  The massive black hole in our galactic center, Sgr A*, accretes only a small
fraction of the gas available at its Bondi radius. The physical processes
determining this accretion rate remain unknown, partly due to a lack of
observational constraints on the gas at distances between ~10 and ~10$^5$
Schwarzschild radii (Rs) from the black hole. Recent infrared observations
identify low-mass gas clouds, G1 and G2, moving on highly eccentric, nearly
co-planar orbits through the accretion flow around Sgr A*. Although it is not
yet clear whether these objects contain embedded stars, their extended gaseous
envelopes evolve independently as gas clouds. In this paper we attempt to use
these gas clouds to constrain the properties of the accretion flow at ~10$^3$
Rs. Assuming that G1 and G2 follow the same trajectory, we model the small
differences in their orbital parameters as evolution resulting from interaction
with the background flow. We find evolution consistent with the G-clouds
originating in the clockwise disk. Our analysis enables the first unique
determination of the rotation axis of the accretion flow: we localize the
rotation axis to within 20 degrees, finding an orientation consistent with the
parsec-scale jet identified in x-ray observations and with the circumnuclear
disk, a massive torus of molecular gas ~1.5 pc from Sgr A*. This suggests that
the gas in the accretion flow comes predominantly from the circumnuclear disk,
rather than the winds of stars in the young clockwise disk. This result will be
tested by the Event Horizon Telescope within the next year. Our model also
makes testable predictions for the orbital evolution of G1 and G2, falsifiable
on a 5-10 year timescale.
