Radiative transfer equation

In plane-parallel optical-depth coordinates without scattering,with source function $S_{\nu }$.

The radiative transfer source function is emission per unit extinction. If $j_{\nu }$ is emission per unit path length and $\rho {\kappa }_{\nu }$ is extinction per unit path length, $S_{\nu }=j_{\nu }/(\rho {\kappa }_{\nu })$ and the transfer equation is $\mu {\partial }_{{\tau }_{\nu }}{I}_{\nu }=I_{\nu }-S_{\nu }$. Under thermal absorption in local thermodynamic equilibrium it is the Planck function; for conservative coherent isotropic scattering it is the mean intensity.

In an optically thick medium whose radiation field is nearly isotropic and thermal, the first angular correction to the Planck function produces the diffusive fluxwhere ${\kappa }_R$ is the Rosseland mean opacity.

The Rosseland mean opacity is the harmonic frequency average weighted by $\partial B_{\nu }/\partial T$. It controls radiative diffusion because flux preferentially travels through low-opacity frequencies.

It is a frequency-averaged opacity appropriate to the radiative-diffusion limit, not the general opacity concept itself.

In local thermodynamic equilibrium, matter populations are thermal at the local temperature and the source function is the Planck function, $S_{\nu }=B_{\nu }(T)$.

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.

Coherent isotropic scattering preserves photon frequency and redistributes direction uniformly. Its source function is the mean intensity $J_{\nu }$, so conservative scattering removes light from a pencil beam while preserving luminosity integrated over all escaping directions.

The single-scattering albedo is the fraction of total extinction due to scattering, ${\omega }_{\nu }={\alpha }_{\nu ,\mathrm{sca}}/{\alpha }_{\nu ,\mathrm{ext}}$. Its complement is the absorptive fraction.

Kirchhoff's law of thermal radiation states that a body in thermal equilibrium has equal absorptivity and emissivity at each frequency and direction. A layer with infrared absorptivity $ϵ$ therefore emits $ϵB_{\nu }(T)$.