Unveiling the beta Pictoris system, coupling high contrast imaging, interferometric, and radial velocity data

Context. The nearby and young beta Pictoris system hosts a well resolved disk, a directly imaged massive giant planet orbiting at similar or equal to 9 au, as well as an inner planet orbiting at similar or equal to 2.7 au, which was recently detected through radial velocity (RV). As such, it offers several unique opportunities for detailed studies of planetary system formation and early evolution. Aims. We aim to further constrain the orbital and physical properties of beta Pictoris b and c using a combination of high contrast imaging, long base-line interferometry, and RV data. We also predict the closest approaches or the transit times of both planets, and we constrain the presence of additional planets in the system. Methods. We obtained six additional epochs of SPHERE data, six additional epochs of GRAVITY data, and five additional epochs of RV data. We combined these various types of data in a single Markov-chain Monte Carlo analysis to constrain the orbital parameters and masses of the two planets simultaneously. The analysis takes into account the gravitational influence of both planets on the star and hence their relative astrometry. Secondly, we used the RV and high contrast imaging data to derive the probabilities of presence of additional planets throughout the disk, and we tested the impact of absolute astrometry. Results. The orbital properties of both planets are constrained with a semi-major axis of 9.8 0.4 au and 2.7 +/- 0.02 au for b and c, respectively, and eccentricities of 0.09 +/- 0.1 and 0.27 +/- 0.07, assuming the HIPPARCOS distance. We note that despite these low fitting error bars, the eccentricity of beta Pictoris c might still be over-estimated. If no prior is provided on the mass of beta Pictoris b, we obtain a very low value that is inconsistent with what is derived from brightness-mass models. When we set an evolutionary model motivated prior to the mass of beta Pictoris b, we find a solution in the 10-11 M-Jup range. Conversely, beta P