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.
