(i) A spherically symmetric star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is distance from the star's centre. Stating without proof any theorems you may need, show that mechanical equilibrium implies the Newtonian pressure support equation

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

where $m(r)$ is the mass within radius $r$ and $P^{\prime}=d P / d r$.

Write down an integral expression for the total gravitational potential energy, $E_{g r}$. Use this to derive the "virial theorem"

$$E_{g r}=-3\langle P\rangle V$$

when $\langle P\rangle$ is the average pressure.

(ii) Given that the total kinetic energy, $E_{k i n}$, of a spherically symmetric star is related to its average pressure by the formula

$$E_{k i n}=\alpha\langle P\rangle V$$

for constant $\alpha$, use the virial theorem (stated in part (i)) to determine the condition on $\alpha$ needed for gravitational binding. State the relation between pressure $P$ and "internal energy" $U$ for an ideal gas of non-relativistic particles. What is the corresponding relation for ultra-relativistic particles? Hence show that the formula $(*)$ applies in these cases, and determine the values of $\alpha$.

Why does your result imply a maximum mass for any star, whatever the source of its pressure? What is the maximum mass, approximately, for stars supported by

(a) thermal pressure,

(b) electron degeneracy pressure (White Dwarf),

(c) neutron degeneracy pressure (Neutron Star).

A White Dwarf can accrete matter from a companion star until its mass exceeds the Chandrasekar limit. Explain briefly the process by which it then evolves into a neutron star.