Interstellar H$_2$O Masers from J Shocks
  We present a model in which the 22 GHz H$_2$O masers observed in star-forming
regions occur behind shocks propagating in dense regions (preshock density $n_0
\sim 10^6 - 10^8$ cm$^{-3}$). We focus on high-velocity ($v_s > 30$ km
s$^{-1}$) dissociative J shocks in which the heat of H$_2$ re-formation
maintains a large column of $\sim 300-400$ K gas; at these temperatures the
chemistry drives a considerable fraction of the oxygen not in CO to form
H$_2$O. The H$_2$O column densities, the hydrogen densities, and the warm
temperatures produced by these shocks are sufficiently high to enable powerful
maser action. The observed brightness temperatures (generally $\sim 10^{11} -
10^{14}$ K) are the result of coherent velocity regions that have dimensions in
the shock plane that are 10 to 100 times the shock thickness of $\sim 10^{13}$
cm. The masers are therefore beamed towards the observer, who typically views
the shock "edge-on", or perpendicular to the shock velocity; the brightest
masers are then observed with the lowest line of sight velocities with respect
to the ambient gas. We present numerical and analytic studies of the dependence
of the maser inversion, the resultant brightness temperature, the maser spot
size and shape, the isotropic luminosity, and the maser region magnetic field
on the shock parameters and the coherence path length; the overall result is
that in galactic H$_2$O 22 GHz masers these observed parameters can be produced
in J shocks with $n_0\sim 10^6 - 10^8$ cm$^{-3}$ and $v_s \sim 30 -200$ km
s$^{-1}$. A number of key observables such as maser shape, brightness
temperature, and global isotropic luminosity depend only on the particle flux
into the shock, $j=n_0v_s$, rather than on $n_0$ and $v_s$ separately.
