Upper panel: theoretical distribution of masses 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 in 10 Gyr, while open squares (blue) represent planets that survive significant loss of the water envelopes via photo-evaporation. The green line is the minimum threshold masses, . Here, we have adopted ε = 0.1. Lower panel: distribution of masses and semi-major axes (or incident fluxes) of detected exoplanets compared to the minimum threshold mass, , derived in this study (see Sect. 3.3 for definition). We have shown three relationships for different heating efficiencies: ε = 1 (solid line), ε = 0.1 (dashed line), and ε = 0.01 (dotted line). Filled circles with error bars represent observational data (from http://exoplanet.org (Wright et al. 2011)) for planets orbiting host stars with effective temperature of 5000–6000 K (relatively early K-type stars and G-type stars). Planets are colored according to their zero-albedo equilibrium temperatures in K. In the planet names, “CoR” and “Kep” stand for CoRoT and Kepler, respectively.