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The Kepler satellite has revealed a gap between sub-Neptunes and super-Earthsthat atmospheric escape models had predicted as an evaporation valley. We seekto contrast results from a simple XUV-driven energy-limited (ELIM) escape modelagainst those from a direct hydrodynamic (HYDRO) model. Besides XUV-drivenescape, the latter also includes the boil-off regime. We couple the two modelsto an internal structure model and follow the planets' temporal evolution overGyr. To see the population-wide imprint of the two models, we first employ arectangular grid in initial conditions. We then study the slope of the valleyalso for initial conditions derived from the Kepler planets. For therectangular grid, we find that the power-law slope of the valley with respectto orbital period is -0.18 and -0.11 in the ELIM and HYDRO model, respectively.For the initial conditions derived from the Kepler planets, the results aresimilar (-0.16 and -0.10). While the slope found with the ELIM model is steeperthan observed, the one of the HYDRO model is in excellent agreement withobservations. The reason for the shallower slope is caused by the two regimesin which the ELIM model fails: First, puffy planets at low stellar irradiation.For them, boil-off dominates mass loss. However, boil-off is absent in the ELIMmodel, thus it underestimates escape relative to HYDRO. Second, massive compactplanets at high XUV irradiation. For them, the ELIM approximation overestimatesescape relative to the HYDRO case because of cooling by thermal conduction,neglected in the ELIM model. The two effects act together in concert to yieldin the HYDRO model a shallower slope of the valley that agrees very well withobservations. We conclude that an escape model that includes boil-off and amore realistic treatment of cooling mechanisms can reproduce one of the mostimportant constraints, the valley slope.

Planetary evolution with atmospheric photoevaporation II: Fitting the slope of the radius valley by combining boil-off and XUV-driven escape