Probabilistic Mass–radius Relationship for Sub-neptune-sized Planets

The Kepler Mission has discovered thousands of planets with radii <4 Å R , paving the way for the first statistical studies of the dynamics, formation, and evolution of these sub-Neptunes and super-Earths. Planetary masses are an important physical property for these studies, and yet the vast majority of Kepler planet candidates do not have theirs measured. A key concern is therefore how to map the measured radii to mass estimates in this Earth-toNeptune size range where there are no Solar System analogs. Previous works have derived deterministic, one-toone relationships between radius and mass. However, if these planets span a range of compositions as expected, then an intrinsic scatter about this relationship must exist in the population. Here we present the first probabilistic mass–radius relationship (M–R relation) evaluated within a Bayesian framework, which both quantifies this intrinsic dispersion and the uncertainties on the M–R relation parameters. We analyze how the results depend on the radius range of the sample, and on how the masses were measured. Assuming that the M–R relation can be described as a power law with a dispersion that is constant and normally distributed, we find that = Å Å M M R R 2.7 1.3 ( ) , a scatter in mass of Å M 1.9 , and a mass constraint to physically plausible densities, is the “best-fit” probabilistic M–R relation for the sample of RV-measured transiting sub-Neptunes (Rpl< 4 Å R ). More broadly, this work provides a framework for further analyses of the M–R relation and its probable dependencies on period and stellar properties.