We show that with a Next Generation Large Telescope one can detect the accelerated motions of ~100 stars orbiting the massive black hole at the Galactic center. The positions and velocities of these stars will be measured to astrometric and spectroscopic precision several times better than currently attainable enabling detailed measurements of the gravitational potential in the neighborhood of the massive black hole. We show that the monitoring of stellar motions with such a telescopes enables: (1) a measurement of the Galactic center distance R_0 to better than 0.1% accuracy, (2) a measurement of the extended matter distribution near the black hole, including that of the exotic dark matter, (3) a detection of general relativistic effects due to the black hole including the prograde precession of stars and possibly the black hole spin, and (4) a detection of gravitational encounters between monitored stars and stellar remnants that accumulate near the Galactic center. Such encounters probe the mass function of the remnants.
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