The pressure support equation for stars is

$$\frac{1}{r^{2}} \frac{d}{d r}\left[\frac{r^{2}}{\rho} \frac{d P}{d r}\right]=-4 \pi G \rho$$

where $\rho$ is the density, $P$ is the pressure, $r$ is the radial distance, and $G$ is Newton's constant.

(a) What two boundary conditions should we impose on the above equation for it to describe a star?

(b) By assuming a polytropic equation of state,

$$P(r)=K \rho^{1+\frac{1}{n}}(r)$$

where $K$ is a constant, derive the Lane-Emden equation

$$\frac{1}{\xi^{2}} \frac{d}{d \xi}\left[\xi^{2} \frac{d \theta}{d \xi}\right]=-\theta^{n}$$

where $\rho=\rho_{c} \theta^{n}$, with $\rho_{c}$ the density at the centre of the star, and $r=a \xi$, for some $a$ that you should determine.

(c) Show that the mass of a polytropic star is

$$M=\frac{1}{2 \sqrt{\pi}}\left(\frac{(n+1) K}{G}\right)^{\frac{3}{2}} \rho_{c}^{\frac{3-n}{2 n}} Y_{n}$$

where $Y_{n} \equiv-\left.\xi_{1}^{2} \frac{d \theta}{d \xi}\right|_{\xi=\xi_{1}}$ and $\xi_{1}$ is the value of $\xi$ at the surface of the star.

(d) Derive the following relation between the mass, $M$, and radius, $R$, of a polytropic star

$$M=A_{n} K^{\frac{n}{n-1}} R^{\frac{3-n}{1-n}}$$

where you should determine the constant $A_{n}$. What type of star does the $n=3$ polytrope represent and what is the significance of the mass being constant for this star?