Probing Planet Forming Zones with Rare CO Isotopologues

The gas near the midplanes of planet-forming protostellar disks remains largely unprobed by observations due to the high optical depth of commonly observed molecules such as CO and H$_2$O. However, rotational emission lines from rare molecules may have optical depths near unity in the vertical direction, so that the lines are strong enough to be detected, yet remain transparent enough to trace the disk midplane. Here we present a chemical model of an evolving T-Tauri disk and predict the optical depths of rotational transitions of $^{12}$C$^{16}$O, $^{13}$C$^{16}$O, $^{12}$C$^{17}$O and $^{12}$C$^{18}$O. The MRI-active disk is primarily heated by the central star due to the formation of the dead zone. CO does not freeze out in our modeled region within $70$AU around a sunlike star. However, the abundance of CO decreases because of the formation of complex organic molecules (COM), producing an effect that can be misinterpreted as the "snow line". These results are robust to variations in our assumptions about the evolution of the gas to dust ratio. The optical depths of low-order rotational lines of C$^{17}$O are around unity, making it possible to see into the disk midplane using C$^{17}$O. Combining observations with modeled C$^{17}$O$/$H$_2$ ratios, like those we provide, can yield estimates of protoplanetary disks' gas masses.