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Register a new userQuantifying the Bayesian Evidence for a Planet in Radial Velocity Data
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We present results from a data challenge posed to the radial velocity (RV) community: namely, to quantify the Bayesian evidence for n={0,1,2,3} planets in a set of synthetically generated RV datasets containing a range of planet signals. Participating teams were provided the same likelihood function and set of priors to use in their analysis. They applied a variety of methods to estimate Z, the marginal likelihood for each n-planet model, including cross-validation, the Laplace approximation, importance sampling, and nested sampling. We found the dispersion in Z across different methods grew with increasing n-planet models: ~3 for 0-planets, ~10 for 1-planet, ~100-1000 for 2-planets, and >10,000 for 3-planets. Most internal estimates of uncertainty in Z for individual methods significantly underestimated the observed dispersion across all methods. Methods that adopted a Monte Carlo approach by comparing estimates from multiple runs yielded plausible uncertainties. Finally, two classes of numerical algorithms (those based on importance and nested samplers) arrived at similar conclusions regarding the ratio of Zs for n and (n+1)-planet models. One analytic method (the Laplace approximation) demonstrated comparable performance. We express both optimism and caution: we demonstrate that it is practical to perform rigorous Bayesian model comparison for <=3-planet models, yet robust planet discoveries require researchers to better understand the uncertainty in Z and its connections to model selection.
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Methods used to detect giant exoplanets can be broadly divided into two categories: indirect and direct. Indirect methods are more sensitive to planets with a small orbital period, whereas direct detection is more sensitive to planets orbiting at a large distance from their host star. %, and thus on long orbital period. This dichotomy makes it difficult to combine the two techniques on a single target at once. Simultaneous measurements made by direct and indirect techniques offer the possibility of determining the mass and luminosity of planets and a method of testing formation models. Here, we aim to show how long-baseline interferometric observations guided by radial-velocity can be used in such a way. We observed the recently-discovered giant planet $beta$ Pictoris c with GRAVITY, mounted on the Very Large Telescope Interferometer (VLTI). This study constitutes the first direct confirmation of a planet discovered through radial velocity. We find that the planet has a temperature of $T = 1250pm50$,K and a dynamical mass of $M = 8.2pm0.8,M_{rm Jup}$. At $18.5pm2.5$,Myr, this puts $beta$ Pic c close to a hot start track, which is usually associated with formation via disk instability. Conversely, the planet orbits at a distance of 2.7,au, which is too close for disk instability to occur. The low apparent magnitude ($M_{rm K} = 14.3 pm 0.1$) favours a core accretion scenario. We suggest that this apparent contradiction is a sign of hot core accretion, for example, due to the mass of the planetary core or the existence of a high-temperature accretion shock during formation.
The hunt for Earth analogue planets orbiting Sun-like stars has forced the introduction of novel methods to detect signals at, or below, the level of the intrinsic noise of the observations. We present a new global periodogram method that returns more information than the classic Lomb-Scargle periodogram method for radial velocity signal detection. Our method uses the Minimum Mean Squared Error as a framework to determine the optimal number of genuine signals present in a radial velocity timeseries using a global search algorithm, meaning we can discard noise spikes from the data before follow-up analysis. This method also allows us to determine the phase and amplitude of the signals we detect, meaning we can track these quantities as a function of time to test if the signals are stationary or non-stationary. We apply our method to the radial velocity data for GJ876 as a test system to highlight how the phase information can be used to select against non-stationary sources of detected signals in radial velocity data, such as rotational modulation of star spots. Analysis of this system yields two new statistically significant signals in the combined Keck and HARPS velocities with periods of 10 and 15 days. Although a planet with a period of 15 days would relate to a Laplace resonant chain configuration with three of the other planets (8:4:2:1), we stress that follow-up dynamical analyses are needed to test the reliability of such a six planet system.
Planet yield calculations may be used to inform the target selection strategy and science operations of space observatories. Forthcoming and proposed NASA missions, such as the Wide-Field Infrared Survey Telescope (WFIRST), the Habitable Exoplanet Imaging Mission (HabEx), and the Large UV/Optical/IR Surveyor (LUVOIR), are expected to be equipped with sensitive coronagraphs and/or starshades. We are developing a suite of numerical simulations to quantify the extent to which ground-based radial velocity (RV) surveys could boost the detection efficiency of direct imaging missions. In this paper, we discuss the first step in the process of estimating planet yields: generating synthetic planetary systems consistent with observed occurrence rates from multiple detection methods. In an attempt to self-consistently populate stars with orbiting planets, it is found that naive extrapolation of occurrence rates (mass, semi-major axis) results in an unrealistically large number-density of Neptune-mass planets beyond the ice-line ($a gtrsim 5$au), causing dynamic interactions that would destabilize orbits. We impose a stability criterion for multi-planet systems based on mutual Hill radii separation. Considering the influence of compact configurations containing Jovian-mass and Neptune-mass planets results in a marked suppression in the number of terrestrial planets that can exist at large radii. This result has a pronounced impact on planet yield calculations particularly in regions accessible to high-contrast imaging and microlensing. The dynamically compact configurations and occurrence rates that we develop may be incorporated as input into joint RV and direct imaging yield calculations to place meaningful limits on the number of detectable planets with future missions.
GJ667C is the least massive component of a triple star system which lies at a distance of about 6.8 pc (22.1 light-years) from Earth. GJ667C has received much attention recently due to the claims that it hosts up to seven planets including three super-Earths inside the habitable zone. We present a Bayesian technique for the analysis of radial velocity (RV) data-sets in the presence of correlated noise component (red noise), with unknown parameters. We also introduce hyper-parameters in our model in order to deal statistically with under or over-estimated error bars on measured RVs as well as inconsistencies between different data-sets. By applying this method to the RV data-set of GJ667C, we show that this data-set contains a significant correlated (red) noise component with correlation timescale for HARPS data of order 9 days. Our analysis shows that the data only provides strong evidence for the presence of two planets: GJ667Cb and c with periods 7.19d and 28.13d respectively, with some hints towards the presence of a third signal with period 91d. The planetary nature of this third signal is not clear and additional RV observations are required for its confirmation. Previous claims of the detection of additional planets in this system are due the erroneous assumption of white noise. Using the standard white noise assumption, our method leads to the detection of up to five signals in this system. We also find that with the red noise model, the measurement uncertainties from HARPS for this system are under-estimated at the level of ~50 per cent.
Eccentricity is a parameter of particular interest as it is an informative indicator of the past of planetary systems. It is however not always clear whether the eccentricity fitted on radial velocity data is real or if it is an artefact of an inappropriate modelling. In this work, we address this question in two steps: we first assume that the model used for inference is correct and present interesting features of classical estimators. Secondly, we study whether the eccentricity estimates are to be trusted when the data contain incorrectly modelled signals, such as missed planetary companions, non Gaussian noises, correlated noises with unknown covariance, etc. Our main conclusion is that data analysis via posterior distributions, with a model including a free error term gives reliable results provided two conditions. First, convergence of the numerical methods needs to be ascertained. Secondly, the noise power spectrum should not have a particularly strong peak at the semi period of the planet of interest. As a consequence, it is difficult to determine if the signal of an apparently eccentric planet might be due to another inner companion in 2:1 mean motion resonance. We study the use of Bayes factors to disentangle these cases. Finally, we suggest methods to check if there are hints of an incorrect model in the residuals. We show on simulated data the performance of our methods and comment on the eccentricities of Proxima b and 55 Cnc f.
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