Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 317 1 i Solution 2026-09-28
A -- main sequence star burns hydrogen mainly through the CNO cycle. Its strong temperature sensitivity produces a convective core, which leaves a helium-rich core with a relatively sharp composition discontinuity as it retreats. After central hydrogen exhaustion, hydrogen burning continues in a shell while the helium core contracts and the envelope expands toward the red-giant branch.
When the core becomes hot enough, the Triple-alpha process starts core helium burning. Alpha capture on the newly made carbon also produces oxygen, so central helium exhaustion leaves a carbon-oxygen core. The star then enters the Asymptotic giant branch with an inert core, a helium-burning shell, a hydrogen-burning shell, and a deep convective envelope. At the high-mass end, off-centre carbon burning converts the carbon-oxygen core into an oxygen-neon-magnesium core.
Past exam of the mathematics course of the University of Cambridge 2021 iii Paper 317 1 iv Solution 2026-09-28
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.