Thermal Structure and Radius Evolution of Irradiated Gas Giant Planets

We consider the thermal structure and radii of strongly irradiated gas giant planets over a range in mass and irradiating flux. The cooling rate of the planet is sensitive to the surface boundary condition, which depends on the detailed manner in which starlight is absorbed and energy redistributed by fluid motion. We parametrize these effects by imposing an isothermal boundary condition $T \equiv T_{\rm deep}$ below the photosphere, and then constrain $T_{\rm deep}$ from the observed masses and radii. We compute the dependence of luminosity and core temperature on mass, $T_{\rm deep}$ and core entropy, finding that simple scalings apply over most of the relevant parameter space. These scalings yield analytic cooling models which exhibit power-law behavior in the observable age range $0.1-10 {\rm Gyr}$, and are confirmed by time-dependent cooling calculations. We compare our model to the radii of observed transiting planets, and derive constraints on $T_{\rm deep}$. Only HD 209458 has a sufficiently accurate radius measurement that $T_{\rm deep}$ is tightly constrained; the lower error bar on the radii for other planets is consistent with no irradiation. More accurate radius and age measurements will allow for a determination of the correlation of $T_{\rm deep}$ with the equilibrium temperature, informing us about both the greenhouse effect and day-night asymmetries.