Context. Observations have revealed in the Kepler data a depleted region separating smaller super-Earths from larger sub-Neptunes. This can be explained as an evaporation valley between planets with and without H/He that is caused by atmospheric escape.
Aims. We want to analytically derive the valley’s locus and understand how it depends on planetary properties and stellar X-ray and ultraviolet (XUV) luminosity. We also want to derive constraints for planet formation models.
Methods. First, we conducted numerical simulations of the evolution of close-in low-mass planets with H/He undergoing escape. We performed parameter studies with grids in core mass and orbital separation, and we varied the postformation H/He mass, the strength of evaporation, and the atmospheric and core composition. Second, we developed an analytical model for the valley locus.
Results. We find that the bottom of the valley quantified by the radius of the largest stripped core, Rbare, at a given orbital distance depends only weakly on postformation H/He mass. The reason is that a high initial H/He mass means that more gas needs to evaporate, but also that the planet density is lower, increasing mass loss. Regarding the stellar XUV-luminosity, Rbare is found to scale as LXUV0.135. The same weak dependency applies to the efficiency factor ε of energy-limited evaporation. As found numerically and analytically, Rbare varies a function of orbital period P for a constant ε as P−2pc∕3 ≈ P−0.18, where Mc ∝ Rcpc is the mass-radius relation of solid cores. We note that Rbare is about 1.7 R⊕ at a ten-day orbital period for an Earth-like composition.
Conclusions. The numerical results are explained very well with the analytical model where complete evaporation occurs if the temporal integral over the stellar XUV irradiation that is absorbed by the planet is larger than the binding energy of the envelope in the gravitational potential of the core. The weak dependency on the postformation H/He means that the valley does not strongly constrain gas accretion during formation. But the weak dependency on primordial H/He mass, stellar LXUV, and ε could be the reason why the valley is so clearly visible observationally, and why various models find similar results theoretically. At the same time, given the large observed spread of LXUV, the dependency on it is still strong enough to explain why the valley is not completely empty.
Analytical model of strange star in the low-mass X-ray binary 4U 1820-30
