Dynamical Mass of the Exoplanet Host Star HR 8799

HR 8799 is a young A5/F0 star hosting four directly imaged giant planets at wide separations ($\sim$16-78 au) which are undergoing orbital motion and have been continuously monitored with adaptive optics imaging since their discovery over a decade ago. We present a dynamical mass of HR 8799 using 130 epochs of relative astrometry of its planets, which include both published measurements and new medium-band 3.1 $\mu$m observations that we acquired with NIRC2 at Keck Observatory. For the purpose of measuring the host star mass, each orbiting planet is treated as a massless particle and is fit with a Keplerian orbit using Markov chain Monte Carlo. We then use a Bayesian framework to combine each independent total mass measurement into a cumulative dynamical mass using all four planets. The dynamical mass of HR 8799 is 1.47$^{+0.12}_{-0.17}$ \Msun assuming a uniform stellar mass prior, or 1.46$^{+0.11}_{-0.15}$ \Msun with a weakly informative prior based on spectroscopy. There is a strong covariance between the planets' eccentricities and the total system mass; when the constraint is limited to low eccentricity solutions of $e<0.1$, which is motivated by dynamical stability, our mass measurement improves to 1.43$^{+0.06}_{-0.07}$ \Msun. Our dynamical mass and other fundamental measured parameters of HR 8799 together with MESA Isochrones & Stellar Tracks grids yields a bulk metallicity most consistent with [Fe/H]$\sim$ -0.25-0.00 dex and an age of 10-23 Myr for the system. This implies hot start masses of 2.7-4.9 \Mjup for HR 8799 b and 4.1-7.0 \Mjup for HR 8799 c, d, and e, assuming they formed at the same time as the host star.