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.