Mass-radius relation of intermediate-mass planets outlined by the hydrodynamic escape of planetary atmospheres and formation
Authors: Kubyshkina Daria , Fossati Luca 
Published: September 2022

&lt;p&gt;Exoplanets in the mass range between Earth and Saturn show a large spread in radii/densities for a given planetary mass. The most approaches to explain this spread and the distribution of planetary properties therein can be split into two groups. The first considers the planetary formation paths as the primary mechanism shaping this distribution, and the second group considers the radius spread as a consequence of the atmospheric evolution driven by the atmospheric mass loss. The majority of the latter studies, however, consider only the observed radius spread with some theoretical underlying mass distribution, as for most of the Kepler planets the mass is unknown.&lt;br /&gt;In this study, we examine the mass-radius distribution of the observed planets with masses between 1 and 108 Earth masses with the aim to understand to which extent it can be explained by the evolution of planetary atmospheres driven by thermal contraction and the hydrodynamic escape, and in which regions of the parameters state the initial parameters of planets set up by specific formation processes are critical for the final (gygayears old) state.&lt;br /&gt;Our modeling framework accounts simultaneously for the realistic atmospheric mass loss by interpolating within the grid of upper atmosphere models and for the thermal evolution of planets by means of the MESA code. As the atmospheric mass loss on the long timescales is strongly affected by high energy stellar radiation, we also account for the whole range of different possible stellar evolution histories as represented by the Mors code.&amp;#160;&lt;br /&gt;We consider the grid of model planets in the mass range given above evolving at different orbital separations (corresponding to the equilibrium temperatures of ~500-1700 K) around the solar mass star. As initial parameters for our atmosphere evolution models, we adopt the predictions of the analytical approximations based on formation models (Mordasini 2020) and consider the two possible scenarios: planets formed in the inner disk (relatively small initial atmospheres) and beyond the snow line (large initial atmospheres) with consequent inward migration at the early phase of the planetary system evolution.&lt;br /&gt;The whole radius spread predicted using this approach outlines well the observed distribution (including about 240 planets with mass and radius uncertainties below 45% and 15% respectively), except for a group of very close in (within ~0.1 AU) massive (~70-110 Mearth) planets with radii comparable to the Jupiter radius. The radii of these planets can not be reproduced by our models even by assuming the atmospheric mass fractions above 80% without some additional heating source. A strong correlation of the radii with equilibrium temperature (Rpl~Teq&lt;sup&gt;0.7&lt;/sup&gt;) suggests that the inflation mechanism is similar to that of the so-called &quot;inflated Jupiters&quot;, where a range of possible explanations was suggested including the tidal interaction with the host star, vertical heat transport towards the deep atmospheric levels or the Ohmic dissipation.&lt;br /&gt;The more detailed analysis shows that the low-mass end of the mass-radius distribution (below 10-15 Earth masses) is dominated by the effect of the atmospheric mass loss (and thus extremely dependent on the activity evolution history of the host star) and weakly depend on the initial parameters, and thus, on the specific formation mechanism of the exoplanets. For more massive planets, though some of them can be significantly affected by the atmospheric mass loss, the initial conditions become important and the variability in the possible stellar histories can only explain about one fourth of the whole spread. Thus, for explaining the upper boundary of the spread above ~20 Mearth one needs to consider the voluminous initial atmospheres which can be explained by the formation at the large distance from the host star. However, the activity history of the host star can be theoretically resolved using the present-day radii of the companion planets for the significant fraction of planets with masses up to ~60 Mearth.&lt;br /&gt;Finally, the detailed comparison between the model predictions and the observations within different Teq intervals reveals a relatively small (~6%) but a presumably systematic group of the outliers with radii considerably smaller than the lower boundary predicted by our models for Teq&lt;~800 K. Assuming the hydrogen dominated atmospheres surrounding rocky cores, these planets would not have more than ~1% of their mass in the envelope, while for their masses (&gt;10 Mearth) the accretion models predict the initial atmospheric mass fraction order of 10%, and the total atmospheric mass loss throughout the evolution according to our models is insufficient to remove this much of the atmospheric material. This suggests, that the formation mechanisms and structures of these planets are considerably different from our assumption of the hydrogen-dominated atmospheres accreted onto the rocky core.&lt;/p&gt;