Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 2 / a

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- Solution a

Solution

Hydrostatic equilibrium requires

\frac{d p}{d r} = - ρ \frac{d Φ}{d r} = - ρ Ω^{2} r .

For the polytrope

p = K ρ^{1 + 1/ n}

, the specific enthalpy is

h (ρ) = \int \frac{d p}{ρ} = (n + 1) K ρ^{1/ n} .

Thus

h + Φ

is constant. Since

ρ = p = 0

at

r = R

, this constant is

Ω^{2} R^{2} /2

, and

ρ (r) = [\frac{Ω ^{2} ( R ^{2} - r ^{2} )}{2 ( n + 1 ) K}]^{n},

p (r) = K [\frac{Ω ^{2} ( R ^{2} - r ^{2} )}{2 ( n + 1 ) K}]^{n + 1}

for

0 \leq r \leq R

, with both fields zero outside the model star.

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