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

The Schönberg-Chandrasekhar limit is the largest mass fraction that an approximately isothermal inert core can have while remaining matched in hydrostatic and thermal equilibrium to a hydrogen-rich envelope. For a simple isothermal core and polytropic envelope,Because a helium core has larger mean molecular weight than its hydrogen-rich envelope, the limiting fraction is typically near ten per cent. Once shell burning grows the core beyond this limit, no neighboring equilibrium with an isothermal core exists: the core contracts and heats by Kelvin-Helmholtz contraction, the hydrogen-burning shell brightens, and the envelope expands rapidly toward a red giant.