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Water in star-forming regions with Herschel (WISH) V. The physical conditions in low-mass protostellar outflows revealed by multi-transition water observations

Context: Outflows are an important part of the star formation process as both the result of ongoing active accretion and one of the main sources of mechanical feedback on small scales. Water is the ideal tracer of these effects because it is present in high abundance in various parts of the protostar. Method: We present \textit{Herschel} HIFI spectra of multiple water-transitions towards 29 nearby Class 0/I protostars as part of the WISH Survey. These are decomposed into different Gaussian components, with each related to one of three parts of the protostellar system; quiescent envelope, cavity shock and spot shocks in the jet and at the base of the outflow. We then constrain the excitation conditions present in the two outflow-related components. Results: Water emission is optically thick but effectively thin, with line ratios that do not vary with velocity, in contrast to CO. The physical conditions of the cavity and spot shocks are similar, with post-shock H$_{2}$ densities of order 10$^{5}-$10$^{8}$\,cm$^{-3}$ and H$_{2}$O column densities of order 10$^{16}-$10$^{18}$\,cm$^{-2}$. H$_{2}$O emission originates in compact emitting regions: for the spot shocks these correspond to point sources with radii of order 10-200\,AU, while for the cavity shocks these come from a thin layer along the outflow cavity wall with thickness of order 1-30\,AU. Conclusions: Water emission at the source position traces two distinct kinematic components in the outflow; J shocks at the base of the outflow or in the jet, and C shocks in a thin layer in the cavity wall. Class I sources have similar excitation conditions to Class 0 sources, but generally smaller line-widths and emitting region sizes. We suggest that it is the velocity of the wind driving the outflow, rather than the decrease in envelope density or mass, that is the cause of the decrease in H$_{2}$O intensity between Class 0 and I.