The Mass Distribution Function of Planets

TL;DR

This paper analyzes the distribution of orbital period ratios of exoplanets, infers a log-normal distribution of orbital separations, and estimates the planet mass distribution using Hill's stability criterion, revealing a peaked mass function.

Contribution

It introduces a method to estimate the planet mass distribution from orbital separation data using Hill's criterion and empirical distributions, providing new insights into planet mass functions.

Findings

Orbital separations follow an approximately log-normal distribution.

Most common planet mass does not exceed about two-thirds Jupiter mass.

The planet mass function peaks in logarithmic mass with specific mean and standard deviation.

Abstract

The distribution of orbital period ratios of adjacent planets in extra-solar planetary systems discovered by the {\it Kepler} space telescope exhibits a peak near $\sim1.5$--$2$, a long tail of larger period ratios, and a steep drop-off in the number of systems with period ratios below $\sim1.5$. We find from this data that the dimensionless orbital separations have an approximately log-normal distribution. Using Hill's criterion for the dynamical stability of two planets, we find an upper bound on planet masses such that the most common planet mass does not exceed $10^{-3.2}m_*$, or about two-thirds Jupiter mass for solar mass stars. Assuming that the mass ratio and the dynamical separation (orbital spacings in units of mutual Hill radius) of adjacent planets are independent random variates, and adopting empirical distributions for these, we use Hill's criterion in a statistical way to estimate the planet mass distribution function from the observed distribution of orbital separations. We find that the planet mass function is peaked in logarithm of mass, with a peak value and standard deviation of $\log m/M_\oplus$ of $\sim(0.6-1.0)$ and $\sim(1.1-1.2)$, respectively.