= Solution
The <Rosseland mean opacity> is the transport opacity appropriate to an optically thick region close to <local thermodynamic equilibrium>. If $\kappa_\nu$ is monochromatic transport opacity and $B_\nu(T)$ the <Planck function>, then
$$
\boxed{\frac1{\kappa_R}=\frac{\int_0^\infty\kappa_\nu^{-1}(\partial B_\nu/\partial T)\,d\nu}{\int_0^\infty(\partial B_\nu/\partial T)\,d\nu}.}
$$
It is a weighted harmonic mean: low-opacity frequency windows carry much of the diffusive flux. One must combine the monochromatic processes before averaging, rather than adding their separately averaged opacities.
In hot fully ionized material, <electron-scattering opacity> supplies an almost density- and temperature-independent contribution at fixed composition. In the nonrelativistic <Thomson scattering> limit, $\kappa_{\rm es}=\sigma_Tn_e/\rho\simeq0.2(1+X)\ {\rm cm^2\,g^{-1}}$. <Free-free opacity>, the absorption counterpart of <thermal bremsstrahlung>, comes from photon absorption during encounters between free electrons and ions. Its approximate mean obeys <Kramers' opacity law>, $\kappa_{\rm ff}\propto\rho T^{-7/2}$, with composition and quantum correction factors.
When ions retain bound electrons, bound-free absorption is <photoionization>; edges, occupations and ionization fractions strongly affect its frequency dependence. Bound-bound absorption produces <absorption lines>. Numerous metal lines can close low-opacity windows and produce important opacity features in partially ionized layers. Line populations depend on excitation as well as ionization, while <Doppler broadening>, <pressure broadening> and overlapping lines change the transport windows. Stimulated inverse processes must be included consistently with detailed balance and the local thermal radiation field.
In cooler hydrogen-rich atmospheres, <negative hydrogen ion opacity> from both bound-free and free-free processes is often important. Its strong temperature dependence helps determine convective surface relations. In still cooler layers, molecular bands, and at sufficiently low temperatures dust absorption/scattering, may be important. These processes cannot be represented by one Kramers power law over the whole density-temperature plane.
An opacity calculation therefore needs the chemical abundances, the <equation of state>, electron density and bound-state populations. <Saha ionization equation> and thermal excitation populations give useful dilute LTE estimates; high-density pressure ionization, nonideal effects and altered level populations require corrections. Composition matters even when one writes only $\kappa(\rho,T)$: hydrogen, helium and the metal abundance pattern are additional fixed inputs.
In dense degenerate cores, <thermal conduction> by electrons can dominate energy transport. An equivalent conductive opacity may be introduced so that $1/\kappa_{\rm eff}=1/\kappa_R+1/\kappa_{\rm cond}$, because the two diffusive fluxes add; conduction is not itself a photon absorption process. Very hot regimes also require departures from the simple nonrelativistic scattering approximation. Near the optically thin photosphere, a Rosseland diffusion description alone is inadequate, and frequency-dependent <radiative transfer> with appropriate boundary conditions is required. These regime changes are why realistic stellar opacity tables, rather than one global power law, enter <stellar structure>.
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