Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 4 / iv

Table of contents
- Solution iv

Solution

Stellar homology gives

P_{c} \sim \frac{G M ^{2}}{R ^{4}}, ρ_{c} \sim \frac{M}{R ^{3}} .

For an

n = 3

P = K ρ^{4/3}

, so these scalings imply

K \sim G M^{2/3}, M \propto (K / G)^{3/2} .

The equation of state in part i gives

K^{3} \propto \frac{1 - β}{β ^{4}}

at fixed chemical composition. Substitution into the homologous mass scale proves the Eddington quartic relation

M \propto \frac{( 1 - β ) ^{1/2}}{β ^{2}} .

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