FENRIR: A statistical model of stellar variability

Context. Stellar surfaces exhibit magnetic activity, which manifests in photometric and spectroscopic observations as a stochastic process. Precisely understanding its statistical structure is crucial for distinguishing stellar variability from signals of potential exoplanets. Furthermore, it could provide insights into the star itself. Aims. Photometric and spectroscopic observations – including radial velocities (RVs) – can be described by their joint statistical distribution as a function of model parameters, also called the likelihood function. We aim to derive a likelihood function from a quantitative physical model. Methods. We modeled stellar activity as a stochastic process and analytically derived its Gaussian process (GP) approximation in two variants: a fully physics-based joint model of RVs and photometry, and a model that retains the physical motivation while incorporating data-driven assumptions, applicable to any combination of photometric and spectroscopic measurements. The GP kernels are implemented in a public Python package using the S+LEAF framework, ensuring that likelihood evaluations scale linearly with data size. Results. We applied our method to solar observations, HARPS-N spectroscopy and SORCE photometry. We show that the FENRIR GPs significantly outperform existing ones in terms of cross-validation. We give a proof of concept of “statistical Doppler imaging,” constraining the average properties of stellar spots and faculae even when they are too small to be individually resolved. Using only the statistical properties of RVs and photometry, we estimate the solar sky-projected obliquity with a precision of ~5°, Finally, we discuss the limitations of our model and exhibit non-Gaussianity in solar HARPS-N RVs.