Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 9C (Cosmology)

For a thin spherical shell, the radial pressure force per unit volume is $-dP/dr$. By the shell theorem, the inward gravitational acceleration is $GM(r)/r^2$, because exterior spherical shells contribute no force. Balancing force densities in hydrostatic equilibrium givesThe second equation is mass conservation; it closes the pressure-support equation once an equation of state is supplied.

Regularity at the centre requires finite ${\rho }_c,P_c$ and $M(0)=0$. Then $M(r)=4\pi {\rho }_c{r}^3/3+o(r^3)$, so $P^{\prime }(0)=0$. At a free stellar surface, $P(R)=0$; more generally it matches the external pressure. For $P=K{\rho }^2$ with $K>0$ and vacuum outside, this also gives $\rho (R)=0$.

Inside the star, cancel positive $\rho$ to obtain $2K{\rho }^{\prime }=-GM/r^2$. Differentiate and use mass conservation:Thus the Lane-Emden equation of polytropic index one isFor example, the regular solution is $\rho ={\rho }_c\sin x/x$, with its central value interpreted by continuity, and the first vacuum surface is at $x=\pi$. This also verifies the stated central and surface boundary conditions.