Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 317 / 1 / b / Solution

In the radiative-zero white-dwarf envelope, $dP/dT=(17/4)P/T=(17/4)\mathcal{R}\rho$. Substitute this derivative into the stellar hydrostatic equation:Integrating from the idealized zero-temperature surface at $R$ givesThis also gives $d\log T/d\log P=4/17$. A finite surface temperature adds $T_s$ to the integrated profile if the pure power-law pressure relation is retained locally. Near the actual photosphere, the integration constant in the pressure law also matters; the displayed expression is the radiative-zero limit implicit in the printed result and accurate in the deep envelope when $T_c\gg T_{\mathrm{eff}}$.