For $P=K\rho^{1+1/n}$, introduce the <Lane-Emden equation> variables
$$
\rho=\lambda\theta^n,
\qquad
r=\alpha\xi,
\qquad
\lambda=\rho_c,
$$
$$
\boxed{
\alpha^2=\frac{(n+1)K}{4\pi G}
\lambda^{1/n-1}}.
$$
Combining mass conservation with hydrostatic equilibrium gives
$$
\boxed{
\frac1{\xi^2}\frac d{d\xi}
\left(\xi^2\frac{d\theta}{d\xi}\right)
=-\theta^n}.
$$
Regularity and normalization impose
$$
\theta(0)=1,
\qquad
\theta'(0)=0.
$$
The stellar surface is the first positive zero $\xi_1$ of $\theta$. Integrating the equation once gives
$$
\boxed{
m(\xi)=-4\pi\alpha^3\lambda\xi^2\theta'(\xi)},
$$
and hence
$$
\boxed{
R=\alpha\xi_1,
\qquad
M=-4\pi\alpha^3\lambda\xi_1^2\theta'(\xi_1)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-317/2/ii/solution