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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-58/2/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 58 / 2 / a / Solution by

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

For a uniform-density stellar model,

ρ = 3 M / (4 π R^{3})

and

m (r) = 4 π ρ r^{3} /3

. With zero surface pressure, integrate the stellar hydrostatic equation:

P (r) = \int_{r}^{R} \frac{4 π G ρ ^{2}}{3} s d s = \frac{2 π G ρ ^{2}}{3} (R^{2} - r^{2}) .

Using the ideal gas equation of state,

T (r) = P (r) / (R ρ)

. Therefore

P_{c} = \frac{2 π G ρ ^{2} R ^{2}}{3} = \frac{1}{2} (\frac{4 π}{3})^{1/3} G M^{2/3} ρ^{4/3}, T_{c} = \frac{P _{c}}{R ρ} = \frac{G}{2 R} (\frac{4 π}{3})^{1/3} M^{2/3} ρ^{1/3} .

Equivalently

P_{c} = 3 G M^{2} / (8 π R^{4})

and

T_{c} = GM / (2 R R)

. The constant mass density is imposed as a structural approximation; a perfect gas can realize it hydrostatically by allowing its temperature and entropy to vary with radius.

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