The detection of terrestrial planets by radial velocity and photometry is hindered by the presence of stellar signals. Those are often modeled as stationary Gaussian processes, whose kernels are based on qualitative considerations, which do not fully leverage the existing physical understanding of stars. Our aim is to build a formalism which allows to transfer the knowledge of stellar activity into practical data analysis methods. In particular, we aim at obtaining kernels with physical parameters. This has two purposes: better modelling signals of stellar origin to find smaller exoplanets, and extracting information about the star from the statistical properties of the data. We consider several observational channels such as photometry, radial velocity, activity indicators, and build a model called FENRIR to represent their stochastic variations due to stellar surface inhomogeneities. We compute analytically the covariance of this multi-channel stochastic process, and implement it in the S+LEAF framework to reduce the cost of likelihood evaluations from $O(N^3)$ to $O(N)$. We also compute analytically higher order cumulants of our FENRIR model, which quantify its non-Gaussianity. We obtain a fast Gaussian process framework with physical parameters, which we apply to the HARPS-N and SORCE observations of the Sun, and constrain a solar inclination compatible with the viewing geometry. We then discuss the application of our formalism to granulation. We exhibit non-Gaussianity in solar HARPS radial velocities, and argue that information is lost when stellar activity signals are assumed to be Gaussian. We finally discuss the origin of phase shifts between RVs and indicators, and how to build relevant activity indicators. We provide an open-source implementation of the FENRIR Gaussian process model with a Python interface.

翻译：通过视向速度测光法探测类地行星受到恒星信号的干扰。这些信号通常被建模为平稳高斯过程，其核函数基于定性考量，未能充分利用现有对恒星的物理认知。本文旨在构建一种形式体系，将恒星活动知识转化为实用的数据分析方法，特别关注获得具有物理参数的核函数。这具有双重目的：通过更精准地模拟恒星起源信号以发现更小系外行星，并从数据统计特性中提取恒星信息。我们考虑多种观测通道（如测光、视向速度、活动指标），构建名为FENRIR的模型以表征由恒星表面不均匀性引起的随机变率。我们解析计算了该多通道随机过程的协方差，并在S+LEAF框架中实现以将似然评估成本从O(N³)降至O(N)。同时解析计算FENRIR模型的高阶累积量以量化其非高斯性。我们建立了具有物理参数的快速高斯过程框架，将其应用于HARPS-N和SORCE太阳观测数据，并约束了与观测几何兼容的太阳倾角。进而讨论该形式体系在米粒组织中的应用。在HARPS太阳视向速度数据中观测到非高斯性，证明当恒星活动信号被假设为高斯过程时会丢失信息。最后讨论视向速度与活动指标间相位差的起源，以及如何构建有效的活动指标。我们提供具备Python接口的FENRIR高斯过程模型开源实现。