Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 315 / 3 / c

For uniform density, Kelvin-Helmholtz contraction releases binding energy $|U|=3G{M}^2/(5R)$. If mass and luminosity are constant and stellar heating is negligible at 90 au, the contraction age isEnergy conservation, $L=-dU/dt$, givesThe virial theorem places roughly half of the released gravitational energy into internal heat. Including that effect gives $t_{\mathrm{KH}}\simeq 3G{M}^2/(10RL)$ and $\dot{R}\simeq -10L{R}^2/(3G{M}^2)$.