Hydrogen-shell burning deposits helium onto a geometrically thin helium layer. Once helium ignites, the strong temperature dependence of the <Triple-alpha process> and the shell's initially weak expansion response produce the <Härm–Schwarzschild instability>. The resulting <AGB thermal pulse> drives a short-lived intershell convection zone, expands the layers above it, and temporarily extinguishes the hydrogen-burning shell before the cycle restarts.

Toward the upper end of the mass range, neutrino cooling keeps the centre cooler than an off-centre shell, so <carbon burning> can ignite off-centre under partial <electron degeneracy pressure>. Repeated flashes and an inward-moving carbon flame consume most carbon and leave a degenerate <oxygen-neon-magnesium core>, surrounded by helium- and hydrogen-burning shells.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 317 / 1 / iv / Solution

At the core-envelope interface, continuity at temperature $T_c$ requires the envelope pressure
$$
P=C_0\left(\frac ML\right)^{1/2}T_c^{17/4},
\qquad
C_0^2=\frac{64\pi a_{\rm rad}cG\mathcal R}{51\kappa_0},
$$
to equal the nonrelativistic <electron degeneracy pressure>
$$
P=\widetilde K\left(\frac{\rho}{\mu_e}\right)^{5/3},
\qquad
\rho=\frac{P}{\mathcal RT_c}.
$$
Eliminating $\rho$ gives
$$
P=(\mathcal R\mu_eT_c)^{5/2}\widetilde K^{-3/2}.
$$
Equating the two pressures and solving for luminosity gives <Mestel's cooling law>
$$
\boxed{
\frac LM
=\frac{64\pi a_{\rm rad}cG}{51\kappa_0}
\frac{\widetilde K^3}{\mathcal R^4\mu_e^5}
T_c^{7/2}}.
$$
In particular, $L/M\propto T_c^{7/2}$.

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 2 / ii / Solution

On a <planetary mass-radius relation>, compressed rocky planets grow sublinearly, approximately $R\propto M^{0.25-0.3}$. Adding a hydrogen-helium envelope produces a rapid radius increase toward sub-Neptunes and gas giants. Around a few Jupiter masses the radius is nearly constant and then decreases as <electron degeneracy pressure> becomes important, approximately approaching the nonrelativistic degenerate scaling $R\propto M^{-1/3}$.

<Brown dwarfs> occupy roughly $13$ to $75$--$80$ Jupiter masses, with deuterium burning near the lower conventional boundary and sustained hydrogen burning beginning at the <hydrogen-burning minimum mass>. Low-mass main-sequence stars then have radii that increase with mass. Thus an isolated-body sketch has a rising rocky branch, a broad giant-planet/brown-dwarf radius maximum and decline, followed by a rising stellar branch.

<Hot-Jupiter radius inflation> places strongly irradiated hot Jupiters above the isolated giant-planet sequence. Irradiation retards cooling and contraction; additional proposed contributions include tidal heating, Ohmic dissipation, atmospheric circulation depositing energy at depth, enhanced opacity, and residual youth.

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 4 / d / Solution

{wiki}

Electron degeneracy pressure supports dense objects when electrons fill momentum states up to a Fermi surface. In the nonrelativistic limit, a fixed-mass self-gravitating object has the characteristic degenerate scaling $R\propto M^{-1/3}$.


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