Water isotopologues in the circumstellar envelopes of M-type AGB stars
  AIM: In this study we examine rotational emission lines of two isotopologues
of water: H$_2$$^{17}$O and H$_2$$^{18}$O. By determining the abundances of
these molecules, we aim to use the derived isotopologue --- and hence oxygen
isotope --- ratios to put constraints on the masses of a sample of M-type AGB
stars that have not been classified as OH/IR stars. METHODS: We use detailed
radiative transfer analysis based on the accelerated lambda iteration method to
model the circumstellar molecular line emission of H$_2$$^{17}$O and
H$_2$$^{18}$O for IK Tau, R Dor, W Hya, and R Cas. The emission lines used to
constrain our models come from Herschel/HIFI and Herschel/PACS observations and
are all optically thick, meaning that full radiative transfer analysis is the
only viable method of estimating molecular abundance ratios. RESULTS: We find
generally low values of the $^{17}$O/$^{18}$O ratio for our sample, ranging
from 0.15 to 0.69. This correlates with relatively low initial masses, in the
range $\sim1.0$ to 1.5 M$_\odot$ for each source, based on stellar evolutionary
models. We also find ortho-to-para ratios close to 3, which are expected from
warm formation predictions. CONCLUSIONS: The $^{17}$O/$^{18}$O ratios found for
this sample are at the lower end of the range predicted by stellar evolutionary
models, indicating that the sample chosen had relatively low initial masses.
