Kelvin-Helmholtz mechanism

Kelvin-Helmholtz contraction releases gravitational binding energy as a self-gravitating body cools and shrinks. It powers young stars and giant planets before nuclear burning or degeneracy dominates their thermal evolution.

For a monatomic gas obeying the ideal gas law, the specific internal energy is $e=3P/(2\rho )$. With no stellar nuclear fusion heating, the first law of thermodynamics makes the effective stellar energy-generation rate $ϵ=-{\partial }_{t}e-P{\partial }_t(1/\rho )$, with the derivatives taken at fixed enclosed mass. Differentiating gives $ϵ=-3{\partial }_{t}P/(2\rho )+5P{\partial }_t\rho /(2{\rho }^2)$. In a fixed-mass stellar homology Kelvin-Helmholtz contraction, $\rho \propto R^{-3}$ and $P\propto R^{-4}$ at each mass label. Hence $ϵ=-3P\dot{R}/(2\rho R)>0$ when $\dot{R}<0$. This includes internal energy storage as well as pressure work; using the gravitational potential-energy decrease alone would overestimate the outgoing luminosity.

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.

The Kelvin-Helmholtz cooling time estimates how long a self-gravitating body can radiate a reservoir of energy of order its gravitational binding energy. It uses intrinsic luminosity $L_{\mathrm{int}}$, excluding reradiated stellar light. At fixed radius and structure, this reservoir scales as $M^2$; the approximation changes when degeneracy or other internal physics changes the accessible energy.