2010MNRAS.403..731G

A Bayesian analysis of 47 Ursae Majoris radial velocity data confirms and refines the properties of two previously reported planets with periods of 1079 and 2325d. The analysis also provides orbital constraints on an additional long-period planet with a period of ∼10000 d. The three-planet model is found to be 10⁵ times more probable than the next most probable model which is a two-planet model. The non-linear model fitting is accomplished with a new hybrid Markov chain Monte Carlo (HMCMC) algorithm which incorporates parallel tempering, simulated annealing and genetic crossover operations. Each of these features facilitate the detection of a global minimum in χ². By combining all three, the HMCMC greatly increases the probability of realizing this goal. When applied to the Kepler problem, it acts as a powerful multiplanet Kepler periodogram.

The measured periods are 1078±2d, 2391⁺¹⁰⁰_–87d and 14002⁺⁴⁰¹⁸_–5095d, and the corresponding eccentricities are 0.032 ±0.014,0.098^+.047_–.096 and 0.16^+.09_–.16. The results favour low-eccentricity orbits for all three. Assuming the three signals (each one consistent with a Keplerian orbit) are caused by planets, the corresponding limits on planetary mass (M sini) and semimajor axis are (2.53^+.07_–.06M_J,2.10 ±0.02au), (0.54±0.07 M_J,3.6±0.1au) and (1.6^+0.3_–0.5M_J,11.6^+2.1_–2.9au), respectively. Based on a three-planet model, the remaining unaccounted for noise (stellar jitter) is 5.7 m/s.

The velocities of model fit residuals were randomized in multiple trials and processed using a one-planet version of the HMCMC Kepler periodogram. In this situation, periodogram peaks are purely the result of the effective noise. The orbits corresponding to these noise-induced periodogram peaks exhibit a well-defined strong statistical bias towards high eccentricity. We have characterized this eccentricity bias and designed a correction filter that can be used as an alternate prior for eccentricity to enhance the detection of planetary orbits of low or moderate eccentricity.