A HIFI view on circumstellar H2O in M-type AGB stars: radiative transfer, velocity profiles, and H2O line cooling

Aims. We aim to constrain the temperature and velocity structures, and H2O abundances in the winds of a sample of M-type asymptotic giant branch (AGB) stars. We further aim to determine the effect of H2O line cooling on the energy balance in the inner circumstellar envelope. Methods. We use two radiative-transfer codes to model molecular emission lines of CO and H2O towards four M-type AGB stars. We focus on spectrally resolved observations of CO and H2O from HIFI aboard the Herschel Space Observatory. The observations are complemented by ground-based CO observations, and spectrally unresolved CO and H2O observations with PACS aboard Herschel. The observed line profiles constrain the velocity structure throughout the circumstellar envelopes (CSEs), while the CO intensities constrain the temperature structure in the CSEs. The H2O observations constrain the o-H2O and p-H2O abundances relative to H-2. Finally, the radiative-transfer modelling allows to solve the energy balance in the CSE, in principle including also H2O line cooling. Results. The fits to the line profiles only set moderate constraints on the velocity profile, indicating shallower acceleration profiles in the winds of M-type AGB stars than predicted by dynamical models, while the CO observations effectively constrain the temperature structure. Including H2O line cooling in the energy balance was only possible for the low-mass-loss-rate objects in the sample, and required an ad hoc adjustment of the dust velocity profile in order to counteract extreme cooling in the inner CSE. H2O line cooling was therefore excluded from the models. The constraints set on the temperature profile by the CO lines nevertheless allowed us to derive H2O abundances. The derived H2O abundances confirm previous estimates and are consistent with chemical models. However, the uncertainties in the derived abundances are relatively large, in particular for p-H2O, and consequently the derived o/p-H2O ratios are not well constrained.