The radial velocity method is one of the most successful techniques for the discovery and characterization of exoplanets. Modern spectrographs promise measurement precision of ~0.2-0.5 m/s for an ideal target star. However, the intrinsic variability of stellar spectra can mimic and obscure true planet signals at these levels. Rajpaul et al. (2015) and Jones et al. (2017) proposed applying a physically motivated, multivariate Gaussian process (GP) to jointly model the apparent Doppler shift and multiple indicators of stellar activity as a function of time, so as to separate the planetary signal from various forms of stellar variability. These methods are promising, but performing the necessary calculations can be computationally intensive and algebraically tedious. In this work, we present a flexible and computationally efficient software package, GPLinearOdeMaker.jl, for modeling multivariate time series using a linear combination of univariate GPs and their derivatives. The package allows users to easily and efficiently apply various multivariate GP models and different covariance kernel functions. We demonstrate GPLinearOdeMaker.jl by applying the Jones et al. (2017) model to fit measurements of the apparent Doppler shift and activity indicators derived from simulated active solar spectra time series affected by many evolving star spots. We show how GPLinearOdeMaker.jl makes it easy to explore the effect of different choices for the GP kernel. We find that local kernels could significantly increase the sensitivity and precision of Doppler planet searches relative to the widely used quasi-periodic kernel.

The angular momentum deficit (AMD) of a planetary system is a measure of its orbital excitation and a predictor of long–term stability. We adopt the AMD-stability criteria to constrain the orbital architectures for exoplanetary systems. Previously, He, Ford, & Ragozzine (2019) showed through forward modelling (SysSim) that the observed multiplicity distribution can be well reproduced by two populations consisting of a low and a high mutual inclination component. Here, we show that a broad distribution of mutual inclinations arising from systems at the AMD–stability limit can also match the observed Kepler population. We show that distributing a planetary system’s maximum AMD amongst its planets results in a multiplicity–dependent distribution of eccentricities and mutual inclinations. Systems with intrinsically more planets have lower median eccentricities and mutual inclinations, and this trend is well described by power–law functions of the intrinsic planet multiplicity ($n$). We also find that intrinsic single planets have higher eccentricities ($\sigma_{e,1} \sim 0.25$) than multi-planet systems, and that the trends with multiplicity appear in the observed distributions of period–normalized transit duration ratios. We show that the observed preferences for planet size orderings and uniform spacings are more extreme than what can be produced by the detection biases of the Kepler mission alone. Finally, we find that for systems with detected transiting planets between 5 and 10d, there is another planet with a greater RV signal ~53% of the time.

To robustly detect and accurately characterize low-mass planets via Doppler planet surveys, the exoplanet community must develop statistical models capable of jointly modeling planetary perturbations and intrinsic stellar variability. Toward this effort, this note presents simulations of extremely high resolution, solar-like spectra created with SOAP 2.0 (Dumusque et al. 2014) that includes multiple evolving star spots. We anticipate this data set will contribute to future studies developing, testing, and comparing statistical methods for measuring physical radial velocities amid contamination by stellar variability.

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. 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.

Press Coverage

• Extrasolar Earths May Circle 1 In 6 Sunlike Stars, Says New NASA Kepler Study: Forbes, Aug 15, 2019.
• Wow! What if 1 in 4 sunlike stars has an Earth?: EarthSky, Aug 23, 2019.
• Analysis shows 1-in-6 stars may host Earth-size planets: Astronomy Now, Aug 16, 2019.
• How many Earth-size planets are around sun-like stars?: Penn State, Aug 14, 2019.