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, $L/M\propto T_c^{7/2}$.

Solution

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

{c}

Mestel's cooling law matches a degenerate isothermal white-dwarf core to a nondegenerate radiative envelope. With Kramers opacity it gives $L/M\propto T_c^{7/2}$ in terms of core temperature.


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 Incoming links: Mestel's cooling law