Context :

Aims : In this study we examine rotational emission lines of two isotopologues of water : \mathrm { H } _ { 2 } ^ { 17 } O and \mathrm { 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 \mathrm { H } _ { 2 } ^ { 17 } O and \mathrm { 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 \sim 1.0 to 1.5 \mathrm { 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 .