Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 15A (Cosmology)

By the gravitational shell theorem, the inward acceleration at radius $r$ is $Gm(r)/r^2$, where $m(r)=4\pi {\int }_0^r\rho (s)s^{2}ds$. Force balance for a thin fluid shell gives $-\nabla P=\rho \nabla \Phi$, and henceAssembling shells from infinity gives $E_{\mathrm{grav}}=-{\int }_0^{R}Gm(r)dm(r)/r=-4\pi G{\int }_0^{R}m\rho rdr$. Using the pressure equation and integration by parts yieldsFor a star with negligible external pressure, $P(R)=0$, this is $E_{\mathrm{grav}}=-3⟨P⟩V$. Nonzero external pressure would add the displayed surface term.

For a nonrelativistic ideal gas, the virial relation is $E_{\mathrm{grav}}=-2E_{\mathrm{kin}}$ and total energy is negative. Under adiabatic compression its kinetic energy scales as $R^{-2}$, faster than the magnitude $R^{-1}$ of gravity, providing restoring support; its adiabatic index $5/3$ exceeds $4/3$. For an ultrarelativistic gas, $E_{\mathrm{grav}}=-E_{\mathrm{kin}}$ and both terms scale as $R^{-1}$, giving marginal stability at adiabatic index $4/3$ in this idealized Newtonian model. Corrections such as general relativity can then destabilize it.

Integrating the kinetic energy $p^2/(2m_p)$ over filled momentum states givesSince $p_F=h[3n/(4\pi g_s){]}^{1/3}$, the pressure isFinally $|E_{\mathrm{grav}}|\sim G{M}^2/R$, whereas $PV\sim (h^2/m_p)[M/(m_p{R}^3){]}^{5/3}{R}^3$. Equating these by the virial theorem givesThis is an order-of-magnitude estimate for nonrelativistic degeneracy support, not a uniform-density exact stellar solution.