Candidate Water Vapor Lines to Locate the $\mathrm{H_2O}$ Snowline Through High-dispersion Spectroscopic Observations. III. Sub-millimeter $\mathrm{H_2}$$^{16}\mathrm{O}$ and $\mathrm{H_2}$$^{18}\mathrm{O}$ Lines

In this paper, we extend the results presented in our former papers (Notsu et al. 2016, 2017) on using ortho-$mathrm{H_2}$$^{16}mathrm{O}$ line profiles to constrain the location of the $mathrm{H_2O}$ snowline in T Tauri and Herbig Ae disks, to include sub-millimeter para-$mathrm{H_2}$$^{16}mathrm{O}$ and ortho- and para-$mathrm{H_2}$$^{18}mathrm{O}$ lines. Since the number densities of the ortho- and para-H$_{2}$$^{18}$O molecules are about 560 times smaller than their $^{16}$O analogues, they trace deeper into the disk than the ortho-H$_{2}$$^{16}$O lines (down to $z=0$, i.e., the midplane). Thus these H$_{2}$$^{18}$O lines are potentially better probes of the position of the H$_{2}$O snowline at the disk midplane, depending on the dust optical depth. The values of the Einstein $A$ coefficients of sub-millimeter candidate water lines tend to be lower (typically $u003c$$10^{-4}$ s$^{-1}$) than infrared candidate water lines (Notsu et al. 2017). Thus in the sub-millimeter candidate water line cases, the local intensity from the outer optically thin region in the disk is around $10^{4}$ times smaller than that in the infrared candidate water line cases. Therefore, in the sub-millimeter lines, especially H$_{2}$$^{18}$O and para-H$_{2}$$^{16}$O lines with relatively lower upper state energies ($sim$ a few 100K) can also locate the position of the $mathrm{H_2O}$ snowline. We also investigate the possibility of future observations with ALMA to identify the position of the water snowline. There are several candidate water lines that trace the hot water vapor inside the $mathrm{H_2O}$ snowline in ALMA Bands $5-10$.