  The Kepler satellite has revealed a gap between sub-Neptunes and super-Earths
that atmospheric escape models had predicted as an evaporation valley. We seek
to contrast results from a simple XUV-driven energy-limited (ELIM) escape model
against those from a direct hydrodynamic (HYDRO) model. Besides XUV-driven
escape, the latter also includes the boil-off regime. We couple the two models
to an internal structure model and follow the planets' temporal evolution over
Gyr. To see the population-wide imprint of the two models, we first employ a
rectangular grid in initial conditions. We then study the slope of the valley
also for initial conditions derived from the Kepler planets. For the
rectangular grid, we find that the power-law slope of the valley with respect
to 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 are
similar (-0.16 and -0.10). While the slope found with the ELIM model is steeper
than observed, the one of the HYDRO model is in excellent agreement with
observations. The reason for the shallower slope is caused by the two regimes
in 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 ELIM
model, thus it underestimates escape relative to HYDRO. Second, massive compact
planets at high XUV irradiation. For them, the ELIM approximation overestimates
escape relative to the HYDRO case because of cooling by thermal conduction,
neglected in the ELIM model. The two effects act together in concert to yield
in the HYDRO model a shallower slope of the valley that agrees very well with
observations. We conclude that an escape model that includes boil-off and a
more realistic treatment of cooling mechanisms can reproduce one of the most
important constraints, the valley slope.
