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Hot-Jupiter Inflation due to Deep Energy Deposition

Some extrasolar giant planets in close orbits---"hot Jupiters"---exhibit larger radii than that of a passively cooling planet. The extreme irradiation $L_{\rm eq}$ these hot Jupiters receive from their close in stars creates a thick isothermal layer in their envelopes, which slows down their convective cooling, allowing them to retain their inflated size for longer. This is yet insufficient to explain the observed sizes of the most inflated planets. Some models invoke an additional power source, deposited deep in the planet's envelope. Here we present an analytical model for the cooling of such irradiated, and internally heated gas giants. We show that a power source $L_{\rm dep}$, deposited at an optical depth $τ_{\rm dep}$, creates an exterior convective region, between optical depths $L_{\rm eq}/L_{\rm dep}$ and $τ_{\rm dep}$, beyond which a thicker isothermal layer exists, which in extreme cases may extend to the center of the planet. This convective layer, which occurs only for $L_{\rm dep}τ_{\rm dep}>L_{\rm eq}$, further delays the cooling of the planet. Such a planet is equivalent to a planet irradiated with $L_{\rm eq}\left(1+L_{\rm dep}τ_{\rm dep}/L_{\rm eq}\right)^β$, where $β\approx 0.35$ is an effective power-law index describing the radiative energy density as function of the optical depth for a convective planet $U\proptoτ^β$. Our simple analytical model reproduces the main trends found in previous numerical works, and provides an intuitive understanding. We derive scaling laws for the cooling rate of the planet, its central temperature, and radius. These scaling laws can be used to estimate the effects of tidal or Ohmic dissipation, wind shocks, or any other mechanism involving energy deposition, on sizes of hot Jupiters.