Kramers' opacity law

Remarks

The law comes from H. A. Kramers' semiclassical formula for the absorption cross-section of an electron passing an ion. This formula is averaged over a Maxwellian distribution of electrons and over frequencies with the Rosseland weight. The factor $\rho$ appears because absorption needs an electron and an ion, so it grows like the square of the density, and the opacity is per unit mass. Example: if the temperature doubles at fixed density, the opacity drops by the factor $2^{7/2} \approx 11$. The law fails at low temperature, where absorption by negative hydrogen ions and by molecules dominates. It also fails at high temperature, where electron scattering dominates and the opacity becomes nearly independent of $\rho$ and $T$. In the equations of stellar structure it gives, together with the ideal gas law, a mass–luminosity scaling close to $L \propto M^{5.5} R^{-0.5}$.

Going deeper

For a student

Opacity enters the radiative temperature gradient, so it decides how steep the temperature profile must be and where convection sets in. Power laws of the form $\kappa \propto \rho^a T^{-b}$ allow homologous solutions of the stellar structure equations, so that scalings between mass, radius and luminosity can be derived without solving them. Detailed models use tables of opacities instead, but Kramers' law remains the reference case.