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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-58/1/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 58 / 1 / b / Solution by

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

Evaluate radiation pressure on the gas-supported hydrostatic equilibrium solution. At the centre,

\frac{P _{rad, c}}{P _{gas, c}} = \frac{a T _{c}^{4}}{3 P _{c}} = \frac{125 πa G ^{3}}{15552 R ^{4}} M^{2} .

The factors

R^{- 4}

cancel between

T_{c}^{4}

and

P_{c}

, so at fixed mean molecular weight the ratio depends only on mass. Equivalently it is

(M / M_{eq})^{2}

, with

M_{eq} = \frac{15552}{125 πa} \frac{R ^{2}}{G ^{3/2}} = \frac{15552}{125 πa} \frac{k ^{2}}{G ^{3/2} μ ^{2} H ^{2}} .

For

H ≃ 1.6605 \times 1 0^{- 27} kg

this is approximately

14.6 μ^{- 2} M_{⊙}

, or

40.5 M_{⊙}

if

μ = 0.6

. If

H

denotes hydrogen-atom mass instead, use the corresponding mean molecular weight convention consistently; the coefficient is about

14.4 μ^{- 2} M_{⊙}

. Equality marks the breakdown of the gas-pressure approximation. It is a diagnostic based on the gas-only model, not an exact mixed-pressure stellar solution.

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