Evolution of Gas Giant Entropy During Formation by Runaway Accretion

We calculate the evolution of gas giant planets during the runaway gas accretion phase of formation, to understand how the luminosity of young giant planets depends on the accretion conditions. We construct steady-state envelope models, and run time-dependent simulations of accreting planets with the Modules for Experiments in Stellar Astrophysics (MESA) code. We show that the evolution of the internal entropy depends on the contrast between the internal adiabat and the entropy of the accreted material, parametrized by the shock temperature $T_0$ and pressure $P_0$. At low temperatures ($T_0\lesssim 300$--$1000\ {\rm K}$, depending on model parameters), the accreted material has a lower entropy than the interior. The convection zone extends to the surface and can drive a large luminosity, leading to rapid cooling and cold starts. For higher temperatures, the accreted material has a larger entropy than the interior, giving a radiative zone that stalls cooling. For $T_0\gtrsim 2000\ {\rm K}$, the surface--interior entropy contrast cannot be accommodated by the radiative envelope, and the accreted matter accumulates with high entropy, forming a hot start. The final state of the planet depends on the shock temperature, accretion rate, and starting entropy at the onset of runaway accretion. Cold starts with $L\lesssim 5\times 10^{-6}\ L_\odot$ require low accretion rates and starting entropy, and that the temperature of the accreting material is maintained close to the nebula temperature. If instead the temperature is near the value required to radiate the accretion luminosity, $4\pi R^2\sigma T_0^4\sim (GM\dot M/R)$, as suggested by previous work on radiative shocks in the context of star formation, gas giant planets form in a hot start with $L\sim 10^{-4}\ L_\odot$.