<Stellar homology> gives
$$
P_c\sim\frac{GM^2}{R^4},\qquad
\rho_c\sim\frac{M}{R^3}.
$$
For an $n=3$ polytrope, $P=K\rho^{4/3}$, so these scalings imply
$$
K\sim GM^{2/3},\qquad M\propto(K/G)^{3/2}.
$$
The equation of state in part i gives
$$
K^3\propto\frac{1-\beta}{\beta^4}
$$
at fixed chemical composition. Substitution into the homologous mass scale proves the <Eddington quartic relation>
$$
\boxed{M\propto\frac{(1-\beta)^{1/2}}{\beta^2}}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-317/4/iv/solution