A linearized approach to radial velocity extraction

High-precision radial velocity (RV) measurements are crucial for exoplanet detection and characterisation. Efforts to achieve ~10 cm/s precision have been made over the recent decades, with significant advancements in instrumentation, data reduction techniques, and statistical inference methods. However, despite these efforts, RV precision is currently limited to ~50 cm/s. This value exceeds state-of-the-art spectrographs' expected instrumental noise floor and is mainly attributed to RV signals induced by stellar variability. In this work, we propose a factorisation method to overcome this limitation. The factorisation is particularly suitable for controlling the effect of localised changes in the stellar emission profile, assuming some smooth function of a few astrophysical parameters governs them. We use short-time Fourier transforms (STFT) to infer the RV in a procedure equivalent to least-squares minimisation in the wavelength domain and demonstrate the effectiveness of our method in treating arbitrary temperature fluctuations on the star's surface. The proposed prescription can be naturally generalised to account for other effects, either intrinsic to the star, such as magnetic fields, or extrinsic to it, such as telluric contamination. As a proof-of-concept, we empirically derive a set of factorisation terms describing the Solar centre-to-limb variation and apply them to a set of realistic SOAP-GPU spectral simulations. We discuss the method's capability to mitigate variability-induced RV signals and its potential extensions to serve as a tomographic tool.