For the plane-parallel <radiative transfer equation>
$$
\mu\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu,
$$
angular integration gives the zeroth <radiation-field moment> equation
$$
\frac{dH_\nu}{d\tau_\nu}=J_\nu-S_\nu.
$$
The net radiative heating per unit volume is therefore
$$
4\pi\int_0^\infty\alpha_\nu(J_\nu-S_\nu)\,d\nu.
$$
<Radiative equilibrium> requires it to vanish, equivalently that the frequency-integrated <radiative flux> be independent of depth:
$$
\boxed{\int_0^\infty\alpha_\nu(J_\nu-S_\nu)\,d\nu=0}.
$$
In <local thermodynamic equilibrium> with <coherent isotropic scattering>,
$$
S_\nu=(1-\omega_\nu)B_\nu(T)+\omega_\nu J_\nu,
$$
where $\omega_\nu$ is the <single-scattering albedo>. Since $\alpha_\nu(1-\omega_\nu)=\alpha_{\nu,\rm abs}$, the condition becomes
$$
\boxed{\int_0^\infty\alpha_{\nu,\rm abs}
[J_\nu-B_\nu(T)]\,d\nu=0}.
$$
Conservative scattering redistributes directions but contributes no net material heating.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 1 / a / Solution

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.


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