  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.
