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We aim to constrain the temperature and velocity structures, and H2Oabundances in the winds of a sample of M-type AGB stars. We further aim todetermine the effect of H2O line cooling on the energy balance in the innercircumstellar envelope. We use two radiative-transfer codes to model molecularemission lines of CO and H2O towards four M-type AGB stars. We focus onspectrally resolved observations of CO and H2O from HIFI. The observations arecomplemented by ground-based CO observations, and spectrally unresolved CO andH2O observations with PAC. The observed line profiles constrain the velocitystructure throughout the circumstellar envelopes (CSEs), while the COintensities constrain the temperature structure in the CSEs. The H2Oobservations constrain the o-H2O and p-H2O abundances relative to H2. Finally,the radiative-transfer modelling allows to solve the energy balance in the CSE,in principle including also H2O line cooling. The fits to the line profilesonly set moderate constraints on the velocity profile, indicating shalloweracceleration profiles in the winds of M-type AGB stars than predicted bydynamical models, while the CO observations effectively constrain thetemperature structure. Including H2O line cooling in the energy balance wasonly possible for the low-mass-loss-rate objects in the sample, and required anad hoc adjustment of the dust velocity profile in order to counteract extremecooling in the inner CSE. H2O line cooling was therefore excluded from themodels. The constraints set on the temperature profile by the CO linesnevertheless allowed us to derive H2O abundances. The derived H2O abundancesconfirm previous estimates and are consistent with chemical models. However,the uncertainties in the derived abundances are relatively large, in particularfor p-H2O, and consequently the derived o/p-H2O ratios are not wellconstrained.

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