We present an isolated analytical model for the ohmic heating in the interior of hot jupiters, treating the wind zone as a parameterized boundary condition. Under a conserved estimation of the strength of induced field and the assumption of an isothermal-convective planet model, we conclude that the mechanism of ohmic heating may not explain the over-inflated radius of hot jupiters along. We also develop a new time dependent evolution model for hot jupiters with ohmic heating, further show that ohmic heating is important only when the planet mass is small or the planet is at late stage of evolution.

We have established precise planet radii, semimajor axes, incident stellar fluxes, and stellar masses for 909 planets in 355 multi-planet systems discovered by Kepler. In this sample, we find that planets within a single multi-planet system have correlated sizes: each planet is more likely to be the size of its neighbor than a size drawn at random from the distribution of observed planet sizes. In systems with three or more planets, the planets tend to have a regular spacing: the orbital period ratios of adjacent pairs of planets are correlated. Furthermore, the orbital period ratios are smaller in systems with smaller planets, suggesting that the patterns in planet sizes and spacing are linked through formation and/or subsequent orbital dynamics. Yet, we find that essentially no planets have orbital period ratios smaller than 1.2, regardless of planet size. Using empirical mass-radius relationships, we estimate the mutual Hill separations of planet pairs. We find that 93% of the planet pairs are at least 10 mutual Hill radii apart, and that a spacing of ∼20 mutual Hill radii is most common. We also find that when comparing planet sizes, the outer planet is larger in 65% ± 0.4% of cases, and the typical ratio of the outer to inner planet size is positively correlated with the temperature difference between the planets. This could be the result of photo-evaporation.