Waterfalls around protostars Infall motions towards Class 0/I envelopes as probed by water

Context. For stars to form, material must fall inwards from core scales through the envelope towards the central protostar. While theories of how this takes place have been around for some time, the velocity profile around protostars is poorly constrained. The combination of observations in multiple transitions of a tracer which is sensitive to kinematics and radiative transfer modelling of those lines has the potential to break this deadlock. Aims. Seven protostars observed with the Heterodyne Instrument for the Far-Infrared (HIFI) on board the Herschel Space Observatory as part of the Water in star-forming regions with Herschel (WISH) survey show infall signatures in water line observations. We aim to constrain the infall velocity and the radii over which infall is taking place within the protostellar envelopes of these sources. We will also use these data to constrain the chemistry of cold water. Methods. We use 1-D non-LTE ratran radiative transfer models of the observed water lines to constrain the infall velocity and chemistry in the protostellar envelopes of six Class 0 protostars and one Class I source. We assume a free-fall velocity profile and, having found the best fit, vary the radii over which infall takes place. Results. In the well-studied Class 0 protostar NGC1333-IRAS4A we find that our observations probe infall over the whole envelope to which our observations are sensitive (r greater than or similar to 1000 AU). For L1527, L1157, BHR71 and IRAS 15398 infall takes place on core to envelope scales (i.e. similar to 10000-3000 AU). In Serpens-SMM4 and GSS30 the inverse P-Cygni profiles seen in the ground-state lines are more likely due to larger-scale motions or foreground clouds. Models including a simple consideration of the chemistry are consistent with the observations, while using step abundance profiles are not. The non-detection of excited water in the inner envelope in six out of seven protostars is further evidence that water must be heavily depleted from the gas-phase at these radii. Conclusions. Infall in four of the sources is supersonic and in all sources must take place at the outer edge of the envelope, which may be evidence that collapse is global or outside-in rather than inside-out. The mass infall rate in NGC1333-IRAS4A is large (>= 10(-4) M-circle dot yr(-1)), higher than the mass outflow rate and expected mass accretion rates onto the star. This suggests that any flattened disk-like structure on small scales will be gravitationally unstable, potentially leading to rotational fragmentation and/or episodic accretion.