Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 311 / 1 / b / i

Table of contents
- Solution i

Solution

For constant density,

m (r) = \frac{4 π ρ _{0} r ^{3}}{3} = M \frac{r ^{3}}{R ^{3}}, M = \frac{4 π ρ _{0} R ^{3}}{3} .

Substitution in the pressure equation makes it separable. Integrating from

r

to the surface, where

p (R) = 0

, gives the Interior Schwarzschild solution

p (r) = ρ_{0} \frac{1 - 2 M r ^{2} / R ^{3} - 1 - 2 M / R}{3 1 - 2 M / R - 1 - 2 M r ^{2} / R ^{3}} .

Direct differentiation verifies the TOV equation and the surface boundary condition.

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