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Nearly 15 years of radial velocity (RV) monitoring and direct imaging enabledthe detection of two giant planets orbiting the young, nearby star $\beta$Pictoris. The $\delta$ Scuti pulsations of the star, overwhelming planetarysignals, need to be carefully suppressed. In this work, we independentlyrevisit the analysis of the RV data following a different approach than in theliterature to model the activity of the star. We show that a Gaussian Process(GP) with a stochastically driven damped harmonic oscillator kernel can modelthe $\delta$ Scuti pulsations. It provides similar results as parametric modelsbut with a simpler framework, using only 3 hyperparameters. It also enables tomodel poorly sampled RV data, that were excluded from previous analysis, henceextending the RV baseline by nearly five years. Altogether, the orbit and themass of both planets can be constrained from RV only, which was not possiblewith the parametric modelling. To characterize the system more accurately, wealso perform a joint fit of all available relative astrometry and RV data. Ourorbital solutions for $\beta$ Pic b favour a low eccentricity of$0.029^{+0.061}_{-0.024}$ and a relatively short period of $21.1^{+2.0}_{-0.8}$yr. The orbit of $\beta$ Pic c is eccentric with $0.206^{+0.074}_{-0.063}$ witha period of $3.36\pm0.03$ yr. We find model-independent masses of $11.7\pm1.4$and $8.5\pm0.5$ M$_{Jup}$ for $\beta$ Pic b and c, respectively, assumingcoplanarity. The mass of $\beta$ Pic b is consistent with the hottest startevolutionary models, at an age of $25\pm3$ Myr. A direct direction of $\beta$Pic c would provide a second calibration measurement in a coeval system.

Dynamical Mass Estimates of the $\beta$ Pictoris Planetary System Through Gaussian Process Stellar Activity Modelling