We report ALMA Band 7 (350 GHz) imaging at 0.4 - 0.6arcsec resolution and Band 9 (696 GHz) at ~0.25arcsec resolution of the luminous IR galaxies Arp 220 and NGC 6240. The long wavelength dust continuum is used to estimate ISM masses for Arp 220 East, West and NGC 6240 of 1.9, 4.2 and 1.6x10^9 msun within radii of 69, 65 and 190 pc. The HCN emission was modeled to derive the emissivity distribution as a function of radius and the kinematics of each nuclear disk, yielding dynamical masses consistent with the masses and sizes derived from the dust emission. In Arp 220, the major dust and gas concentrations are at radii less than 50 pc in both counter-rotating nuclear disks. The thickness of the disks in Arp 220estimated from the velocity dispersion and rotation velocities are 10-20 pc and the mean gas densities are n_H2 ~10^5 cm^-3 at R < 50 pc. We develop an analytic treatment for the molecular excitation (including photon trapping), yielding volume densities for both the HCN and CS emission with n_H2 ~2x10^5 cm^-3. The agreement of the mean density from the total mass and size with that required for excitation suggests that the volume is essentially filled with dense gas, i.e. it is not cloudy or like swiss cheese.
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