  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.
