Gaussian processes for radial velocity modeling

Better rotation periods and planetary parameters with the quasi-periodic kernel and constrained priors

¹ Landessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany
e-mail: j.kemmer@lsw.uni-heidelberg.de, sstock@lsw.uni-heidelberg.de, sreffert@lsw.uni-heidelberg.de
² Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

Received: 29 July 2022
Accepted: 24 April 2023

Abstract

Context. Instrumental radial velocity (RV) precision has reached a level where the detection of planetary signals is limited by the ability to understand and simultaneously model stellar astrophysical “noise.” A common method for mitigating the effects of stellar activity is Gaussian process (GP) regression.

Aims. In this study we present an analysis of the performance and properties of the quasi-periodic (QP) GP kernel, which is the multiplication of the squared-exponential kernel by the exponential-sine-squared kernel, based on an extensive set of synthetic RVs, into which the signature of activity was injected.

Methods. The stellar activity within our synthetic data sets was simulated using astrophysically motivated models with different spot distributions and spot lifetimes rotating on the surface of a modeled late-type star. We used dynamic nested sampling to fit different model sets, including QP–GPs, Keplerian models, white noise models, and combinations of these, to synthetic RV time series data that in some cases included additional injected planetary signals.

Results. We find that while the QP–GP rotation parameter matches the simulated rotation period of the star, the length scale cannot be directly connected to the spot lifetimes on the stellar surface. Regarding the setup of the priors for the QP–GP, we find that it can be advantageous to constrain the QP–GP hyperparameters in different ways depending on the application and the goal of the analysis. We find that a constraint on the length scale of the QP–GP can lead to a significant improvement in identifying the correct rotation period of the star, while a constraint on the rotation hyperparameter tends to lead to improved planet detection efficiency and more accurately derived planet parameters. Even though for most of the simulations the Bayesian evidence performed as expected, we identified not far-fetched cases where a blind adoption of this metric would lead to wrong conclusions.

Conclusions. We conclude that modeling stellar astrophysical noise by using a QP–GP considerably improves detection efficiencies and leads to precise planet parameters. Nevertheless, there are also cases in which the QP–GP does not perform optimally, for example RV variations dynamically evolving on short timescales or a mixture of a very stable activity component and random variations. Knowledge of these limitations is essential for drawing correct conclusions from observational data.