S(\tau )=\sum a_j{\tau }^j$, the outward specific intensity is ${\int }_0^{\infty }S(\tau )e^{-\tau /\mu }d\tau /\mu$. Substitution $u=\tau /\mu$ gives coefficients $A_j=j!a_j$ in the surface polynomial in $\mu$. The deep boundary term must vanish. A local truncated Taylor series instead gives this contribution plus a remainder, so its surface polynomial is not automatically exact.

For a thermalized radiative transfer source function $S_{\lambda }=B_{\lambda }(T)$ constant along a ray, the specific intensity after optical-depth interval $\Delta \tau$ is $I_{\lambda }=B_{\lambda }+(I_{\mathrm{in}}-B_{\lambda })e^{-\Delta \tau /\mu }$ for $\mu >0$. As $\Delta \tau \to \infty$, it tends to the Planck function, independently of a bounded boundary input. A temperature gradient is unnecessary for blackbody emission. In a scattering medium, local thermodynamic equilibrium alone does not guarantee $S_{\lambda }=B_{\lambda }$; thermalization must be justified.

A non-scattering isothermal atmospheric layer in local thermodynamic equilibrium, of temperature $T_a$ and slant optical depth $t_{\lambda }$, lies above a blackbody background at $T_b$. The specific intensity isThus an increase in opacity shifts the specific intensity toward the layer's Planck function. If a line is more opaque than its continuum, it appears in absorption for $T_a<T_b$, in emission for $T_a>T_b$, and has no contrast for $T_a=T_b$. This is a local explanation, not a claim that all features diagnose one uniform layer.

When the source function varies approximately linearly with optical depth, the emergent intensity obeys $I_{\nu }(0,\mu )\simeq S_{\nu }({\tau }_{\nu }=\mu )$.

The relation is an approximation within radiative transfer, not the entire transfer theory.