Water in low-mass star-forming regions with Herschel The link between water gas and ice in protostellar envelopes

Aims. Our aim is to determine the critical parameters in water chemistry and the contribution of water to the oxygen budget by observing and modelling water gas and ice for a sample of eleven low-mass protostars, for which both forms of water have been observed. Methods. A simplified chemistry network, which is benchmarked against more sophisticated chemical networks, is developed that includes the necessary ingredients to determine the water vapour and ice abundance profiles in the cold, outer envelope in which the temperature increases towards the protostar. Comparing the results from this chemical network to observations of water emission lines and previously published water ice column densities, allows us to probe the influence of various agents (e.g., far-ultraviolet (FUV) field, initial abundances, timescales, and kinematics). Results. The observed water ice abundances with respect to hydrogen nuclei in our sample are 30-80 ppm, and therefore contain only 10-30% of the volatile oxygen budget of 320 ppm. The keys to reproduce this result are a low initial water ice abundance after the pre-collapse phase together with the fact that atomic oxygen cannot freeze-out and form water ice in regions with T-dust greater than or similar to 15K. This requires short prestellar core lifetimes less than or similar to 0.1Myr. The water vapour profile is shaped through the interplay of FUV photodesorption, photodissociation, and freeze-out. The water vapour line profiles are an invaluable tracer for the FUV photon flux and envelope kinematics. Conclusions. The finding that only a fraction of the oxygen budget is locked in water ice can be explained either by a short precollapse time of less than or similar to 0.1Myr at densities of n(H) similar to 10(4) cm(-3), or by some other process that resets the initial water ice abundance for the post-collapse phase. A key for the understanding of the water ice abundance is the binding energy of atomic oxygen on ice.