Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 317 / 2 / Solution

Letbe the scalar moment of inertia and total kinetic energy. Sincesubstitution of the Euler momentum equation reduces the stress contribution, by the divergence theorem and isotropic pressure $\mathbf{P}=P1$, toFor self-gravity, $\int \rho \mathbf{r}\cdot \mathbf{F}dV=\Omega$, where $\Omega <0$ is the gravitational potential energy. Thus the stellar virial theorem is

Apply hydrostatic equilibrium to the isothermal core, taking its boundary pressure to be $P_c$. Its ideal-gas pressure integral scales as $\int P,dV\propto M_c{T}_c$, its volume as $R_c^3$, and its gravitational energy as $-G{M}_c^2/R_c$. The virial theorem therefore givesor, after absorbing fixed dimensional and structural factors into positive constants,Hydrogen-burning reactions are extremely temperature-sensitive, so expansion cools and suppresses burning while contraction heats and enhances it. This stellar thermostat keeps the shell and adjoining isothermal core near an approximately fixed $T_c$.

At fixed $M_c,T_c$, differentiating $P_c(R_c)$ gives

For a homologous envelope, hydrostatic balance gives $P_c\propto G{M}^2/R^4$ and the ideal-gas temperature scale gives $T_c\propto GM/R$. Eliminating $R$ yieldsup to composition and gravitational constants common to the sequence. A matching core exists only if this required pressure does not exceed $P_{c,\max }\propto T_c^4/M_c^2$. ThereforeThis maximum fractional isothermal-core mass is the mechanism behind the Schönberg-Chandrasekhar limit.