Stellar Dynamics at the Galactic Center with an Extremely Large Telescope

We discuss experiments achievable via monitoring of stellar dynamics near the massive black hole at the Galactic center with a next generation, extremely large telescope (ELT). Given the likely observational capabilities of an ELT and current knowledge of the stellar environment at the Galactic center, we synthesize plausible samples of stellar orbits around the black hole. We use the Markov Chain Monte Carlo method to evaluate the constraints that orbital monitoring places on the matter content near the black hole. Results are expressed as functions of the number N of stars with detectable orbital motions and the astrometric precision dtheta and spectroscopic precision dv at which stellar proper motions and radial velocities are monitored. For N = 100, dtheta = 0.5 mas, and dv = 10 km/s -- a conservative estimate of the capabilities of a 30 meter telescope -- the extended matter distribution enclosed by the orbits will produce measurable deviations from Keplerian motion if >1000 Msun is enclosed within 0.01 pc. The black hole mass and distance to the Galactic center will be measured to better than ~0.1%. Lowest-order relativistic effects, such as the prograde precession, will be detectable if dtheta < 0.5 mas. Higher-order effects, including frame dragging due to black hole spin, requires dtheta < 0.05 mas, or the favorable discovery of a compact, highly eccentric orbit. Finally, we calculate the rate at which monitored stars undergo detectable nearby encounters with background stars. Such encounters probe the mass function of stellar remnants that accumulate near the black hole. We find that ~30 encounters will be detected over a 10 yr baseline for dtheta = 0.5 mas.