Local radiative equilibrium ($\int {\kappa }_{\nu }(J_{\nu }-B_{\nu })d\nu =0$)

A material element is in local radiative equilibrium when its absorbed and emitted radiative powers balance, so radiation causes no net local heating. In a non-scattering grey atmosphere in local thermodynamic equilibrium, this reduces to equality of the angular mean specific intensity $J$ and the Planck function $B$. Integrating the radiative transfer equation over solid angle then shows that the radiative flux is constant through a plane-parallel atmosphere. The radiation can remain anisotropic; equilibrium constrains its angular mean, not every ray separately.