The GJ 504 system revisited. Combining interferometric, radial velocity, and high contrast imaging data

The G-type star GJ504A is known to host a 3 to 35 MJup companion whose temperature, mass, and projected separation all contribute to make it a test case for the planet formation theories and for atmospheric models of giant planets and light brown dwarfs. We collected data from the CHARA interferometer, SOPHIE spectrograph, and VLT/SPHERE high contrast imager to revisit the properties of the system. We measure a radius of 1.35+/- 0.04Rsun for GJ504A which yields isochronal ages of 21+/-2Myr or 4.0+/-1.8Gyr for the system and line-of-sight stellar rotation axis inclination of $162.4_{-4.3}^{+3.8}$ degrees or $18.6_{-3.8}^{+4.3}$ degrees. We re-detect the companion in the Y2, Y3, J3, H2, and K1 dual band SPHERE images. The complete 1-4 $\mu$m SED shape of GJ504b is best reproduced by T8-T9.5 objects with intermediate ages ($\leq1.5$Gyr), and/or unusual dusty atmospheres and/or super-solar metallicities. All six atmospheric models used yield $\mathrm{T_{eff}=550 \pm 50}$K for GJ504b and point toward a low surface gravity (3.5-4.0 dex). The accuracy on the metallicity value is limited by model-to-model systematics. It is not degenerate with the C/O ratio. We derive $\mathrm{log\:L/L_{\odot}=-6.15\pm0.15}$ dex for the companion compatible with masses of $\mathrm{M=1.3^{+0.6}_{-0.3}M_{Jup}}$ and $\mathrm{M=23^{+10}_{-9} M_{Jup}}$ for the young and old age ranges, respectively. The semi-major axis (sma) is above 27.8 au and the eccentricity lower than 0.55. The posterior on GJ~504b's orbital inclination suggests a misalignment with GJ~504A rotation axis. We combine the radial velocity and multi-epoch imaging data to exclude additional objects (90\% prob.) more massive than 2.5 and 30 $\mathrm{M_{Jup}}$ with sma in the range 0.01-80 au for the young and old system ages, respectively. The companion is in the envelope of the population of planets synthetized with our core-accretion model.