Warm CO in evolved stars from the THROES catalogue. II. Herschel/PACS
  spectroscopy of C-rich envelopes
  This is the second paper of a series making use of Herschel/PACS spectroscopy
of evolved stars in the THROES catalogue to study the inner regions of their
circumstellar envelopes (CSEs). We analyze the CO emission spectra, including a
large number of high-$J$ CO lines (from $J$=14-13 to $J$=45-44), as a proxy for
the warm molecular gas in the CSEs of a sample of bright carbon-rich stars
spanning different evolutionary stages from the Asymptotic Giant Branch (AGB)
to the young planetary nebulae (PNe) phase. We use the rotational diagram (RD)
to derive rotational temperatures ($T_{\rm rot}$) and masses ($M_{\rm H_2}$) of
the envelope layers where the CO transitions arise. We also obtain a first
order estimate of the mass-loss rates and assess the impact of the opacity
correction for a range of characteristic envelope radii. We use multi-epoch
spectra for the well studied C-rich envelope IRC+10216 to investigate the
impact of CO flux variability on the values of $T_{\rm rot}$ and $M_{\rm H_2}$.
PACS sensitivity allowed the study of higher rotational numbers than before,
indicating the presence of a significant amount of warmer gas ($\sim$200-900 K)
not traceable with lower-$J$ CO observations at sub-mm/mm wavelengths. The
masses are in the range $\sim10^{-2}-10^{-5}\,\rm M_{\odot}$, anti-correlated
with temperature. For some strong CO emitters we infer a double temperature
(warm $T_{\rm rot}\sim$400 K and hot $T_{\rm rot}\sim$820 K) component. From
the analysis of IRC+10216, we corroborate that the effect of line variability
is perceptible on the $T_{\rm rot}$ of the hot component only, and certainly
insignificant on $M_{\rm H_2}$ and, hence, the mass-loss rate. Therefore, the
parameters derived from the RD are robust even when strong line flux
variability occurs, with the major source of uncertainty in the estimate of the
mass-loss rate being the size of the CO-emitting volume.
