Abstract
Abstract We infer the number of planets per star as a function of orbital period and planet size using Kepler archival data products with updated stellar properties from the Gaia Data Release 2. Using hierarchical Bayesian modeling and Hamiltonian Monte Carlo, we incorporate planet radius uncertainties into an inhomogeneous Poisson point process model. We demonstrate that this model captures the general features of the outcome of the planet formation and evolution around GK stars and provides an infrastructure to use the Kepler results to constrain analytic planet distribution models. We report an increased mean and variance in the marginal posterior distributions for the number of planets per GK star when including planet radius measurement uncertainties. We estimate the number of planets per GK star between 0.75 and 2.5 R ⊕ and with orbital periods of 50–300 days to have a 68% credible interval of 0.49–0.77 and a posterior mean of 0.63. This posterior has a smaller mean and a larger variance than the occurrence rate calculated in this work and in Burke et al. for the same parameter space using the Q1−Q16 (previous Kepler planet candidate and stellar catalog). We attribute the smaller mean to many of the instrumental false positives at longer orbital periods being removed from the DR25 catalog. We find that the accuracy and precision of our hierarchical Bayesian model posterior distributions are less sensitive to the total number of planets in the sample, and more so for the characteristics of the catalog completeness and reliability and the span of the planet parameter space.
Highlights
NASA’s Kepler mission was designed to yield an ensemble of planetary systems amenable to statistical analysis (Borucki et al 2010; Jenkins et al 2010; Koch et al 2010)
We find that a bias is introduced into the occurrence rate posterior distributions when using heterogeneous stellar radius measurement uncertainties
We find an upward shift in the occurrence rate posterior mean and a larger posterior variance when including measurement uncertainty in planet radius
Summary
NASA’s Kepler mission was designed to yield an ensemble of planetary systems amenable to statistical analysis (Borucki et al 2010; Jenkins et al 2010; Koch et al 2010). Foreman-Mackey et al (2014) consider the contribution to the planet radius uncertainties from the measured planet-to-star radius ratio and stellar radius uncertainties in their occurrence rate analysis for GK stars They use a nonparametric Bayesian method that makes it difficult to interpret some population-level parameters for planet formation and subsequent evolution theories. We demonstrate the use of standard and advanced diagnostics to assess the application of HMC for performing our hierarchical Bayesian model calculations We use this statistical framework to demonstrate the impact of subtle differences in host star categorization and small differences in selected planet radii and orbital period across varied completeness and reliability parameter spaces.
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