Characterization of exoplanets from their formation

II. The planetary mass-radius relationship^⋆

¹ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
e-mail: mordasini@mpia.de
² Center for space and habitability, Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
³ Centre de Recherche Astrophysique de Lyon (CRAL), Ecole Nationale Supérieure, 46 Allée d’Italie, 69364 Lyon Cedex 07, France

Received: 15 November 2011
Accepted: 26 August 2012

Abstract

Context. The research of extrasolar planets has entered an era in which we characterize extrasolar planets. This has become possible with measurements of the radii of transiting planets and of the luminosity of planets observed by direct imaging. Meanwhile, the precision of radial velocity surveys makes it possible to discover not only giant planets but also very low-mass ones.

Aims. Uniting all these different observational constraints into one coherent picture to better understand planet formation is an important and simultaneously difficult undertaking. One approach is to develop a theoretical model that can make testable predictions for all these observational techniques. Our goal is to have such a model and use it in population synthesis calculations.

Methods. In a companion paper, we described how we have extended our formation model into a self-consistently coupled formation and evolution model. In this second paper, we first continue with the model description. We describe how we calculate the internal structure of the solid core of the planet and include radiogenic heating. We also introduce an upgrade of the protoplanetary disk model. Finally, we use the upgraded model in population synthesis calculations.

Results. We present how the planetary mass-radius relationship of planets with primordial H₂/He envelopes forms and evolves in time. The basic shape of the mass-radius relationship can be understood from the core accretion model. Low-mass planets cannot bind massive envelopes, while super-critical cores necessarily trigger runway gas accretion, leading to “forbidden” zones in the M − R plane. For a given mass, there is a considerable diversity of radii, mainly due to different bulk compositions, reflecting different formation histories. We compare the synthetic M − R plane with the observed one, finding good agreement for a > 0.1 AU. The synthetic planetary radius distribution is characterized by a strong increase towards small R and a second, lower local maximum at about 1   R_X. The increase towards small radii comes from the increase of the mass function towards low M. The second local maximum is due to the fact that radii are nearly independent of mass for giant planets. A comparison of the synthetic radius distribution with Kepler data shows good agreement for R ≳ 2   R_⊕, but divergence for smaller radii. This indicates that for R ≳ 2   R_⊕ the radius distribution can be described with planets with primordial H₂/He atmospheres, while at smaller radii, planets of a different nature dominate. We predict that in the next few years, Kepler will find the second local maximum at about 1 R_X.

Conclusions. With the updated model, we can compute the most important quantities, like mass, semimajor axis, radius, and luminosity, which characterize an extrasolar planet self-consistently from its formation. The comparison of the radii of the synthetic planets with observations makes it possible to better constrain this formation process and to distinguish between fundamental types of planets.