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For a random complex wave amplitude $E$, its mean squared amplitude is $I=\mathbb{E}[|E{|}^2]$. Expanding $E=m+(E-m)$, with $m=\mathbb{E}[E]$, givesThe mixed terms vanish. Thus the squared coherent field is only one part of the mean intensity; the other is the diffuse contribution. A fixed acoustic or optical normalization converts this squared-amplitude quantity to physical energy flux. This ensemble mean differs from the angularly averaged radiative-transfer mean intensity.

The Hopf function encodes the departure of a plane-parallel grey atmosphere from a purely linear source profile. With physical bolometric radiative flux $F$ and standard mean intensity $J$, it is defined by $J=3F[\tau +q(\tau )]/(4\pi )$. In thermal radiative equilibrium, $T^4=3T_{\mathrm{eff}}^4[\tau +q(\tau )]/4$. The Eddington closure approximation with its approximate surface condition gives constant $q=2/3$; exact angular transfer generally gives a depth-dependent function.

Use the Einstein coefficients convention in which $B_{ij}$ multiplies spectral radiation energy density $u_{\nu }$, with $u_{\nu }=c^{-1}\int I_{\nu }d\Omega$. A normalized spectral line profile obeys $\int \phi (\nu )d\nu =1$. The spontaneous emission probability per upper-state atom per unit time is $A_{21}$; the stimulated emission and upward absorption rates per atom are respectively $B_{21}\int u_{\nu }\phi (\nu )d\nu$ and $B_{12}\int u_{\nu }\phi (\nu )d\nu$.

For isotropic spontaneous emission, $n_2{A}_{21}$ transitions occur per unit volume and time, each supplying photon energy $h{\nu }_{12}$. Dividing among $4\pi$ steradians givesFor a small incident beam, its contribution to $u_{\nu }$ is $I_{\nu }d\Omega /c$. Absorption removes and stimulated emission adds photons in the incident direction. Matching the net loss to ${\alpha }_{\nu }{I}_{\nu }d\Omega$ therefore givesReplacing ${\nu }_{12}$ by $\nu$ across the narrow spectral line profile gives the printed expressions. The negative sign is essential: an inverted population can amplify the incident specific intensity. Thermal equilibrium and the Planck function imply $g_1{B}_{12}=g_2{B}_{21}$ and $A_{21}/B_{21}=8\pi h{\nu }_{12}^3/c^3$, where $g_i$ is the state degeneracy. A convention using mean intensity instead changes both $B$ coefficients by $4\pi /c$.

Specify the radiation normalization first. Let $u_{\nu }$ be radiation energy density per ordinary frequency, with SI units $\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$. Then $A_{21}$ is the spontaneous emission rate per upper-state chemical atom, while $B_{12}{u}_{\nu }$ and $B_{21}{u}_{\nu }$ are the induced absorption and stimulated emission rates, evaluated at a sharp transition frequency. For a finite spectral line profile, replace $u_{\nu }$ by its profile-weighted average.

The Einstein coefficients obeyHere $g_i$ are the statistical weights of the two energy levels, and the field is unpolarized and isotropic. The units and physical dimensions areThe apparently time-independent dimension of $B$ follows because a per-hertz density includes an extra unit of time. If instead $B$ multiplies the mean intensity $J_{\nu }$, use $B^{(J)}=(4\pi /c)B$ and $A/B^{(J)}=2h{\nu }_0^3/c^2$. These distinct normalizations must not be mixed in the radiative transfer equation.

Put $\mu =\cos \theta$. For $I_{\nu }=A_{\nu }+B_{\nu }{\mu }^3$, the mean intensity isbecause the cubic term is odd. The K-integral, or second angular moment, isbecause the ${\mu }^5$ contribution is also odd. HenceThis intensity obeys the Eddington closure approximation even though it is not isotropic.

Specific intensity is defined by $dE=I_{\nu }\cos \theta dAd\Omega d\nu dt$. A ray bundle in free space expands in area while its solid angle contracts by the same factor, soThus specific intensity does not obey an inverse-square law; the flux of an unresolved source does because its apparent solid angle scales as distance${}^{-2}$.

For a cold medium with coherent, isotropic, conservative scattering, the source function is the mean intensity $J_{\nu }$. With scattering optical depth increasing along the ray,andThe unscattered pencil beam is attenuated by $e^{-\tau }$ and reappears as a diffuse halo in other directions. Coherent scattering preserves frequency, and conservative scattering preserves total luminosity when all outgoing directions are collected.

Put $\mu =\cos \theta$ and $S=j/\kappa$. Integrating the radiative transfer equation over solid angle givesThe first integral is the radiative flux $F$. A stationary atmosphere with no local energy source has $dF/d\tau =0$, soBy definition, the mean intensity is $J=(4\pi {)}^{-1}\int Id\Omega$. Thereforewhich is radiative equilibrium: the angle-integrated emission and absorption rates are equal.

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.