A spherically symmetric star with outer radius $R$ has mass density $\rho(r)$ and pressure $P(r)$, where $r$ is the distance from the centre of the star. Show that hydrostatic equilibrium implies the pressure support equation,

$$\frac{d P}{d r}=-\frac{G m \rho}{r^{2}}$$

where $m(r)$ is the mass inside radius $r$. State without proof any results you may need.

Write down an integral expression for the total gravitational potential energy $E_{\text {grav }}$ of the star. Hence use $(\dagger)$ to deduce the virial theorem

$$E_{\mathrm{grav}}=-3\langle P\rangle V,$$

where $\langle P\rangle$ is the average pressure and $V$ is the volume of the star.

Given that a non-relativistic ideal gas obeys $P=2 E_{\mathrm{kin}} / 3 \mathrm{~V}$ and that an ultrarelativistic gas obeys $P=E_{\text {kin }} / 3 V$, where $E_{\text {kin }}$ is the kinetic energy, discuss briefly the gravitational stability of a star in these two limits.

At zero temperature, the number density of particles obeying the Pauli exclusion principle is given by

$$n=\frac{4 \pi g_{s}}{h^{3}} \int_{0}^{p_{\mathrm{F}}} p^{2} d p=\frac{4 \pi g_{s}}{3}\left(\frac{p_{\mathrm{F}}}{h}\right)^{3}$$

where $p_{\mathrm{F}}$ is the Fermi momentum, $g_{s}$ is the degeneracy and $h$ is Planck's constant. Deduce that the non-relativistic internal energy $E_{\text {kin }}$ of these particles is

$$E_{\mathrm{kin}}=\frac{4 \pi g_{s} V h^{2}}{10 m_{p}}\left(\frac{p_{\mathrm{F}}}{h}\right)^{5}$$

where $m_{p}$ is the mass of a particle. Hence show that the non-relativistic Fermi degeneracy pressure satisfies

$$P \sim \frac{h^{2}}{m_{p}} n^{5 / 3} .$$

Use the virial theorem $(*)$ to estimate that the radius $R$ of a star supported by Fermi degeneracy pressure is approximately

$$R \sim \frac{h^{2} M^{-1 / 3}}{G m_{p}^{8 / 3}},$$

where $M$ is the total mass of the star.

[Hint: Assume $\rho(r)=m_{p} n(r) \sim m_{p}\langle n\rangle$ and note that $\left.M \approx\left(4 \pi R^{3} / 3\right) m_{p}\langle n\rangle .\right]$