The Kepler Mission has discovered thousands of planets with radii < 4 R _ { \oplus } , 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-to-Neptune size range where there are no Solar System analogs .
Previous works have derived deterministic , one-to-one 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 M / M _ { \oplus } = 2.7 ( R / R _ { \oplus } ) ^ { 1.3 } , a scatter in mass of 1.9 M _ { \oplus } , 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 ( R _ { pl } < 4 R _ { \oplus } ) .
More broadly , this work provides a framework for further analyses of the M-R relation and its probable dependencies on period and stellar properties .