  We present an analysis of gas densities in the central R=300 parsecs of the
Milky Way, focusing on three clouds: GCM-0.02-0.07 (the 50 km/s cloud),
GCM-0.13-0.08 (the 20 km/s cloud), and GCM0.25+0.01 (the "Brick"). Densities
are determined using observations of the J=(3-2), (4-3), (5-4), (10-9),
(18-17), (19-18), (21-20), and (24-23) transitions of the molecule HC3N. We
find evidence of at least two excitation regimes for HC3N and constrain the
low-excitation component to have a density less than 10^4 cm^-3 and the
high-excitation component to have a density between 10^5 and 10^6 cm^-3. This
is much less than densities of 10^7 cm^-3 that are found in Sgr B2, the most
actively star-forming cloud in the Galactic center. This is consistent with the
requirement of a higher density threshold for star formation in the Galactic
center than is typical in the Galactic disk. We are also able to constrain the
column density of each component in order to determine the mass fraction of
'dense' (n>10^5 cm^-3) gas for these clouds. We find that this is ~15% for all
three clouds. Applying the results of our models to ratios of the (10-9) and
(3-2) line across the entire central R=300 pc, we find that the fraction of gas
with n>10^4 cm^-3 increases inward of a radius of ~140 pc, consistent with the
predictions of recent models for the gas dynamics in this region. Our
observations show that HC3N is an excellent molecule for probing the density
structure of clouds in the Galactic center.
