We present new high-quality ALMA observations of the Red Rectangle (a well known post-AGB object) in C17O J=6-5 and H13CN J=4-3 line emission and results from a new reduction of already published 13CO J=3-2 data. A detailed model fitting of all the molecular line data, including previous maps and single-dish spectra, was performed using a sophisticated code. These observations and the corresponding modeling allowed us to deepen the analysis of the nebular properties. We also stress the uncertainties in the model fitting. We confirm the presence of a rotating equatorial disk and an outflow, which is mainly formed of gas leaving the disk. The mass of the disk is ~ 0.01 Mo, and that of the CO-rich outflow is ~ 10 times smaller. High temperatures of ~ 100 K are derived for most components. From comparison of the mass values, we roughly estimate the lifetime of the rotating disk, which is found to be of about 10000 yr. Taking data of a few other post-AGB composite nebulae into account, we find that the lifetimes of disks around post-AGB stars typically range between 5000 and more than 20000 yr. The angular momentum of the disk is found to be high, ~ 9 Mo AU km/s, which is comparable to that of the stellar system at present. Our observations of H13CN show a particularly wide velocity dispersion and indicate that this molecule is only abundant in the inner Keplerian disk, at ~ 60 AU from the stellar system. We suggest that HCN is formed in a dense photodissociation region (PDR) due to the UV excess known to be produced by the stellar system, following chemical mechanisms that are well established for interstellar medium PDRs and disks orbiting young stars. We further suggest that this UV excess could lead to the efficient formation and excitation of PAHs and other C-bearing macromolecules, whose emission is very intense in the optical counterpart.
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