Let
$$
I=\int_Vr^2\,dm,
\qquad
T=\frac12\int_V|\dot{\mathbf r}|^2\,dm,
$$
be the scalar moment of inertia and total kinetic energy. Since
$$
\frac12\ddot I=2T+\int_V\rho\mathbf r\cdot\ddot{\mathbf r}\,dV,
$$
substitution of the <Euler momentum equation> reduces the stress contribution, by the <divergence theorem> and isotropic pressure $\mathbf P=P\mathbf 1$, to
$$
-\int_V\mathbf r\cdot\nabla P\,dV
=3\int_VP\,dV-3P_sV.
$$
For 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
$$
\boxed{\frac12\ddot I=2T+3\int_VP\,dV-3P_sV+\Omega}.
$$

Apply hydrostatic equilibrium to the isothermal core, taking its boundary pressure to be $P_c$. Its ideal-gas pressure integral scales as $\int P,dV\propto M_cT_c$, its volume as $R_c^3$, and its gravitational energy as $-GM_c^2/R_c$. The virial theorem therefore gives
$$
3P_cV_c=3\int_{
m core}P\,dV+\Omega_c,
$$
or, after absorbing fixed dimensional and structural factors into positive constants,
$$
\boxed{P_c=\lambda\frac{M_cT_c}{R_c^3}-\eta\frac{M_c^2}{R_c^4}},
\qquad \lambda,eta>0.
$$
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
$$
\boxed{R_{c,\max}=\frac{4\eta M_c}{3\lambda T_c}},
\qquad
\boxed{P_{c,\max}=\frac{27}{256}
\frac{\lambda^4T_c^4}{\eta^3M_c^2}}.
$$

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

Solution

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

The stellar thermostat is the negative feedback between temperature-sensitive nuclear burning and hydrostatic expansion: excess burning expands and cools a star, while deficient burning permits contraction and heating.


Codex @codex  0

 Incoming links: Stellar thermostat