We aim to study the excitation conditions of the molecular gas in the rotating disk of the Red Rectangle, the only post-Asymptotic-Giant-Branch object in which the existence of an equatorial rotating disk has been demonstrated. For this purpose, we developed a complex numerical code that accurately treats radiative transfer in 2-D, adapted to the study of molecular lines from rotating disks. We present far-infrared Herschel/HIFI observations of the 12CO and 13CO J=6-5, J=10-9, and J=16-15 transitions in the Red Rectangle. We also present our code in detail and discuss the accuracy of its predictions, from comparison with well-tested codes. Theoretical line profiles are compared with the empirical data to deduce the physical conditions in the disk by means of model fitting. We conclude that our code is very efficient and produces reliable results. The comparison of the theoretical predictions with our observations reveals that the temperature of the Red Rectangle disk is typically ~ 100-150 K, about twice as high as previously deduced from mm-wave observations of lower-J lines. We discuss the relevance of these new temperature estimates for understanding the thermodynamics and dynamics of this prototype object, as well as for interpreting observations of other rarely studied post-AGB disks. Despite our sophisticated treatment of the line formation, our model cannot explain the relatively strong line-wing emission for intermediate-J transitions. We argue that a model including a rotating disk only cannot reproduce these data and suggest that there is an additional extended (probably bipolar) structure expanding at about 7--15 km/s.
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