  The radio source Sgr A* at the center of our Galaxy is believed to be a 2.6 x
10^6 solar mass black hole which accretes gas from the winds of nearby stars.
We show that limits on the X-ray and infrared emission from the Galactic Center
provide an upper limit of ~ 8 x 10^{-5} solar masses per year on the mass
accretion rate in Sgr A*. The advection-dominated accretion flow (ADAF) model
favors a rate < 10^{-5} solar masses per year. In comparison, the Bondi
accretion rate onto Sgr A*, estimated using the observed spatial distribution
of mass losing stars and assuming non-interacting stellar winds, is ~ 3 x
10^{-5} solar masses per year. There is thus rough agreement between the Bondi,
the ADAF, and the X-ray inferred accretion rates for Sgr A*. We discuss
uncertainties in these estimates, emphasizing the importance of upcoming
observations by the Chandra X-ray observatory (CXO) for tightening the X-ray
derived limits.
