A spherically symmetric star in hydrostatic equilibrium has density $\rho(r)$ and pressure $P(r)$, which satisfy the pressure support equation,

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

where $m(r)$ is the mass within a radius $r$. Show that this implies

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

Provide a justification for choosing the boundary conditions $d P / d r=0$ at the centre of the $\operatorname{star}(r=0)$ and $P=0$ at its outer radius $(r=R)$.

Use the pressure support equation $(*)$ to derive the virial theorem for a star,

$$\langle P\rangle V=-\frac{1}{3} E_{\mathrm{grav}}$$

where $\langle P\rangle$ is the average pressure, $V$ is the total volume of the star and $E_{\text {grav }}$ is its total gravitational potential energy.