Ohmic Heating Suspends, not Reverses, the Cooling Contraction of Hot Jupiters

We study the radius evolution of close-in extra-solar jupiters under Ohmic heating, a mechanism that was recently proposed to explain the large observed sizes of many of these planets. Planets are born with high entropy and they subsequently cool and contract. We focus on two cases: first, that ohmic heating commences when the planet is hot (high entropy); and second, that it commences after the planet has cooled. In the former case, we use analytical scalings and numerical experiments to confirm that Ohmic heating is capable of suspending the cooling as long as a few percent of the stellar irradiation is converted into Ohmic heating, and the planet has a surface wind that extends to pressures of ~10 bar or deeper. For these parameters, the radii at which cooling is stalled are consistent with (or larger than) the observed radii of most planets. The only two exceptions are WASP-17b and HAT-P-32b. In contrast to the high entropy case, we show that Ohmic heating cannot significantly re-inflate planets after they have already cooled. This leads us to suggest that the diversity of radii observed in hot jupiters may be partially explained by the different epochs at which they are migrated to their current locations.