= Solution
For a quasi-static self-gravitating object of fixed <mass>, negligible surface <pressure>, and no accretion, nuclear source or additional deep heating, <conservation of energy> gives the <intrinsic planetary luminosity>
$$
L_{\rm int}=-\frac{d}{dt}(U+\Omega),\qquad
\Omega=-\int_0^M\frac{Gm}{r(m)}\,dm.
$$
Here $U$ is internal <energy>, and $\Omega$ is negative gravitational <potential energy>. Gravitational contraction makes $\Omega$ more negative and releases power $L_{\rm grav}\equiv-d\Omega/dt>0$. Some of that release raises the internal <energy>; it cannot all be radiated while the object maintains <hydrostatic equilibrium>.
To quantify the split assume thermal gas pressure with constant <specific-heat ratio> $\gamma>4/3$ and negligible rotation or magnetic support. This excludes the dynamically unstable gas-supported regime. The <stellar virial theorem> gives
$$
\Omega+3\int P\,dV=0,\qquad
U=\frac1{\gamma-1}\int P\,dV=-\frac{\Omega}{3(\gamma-1)}.
$$
It follows that
$$
\boxed{L_{\rm int}=\frac{3\gamma-4}{3(\gamma-1)}L_{\rm grav}.}
$$
For a monatomic <ideal gas>, $\gamma=5/3$, so $L_{\rm int}=L_{\rm grav}/2$: half the released gravitational <energy> heats the gas and half escapes. If the density profile remains homologous, write $\Omega=-qGM^2/R$, with $q>0$ a fixed structure constant. Then
$$
L_{\rm grav}=-q\frac{GM^2}{R^2}\dot R,\qquad
L_{\rm int}\sim\frac{GM^2}{R\tau_{\rm KH}},
$$
up to the structure and virial coefficients, where $\tau_{\rm KH}$ is the <Kelvin-Helmholtz cooling time>.
For a diatomic gas with $\gamma=7/5$ the idealized fraction is $1/6$, already demonstrating that the half-factor is not universal. Partial <electron degeneracy pressure>, dissociation, changing structure, accretion and deep heating require the full energy equation. The general link between luminosity and <Kelvin-Helmholtz contraction> is $L_{\rm int}=-d(U+\Omega)/dt$, with any genuine source terms added explicitly.
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