Planetary evolution with atmospheric photoevaporation

II. Fitting the slope of the radius valley by combining boil-off and XUV-driven escape

¹ Physics Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
e-mail: christoph.mordasini@unibe.ch
² Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA
³ Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria
⁴ School of Physics, Trinity College Dublin, the University of Dublin, College Green, Dublin-2, Ireland

Received: 13 September 2021
Accepted: 28 June 2023

Abstract

Context. Observations by the Kepler satellite have revealed a gap between larger sub-Neptunes and smaller super-Earths that atmospheric escape models had predicted as an evaporation valley prior to discovery.

Aims. We seek to contrast results from a simple X-ray and extreme-ultraviolet (XUV)-driven energy-limited escape model against those from a direct hydrodynamic model. The latter calculates the thermospheric temperature structure self-consistently, including cooling effects such as thermal conduction. Besides XUV-driven escape, it also includes the boil-off escape regime where the escape is driven by the atmospheric thermal energy and low planetary gravity, catalysed by stellar continuum irradiation. We coupled these two escape models to an internal structure model and followed the planets’ temporal evolution.

Methods. To examine the population-wide imprint of the two escape models and to compare it to observations, we first employed a rectangular grid, tracking the evolution of planets as a function of core mass and orbital period over gigayear timescales. We then studied the slope of the valley also for initial conditions derived from the observed Kepler planet population.

Results. 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 energy-limited and hydrodynamic 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 energy-limited model is steeper than observed, the one of the hydrodynamic model is in excellent agreement with observations. The reason for the shallower slope is caused by the two regimes in which the energy-limited approximation fails. The first one are low-mass planets at low-to-intermediate stellar irradiation. For them, boil-off dominates mass loss. However, boil-off is absent in the energy-limited model, and thus it underestimates escape relative to the hydrodynamic model. The second one are massive compact planets at high XUV irradiation. For them, the energy-limited approximation overestimates escape relative to the hydrodynamic model because of cooling by thermal conduction, which is neglected in the energy-limited model.

Conclusions. The two effects act together in concert to yield, in the hydrodynamic model, a shallower slope of the valley that agrees very well with observations. We conclude that a hydrodynamic escape model that includes boil-off and a more realistic treatment of cooling mechanisms can reproduce one of the most important constraints for escape models, the valley slope.