Intrinsic planetary luminosity ($L_{\mathrm{int}}$)

The intrinsic planetary luminosity $L_{\mathrm{int}}$ is the power supplied by interior cooling, contraction, nuclear reactions if present, or internal heating, excluding reflected or passively reradiated stellar light. For fixed mass and no accretion or nuclear source, global conservation of energy gives $L_{\mathrm{int}}=Q_{\mathrm{deep}}-d(U+\Omega )/dt$, where $Q_{\mathrm{deep}}$ is any additional power deposited in the interior, such as tidal heating or Joule heating. For $Q_{\mathrm{deep}}=0$ and a gas-supported quasi-static sphere of fixed specific-heat ratio $\gamma$ and negligible surface pressure, the stellar virial theorem gives $\Omega =-3(\gamma -1)U$, henceFor a dynamically stable gas-supported model take $\gamma >4/3$. For $\gamma =5/3$ this is half the gravitational release. Degenerate giants require a different internal energy reservoir; the half-factor is not universal.

With negligible nuclear or external heating, the stellar structure equations give $L_{\mathrm{int}}=-{\int }_0^{M}T\partial s/\partial tdm$. A nearly isentropic convective interior therefore has $\dot{s}=-L_{\mathrm{int}}/\int Tdm<0$ whenever it radiates. Decreasing specific entropy does not require temperature at every depth to decrease immediately: contraction of an ideal gas can raise its central temperature while total energy is lost.

If otherwise comparable isolated planets of one mass follow $L_{\mathrm{int}}=C{t}^{-\alpha }$, their intrinsic luminosity ratio is $L_{\mathrm{int}}(t_1)/L_{\mathrm{int}}(t_2)=(t_2/t_1{)}^{\alpha }$. The normalization depends on composition and initial entropy. Strong stellar irradiation, deep heating, or accretion can invalidate extrapolating the same cooling law.