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Accounting for the effects of stellar magnetic phenomena is indispensable tofully exploit radial velocities (RVs). Correlated time variations are oftenmitigated by Gaussian processes (GP). They rely on fitting kernel functionsthat are motivated on mathematical grounds, and whose physical interpretationis often elusive. We aim to establish a connection between stellar activityaffecting RVs and their correlations with physical parameters, and compare thisconnection with kernels used in the literature. We use simple activity modelsto investigate the relationship between the physical processes generating thesignals and the covariances typically found in data, and to demonstrate thequalitative behaviour of this relationship. We use the StarSim code tocalculate RVs of an M dwarf with different realistic evolving spotconfigurations. Their auto-correlation functions (ACF) show a very specificbehaviour and are related to the kernel. GP regression is performed using aquasi-periodic (QP) and harmonic oscillator kernels. Comparison of theresulting kernels with the exact ACFs allows us to cross-match the kernelhyper-parameters with the introduced physical values, study the overallcapabilities of the kernels, and improve their definition. We find that the QPkernel provides a more straightforward interpretation of the physics. It isable to consistently recover both the introduced rotation period P and the spotlifetime. Our study indicates that the performance can be enhanced by fixingthe form factor w and adding a physically motivated cosine term with periodP/2, where the contribution to the ACF for the different spot configurationsdiffers significantly. The new quasi-periodic with cosine kernel leads tosignificantly better model likelihoods, can potentially distinguish betweendifferent spot configurations, and can thereby improve the sensitivity of RVexoplanet searches.

astronomy and astrophysics

Auto-correlation functions of astrophysical processes, and their  relation to Gaussian processes; Application to radial velocities of different  starspot configurations