Analysis of Low Excitation HDO Transitions Toward the High-Mass
  Star-forming Regions G34.26+0.15, W51e$_{1}$/e$_{2}$, and W49N
  We present observations of the ground state 1$_{0,1}$-0$_{0,0}$ rotational
transition of HDO at 464.925 GHz and the 1$_{1,0}$-1$_{0,1}$ transition at
509.292 GHz toward the three high-mass star forming regions: G34.26+0.15, W49N,
and W51e$_{1}$/e$_{2}$, carried out with the Caltech Submillimeter Observatory.
The latter transition is observed for the first time from the ground. The
spectra are modeled, together with observations of higher-energy HDO
transitions, as well as submillimeter dust continuum fluxes from the
literature, using a spherically symmetric radiative transfer model to derive
the radial distribution of the HDO abundance in the target sources. The
abundance profile is divided into an inner hot core region, with kinetic
temperatures higher than 100~K, and a cold outer envelope with lower kinetic
temperatures. The derived HDO abundance with respect to H$_2$ is
(0.3-3.7)$\times 10^{-8}$ in the hot inner region ($T > 100 \mathrm{K}$) and
(7.0-10.0)$\times 10^{-11}$ in the cold outer envelope. We also used two
H$_{2}^{18}$O fundamental transitions to constrain the H$_{2}$O abundances in
the outer envelopes. The HDO/H$_{2}$O ratios in these cold regions are found to
be (1.8-3.1)$\times 10^{-3}$ and are consequently higher than in the hot inner
regions of these sources.
