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We present a comprehensive orbital analysis to the exoplanets $\beta$Pictoris b and c that resolves previously reported tensions between thedynamical and evolutionary mass constraints on $\beta$ Pic b. We use the MCMCorbit code orvara to fit fifteen years of radial velocities and relativeastrometry (including recent GRAVITY measurements), absolute astrometry fromHipparcos and Gaia, and a single relative radial velocity measurement between$\beta$ Pic A and b. We measure model-independent masses of$9.3^{+2.6}_{-2.5}\, M_{\rm Jup}$ for $\beta$ Pic b and $8.3\pm 1.0\,M_{\rmJup}$ for $\beta$ Pic c. These masses are robust to modest changes to the inputdata selection. We find a well-constrained eccentricity of $0.119 \pm 0.008$for $\beta$ Pic b, and an eccentricity of $0.21^{+0.16}_{-0.09}$ for $\beta$Pic c, with the two orbital planes aligned to within $\sim$0.5$^\circ$. Bothplanets' masses are within $\sim$1$\sigma$ of the predictions of hot-startevolutionary models and exclude cold starts. We validate our approach on$N$-body synthetic data integrated using REBOUND. We show that orvara canaccount for three-body effects in the $\beta$ Pic system down to a level$\sim$5 times smaller than the GRAVITY uncertainties. Systematics in the massesand orbital parameters from orvara's approximate treatment of multiplanetorbits are a factor of $\sim$5 smaller than the uncertainties we derive here.Future GRAVITY observations will improve the constraints on $\beta$ Pic c'smass and (especially) eccentricity, but improved constraints on the mass of$\beta$ Pic b will likely require years of additional RV monitoring andimproved precision from future Gaia data releases.

Precise Dynamical Masses and Orbital Fits for $\beta$ Pic b and $\beta$ Pic c