[Paper Review] Characterization of exoplanets from their formation II. The planetary mass-radius relationship

C. Mordasini, Y. Alibert+4 more|arXiv (Cornell University)|Jan 1, 2012

Stellar, planetary, and galactic studiesPhysics and Astronomy154 references125 citations

TL;DR

This paper presents a self-consistent planet formation and evolution model that predicts the mass-radius relationship of exoplanets by coupling core accretion, radiogenic heating, and updated protoplanetary disk evolution. It reproduces observed planetary radii, predicts a secondary peak at ~1 RJup due to giant planets, and explains the divergence from Kepler data below 2 R⊕ as evidence for non-primordial atmospheres in small planets.

ABSTRACT

The research of exoplanets has entered an era in which we characterize extrasolar planets. This has become possible with measurements of radii and luminosities. Meanwhile, radial velocity surveys discover also very low-mass planets. Uniting all this observational data into one coherent picture to better understand planet formation is an important, but difficult undertaking. Our approach is to develop a model which can make testable predictions for all these observational methods. We continue to describe how we have extended our formation model into a self-consistently coupled formation and evolution model. We show how we calculate the internal structure of the solid core and radiogenic heating. We also improve the protoplanetary disk model. Finally, we conduct population synthesis calculations. We present how the planetary mass-radius relationship of planets with primordial H/He envelopes forms and evolves in time. The basic shape of the M-R relation 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. We compare the synthetic M-R relation with the observed one, finding good agreement for a>0.1 AU. The synthetic radius distribution is characterized by a strong increase towards small R, and a second, lower local maximum at ~1 Jovian radius. The increase towards small radii reflects 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 agreement for R>2 Earth radii, but divergence for smaller radii. We predict that in the next few years, Kepler should find the second, local maximum at ~1 Jovian radius.

Motivation & Objective

• To develop a unified theoretical model that simultaneously predicts planetary mass, radius, semimajor axis, and luminosity from first principles of formation.
• To understand the origin of the observed mass-radius relationship in exoplanets, particularly the diversity in radii for similar masses.
• To test whether planets with primordial H2/He envelopes can explain the observed radius distribution, especially in the context of Kepler data.
• To identify the physical mechanisms—such as core mass, composition, and disk evolution—that drive the observed spread in planetary radii.
• To distinguish between different planetary types based on formation history and internal structure using synthetic population synthesis.

Proposed method

• Extends a core accretion model to include self-consistent internal structure calculations for solid planetary cores, incorporating radiogenic heating from short- and long-lived isotopes.
• Implements an improved protoplanetary disk model with time-dependent evolution, including viscous spreading, photoevaporation, and variable inner boundary conditions (magnetospheric cavity size).
• Uses population synthesis simulations to generate synthetic planetary systems from seed embryos to 1 Gyr, tracking mass, radius, and luminosity evolution over time.
• Calculates planetary luminosity during runaway gas accretion phases, enabling comparison with direct imaging and SED modeling constraints.
• Applies a modified disk evolution model that captures four distinct phases: initial contraction, quasi-self-similar evolution, gap opening, and final dispersal.
• Compares synthetic mass-radius and radius distribution with observed data from Kepler, CoRoT, and radial velocity surveys to validate the model.

Experimental results

Research questions

• RQ1 How does the inclusion of radiogenic heating and updated disk evolution affect the predicted mass-radius relationship of exoplanets?
• RQ2 Why do planets of similar mass exhibit a wide range of radii, and what physical factors—such as core mass or composition—drive this diversity?
• RQ3 Can the observed radius distribution from Kepler data, particularly the bimodal structure with a peak near 1 RJup, be reproduced by a formation model with primordial H2/He envelopes?
• RQ4 What causes the observed divergence between synthetic and observed radius distributions below 2 R⊕, and what does this imply about the nature of small planets?
• RQ5 How do different disk inner boundary conditions (e.g., magnetospheric cavity size) influence planetary migration and final orbital architecture in the synthetic population?

Key findings

• The synthetic mass-radius relationship reproduces the observed 'forbidden' zones in the M-R plane, where low-mass planets cannot bind massive H2/He envelopes, while super-critical cores trigger runaway gas accretion.
• The synthetic radius distribution shows a strong increase toward small radii due to the rising mass function at low masses, and a secondary, lower peak at approximately 1 RJup due to the near-mass-independence of radii in giant planets.
• For planets with radii ≥2 R⊕, the synthetic radius distribution agrees well with Kepler observations, supporting the hypothesis that these planets have primordial H2/He atmospheres.
• The divergence between synthetic and observed radius distributions below 2 R⊕ indicates that planets in this range are unlikely to have retained primordial envelopes, suggesting alternative formation pathways or atmospheric loss.
• The model predicts that Kepler should detect the secondary peak at ~1 RJup in the coming years, providing a key test for the primordial atmosphere hypothesis.
• The inclusion of radiogenic heating in the core model leads to a more accurate internal structure and affects the timing and efficiency of gas accretion, influencing the final planetary radius.