We have used high resolution (~2.3) observations of the local (D = 46 Mpc) luminous infrared galaxy Arp 299 to map out the physical properties of the molecular gas which provides the fuel for its extreme star formation activity. The 12CO J=3-2, 12CO J=2-1 and 13CO J=2-1 lines were observed with the Submillimeter Array and the short spacings of the 12CO J=2-1 and J=3-2 observations have been recovered using James Clerk Maxwell Telescope single dish observations. We use the radiative transfer code RADEX to estimate the physical properties (density, column density and temperature) of the different regions in this system. The RADEX solutions of the two galaxy nuclei, IC 694 and NGC 3690, are consistent with a wide range of gas components, from warm moderately dense gas with T_{kin} > 30 K and n(H_{2}) ~ 0.3 - 3 x 10^{3} cm^{-3} to cold dense gas with T_{kin} ~ 10-30 K and n(H_{2}) > 3 x 10^{3} cm^{-3}. The overlap region is shown to have a better constrained solution with T_{rm{kin}}$ ~ 10-50 K and n(H_{2}) ~ 1-30 x 10^{3} cm^{-3}. We estimate the gas masses and star formation rates of each region in order to derive molecular gas depletion times. The depletion times of all regions (20-60 Myr) are found to be about 2 orders of magnitude lower than those of normal spiral galaxies. This rapid depletion time can probably be explained by a high fraction of dense gas on kiloparsec scales in Arp 299. We estimate the CO-to-H_{2} factor, alpha_{co} to be 0.4 pm 0.3 (3 x 10^{-4}/ x_{CO}) M_{sol} (K km s^{-1} pc^{2})^{-1} for the overlap region. This value agrees well with values determined previously for more advanced merger systems.
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