Upper panel: theoretical distribution of radii and semi-major axes (or incident fluxes) of planets at 10 Gyr with various initial masses and water mass fractions. Crosses (red) represent planets that lost their water envelopes completely due to the photo-evaporation in 10 Gyr, while open squares (blue) represent planets that survive significant loss of the water envelopes. The green line is the minimum threshold radii, . Here, we have adopted ε = 0.1. Lower panel: distribution of radii and semi-major axes (or incident fluxes) of Kepler planetary candidates, compared to the threshold radius, (see Sect. 3.3 for definition). We have shown three relationships for different heating efficiencies: ε = 1 (solid red line), ε = 0.1 (dashed green line), and ε = 0.01 (dotted blue line). Filled squares represent observational data (http://kepler.nasa.gov, as of June 29, 2013) for planets orbiting host stars with effective temperature of 5300–6000 K (G-type stars).
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