A star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the distance from the centre of the star. These quantities are related by the pressure support equation

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

where $P^{\prime}=d P / d r$ and $m(r)$ is the mass within radius $r$. Use this to derive the virial theorem

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

where $E_{\text {grav }}$ is the total gravitational potential energy and $\langle P\rangle$ the average pressure.

The total kinetic energy of a spherically symmetric star is related to $\langle P\rangle$ by

$$E_{\mathrm{kin}}=\alpha\langle P\rangle V$$

where $\alpha$ is a constant. Use the virial theorem to determine the condition on $\alpha$ for gravitational binding. By considering the relation between pressure and 'internal energy' $U$ for an ideal gas, determine $\alpha$ for the cases of a) an ideal gas of non-relativistic particles, b) an ideal gas of ultra-relativistic particles.

Why does your result imply a maximum mass for any star? Briefly explain what is meant by the Chandrasekhar limit.

A white dwarf is in orbit with a companion star. It slowly accretes matter from the other star until its mass exceeds the Chandrasekhar limit. Briefly explain its subsequent evolution.