RUN DMC: An efficient, parallel code for analyzing Radial Velocity Observations using N-body Integrations and Differential Evolution Markov chain Monte Carlo

Abstract: In the 20+ years of Doppler observations of stars, scientists have uncovered a diverse population of extrasolar multi-planet systems. A common technique for characterizing the orbital elements of these planets is Markov chain Monte Carlo (MCMC), using a Keplerian model with random walk proposals and paired with the Metropolis-Hastings algorithm. For approximately a couple of dozen planetary systems with Doppler observations, there are strong planet-planet interactions due to the system being in or near a mean-motion resonance (MMR). An N-body model is often required to accurately describe these systems. Further computational difficulties arise from exploring a high-dimensional parameter space ($\sim$7 x number of planets) that can have complex parameter correlations. To surmount these challenges, we introduce a differential evolution MCMC (DEMCMC) applied to radial velocity data while incorporating self-consistent N-body integrations. Our Radial velocity Using N-body DEMCMC (RUN DMC) algorithm improves upon the random walk proposal distribution of the traditional MCMC by using an ensemble of Markov chains to adaptively improve the proposal distribution. We describe the methodology behind the algorithm, along with results of tests for accuracy and performance. We find that most algorithm parameters have a modest effect on the rate of convergence. However, the size of the ensemble can have a strong effect on performance. We show that the optimal choice depends on the number of planets in a system, as well as the computer architecture used and the resulting extent of parallelization. While the exact choices of optimal algorithm parameters will inevitably vary due to the details of individual planetary systems, we offer recommendations for choosing the DEMCMC algorithm's algorithmic parameters that result in excellent performance for a wide variety of planetary systems.