(a) Consider a spherically symmetric star with outer radius $R$, density $\rho(r)$ and pressure $P(r)$. By balancing the gravitational force on a shell at radius $r$ against the force due to the pressure gradient, derive the pressure support equation

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

where $m(r)=\int_{0}^{r} \rho\left(r^{\prime}\right) 4 \pi r^{\prime 2} d r^{\prime}$. Show that this implies

$$\frac{d}{d r}\left(\frac{r^{2}}{\rho} \frac{d P}{d r}\right)=-4 \pi G r^{2} \rho$$

Suggest appropriate boundary conditions at $r=0$ and $r=R$, together with a brief justification.

(b) Describe qualitatively the endpoint of stellar evolution for our sun when all its nuclear fuel is spent. Your discussion should briefly cover electron degeneracy pressure and the relevance of stability against inverse beta-decay.

[Note that $m_{n}-m_{p} \approx 2.6 m_{e}$, where $m_{n}, m_{p}, m_{e}$ are the masses of the neutron, proton and electron respectively.]