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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-317/1/b/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-06

In the radiative-zero white-dwarf envelope,

d P / d T = (17/4) P / T = (17/4) R ρ

. Substitute this derivative into the stellar hydrostatic equation:

\frac{17}{4} R ρ \frac{d T}{d r} = - \frac{GMρ}{r ^{2}}, \frac{d T}{d r} = - \frac{4}{17} \frac{GM}{R r ^{2}} .

Integrating from the idealized zero-temperature surface at

R

gives

T (r) = \frac{4}{17} \frac{GM}{R} (\frac{1}{r} - \frac{1}{R}) = \frac{4}{17} \frac{μ m _{H}}{k} GM (\frac{1}{r} - \frac{1}{R}) .

This also gives

d lo g T / d lo g 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} ≫ T_{eff}

.

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