3D stellar kinematics at the Galactic center: measuring the nuclear star cluster spatial density profile, black hole mass, and distance

We present 3D kinematic observations of stars within the central 0.5 pc of the Milky Way nuclear star cluster using adaptive optics imaging and spectroscopy from the Keck telescopes. Recent observations have shown that the cluster has a shallower surface density profile than expected for a dynamically relaxed cusp, leading to important implications for its formation and evolution. However, the true three dimensional profile of the cluster is unknown due to the difficulty in de-projecting the stellar number counts. Here, we use spherical Jeans modeling of individual proper motions and radial velocities to constrain for the first time, the de-projected spatial density profile, cluster velocity anisotropy, black hole mass ($M_mathrm{BH}$), and distance to the Galactic center ($R_0$) simultaneously. We find that the inner stellar density profile of the late-type stars, $rho(r)propto r^{-gamma}$ to have a power law slope $gamma=0.05_{-0.60}^{+0.29}$, much more shallow than the frequently assumed Bahcall $&$ Wolf slope of $gamma=7/4$. The measured slope will significantly affect dynamical predictions involving the cluster, such as the dynamical friction time scale. The cluster core must be larger than 0.5 pc, which disfavors some scenarios for its origin. Our measurement of $M_mathrm{BH}=5.76_{-1.26}^{+1.76}times10^6$ $M_odot$ and $R_0=8.92_{-0.55}^{+0.58}$ kpc is consistent with that derived from stellar orbits within 1$^{primeprime}$ of Sgr A*. When combined with the orbit of S0-2, the uncertainty on $R_0$ is reduced by 30% ($8.46_{-0.38}^{+0.42}$ kpc). We suggest that the MW NSC can be used in the future in combination with stellar orbits to significantly improve constraints on $R_0$.