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High-pressure, low-abundance water in bipolar outflows. Results from a Herschel-WISH survey

(Abridged) We present a survey of the water emission in a sample of more than 20 outflows from low mass young stellar objects with the goal of characterizing the physical and chemical conditions of the emitting gas. We have used the HIFI and PACS instruments on board the Herschel Space Observatory to observe the two fundamental lines of ortho-water at 557 and 1670 GHz. These observations were part of the "Water In Star-forming regions with Herschel" (WISH) key program, and have been complemented with CO and H2 data. We find that the emission from water has a different spatial and velocity distribution from that of the J=1-0 and 2-1 transitions of CO, but it has a similar spatial distribution to H2, and its intensity follows the H2 intensity derived from IRAC images. This suggests that water traces the outflow gas at hundreds of kelvins responsible for the H2 emission, and not the component at tens of kelvins typical of low-J CO emission. A warm origin of the water emission is confirmed by a remarkable correlation between the intensities of the 557 and 1670 GHz lines, which also indicates the emitting gas has a narrow range of excitations. A non-LTE radiative transfer analysis shows that while there is some ambiguity on the exact combination of density and temperature values, the gas thermal pressure nT is constrained within less than a factor of 2. The typical nT over the sample is 4 10^{9} cm^{-3}K, which represents an increase of 10^4 with respect to the ambient value. The data also constrain within a factor of 2 the water column density. When this quantity is combined with H2 column densities, the typical water abundance is only 3 10^{-7}, with an uncertainty of a factor of 3. Our data challenge current C-shock models of water production due to a combination of wing-line profiles, high gas compressions, and low abundances.