Journal Article

A Bayesian periodogram finds evidence for three planets in 47 Ursae Majoris

Philip C. Gregory and Debra A. Fischer

in Monthly Notices of the Royal Astronomical Society

Published on behalf of The Royal Astronomical Society

Volume 403, issue 2, pages 731-747
Published in print April 2010 | ISSN: 0035-8711
Published online March 2010 | e-ISSN: 1365-2966 | DOI:
A Bayesian periodogram finds evidence for three planets in 47 Ursae Majoris

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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 2325 d. The analysis also provides orbital constraints on an additional long-period planet with a period of ∼10 000 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 ± 2 d, 2391+100−87 d and 14002+4018−5095 d, 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 sin i) and semimajor axis are (2.53+.07−.06MJ, 2.10 ± 0.02 au), (0.54 ± 0.07 MJ, 3.6 ± 0.1 au) and (1.6+0.3−0.5MJ, 11.6+2.1−2.9 au), respectively. Based on a three-planet model, the remaining unaccounted for noise (stellar jitter) is 5.7 m s−1.

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.

Keywords: methods: numerical; methods: statistical; techniques: radial velocities; stars: individual: 47 Ursae Majoris; planetary systems

Journal Article.  10467 words.  Illustrated.

Subjects: Astronomy and Astrophysics

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