Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 315 / 4 / b / Solution

Let $L_J$ be Jupiter's current total emitted luminosity. Under the stipulated split, its intrinsic luminosity after removal of sunlight is approximately $L_J/2$. If the accessible energy reservoir down to $T_{\mathrm{cold}}$ is $E_J$, the constant-luminosity estimate gives $\tau \simeq E_J/(L_J/2)$.

Assume the more massive giant has approximately the same radius as Jupiter, a comparable structure factor, and an accessible cooling reservoir scaling as $G{M}^2/R$, as in the Kelvin-Helmholtz cooling time. Assume also that the final cold state contributes negligibly to that reservoir and that the giant radiates at the constant stipulated luminosity $L_J$. ThenThereforeThis is a characteristic scaling under the stated assumptions, not a detailed evolutionary age. A different mass-radius relation, a different accessible internal-energy fraction, or time-dependent luminosity changes the result; the mass and luminosity alone do not uniquely determine a cooling time.