A Multi-line Analysis of the Distribution and Excitation of CS and H$_2$CS in the HD 163296 Disk

The abundance and distribution of sulfur-bearing molecules in protoplanetary disks directly influences the composition and potential habitability of nascent planets in addition to providing powerful probes of the physical gas conditions in the disks themselves. Here, we present new and archival ALMA and SMA observations of CS and H$_2$CS, and their C$^{34}$S and H$_2$C$^{34}$S isotopologues, at high-angular resolution (${\approx}$0."2-0."4; 20-40 au) in the HD 163296 disk, which reveal a central cavity and multi-ringed emission structure. These observations comprise the most comprehensive, multi-line CS data in a planet-forming disk to date, spanning a wide range of excitation conditions from E$_{\rm{u}}$=7.1 K to 129.3 K, and include new detections of C$^{34}$S, H$_2$CS, and H$_2$C$^{34}$S in this system. Using these data, we derive spatially-resolved rotational temperature and column density profiles for all species. We find a column density ratio N(H$_2$CS)/N(CS) $\approx$ 0.5, which is comparable to that of the similar MWC 480 disk and suggests that organic sulfur compounds may constitute a large fraction of the volatile sulfur reservoir in disks around Herbig stars generally. We derive $^{32}$S/$^{34}$S ratios of ${\approx}$5 (CS/C$^{34}$S) and ${\approx}$2 (H$_2$CS/H$_2$C$^{34}$S) based on disk-averaged and spatially-resolved analyses. Both values are consistent across these two pairs of optically-thin molecules and are well-below the expected ISM ratio of ${\approx}$22, suggesting significant sulfur fractionation. We also constrain the CS emitting layer ($z/r\lesssim 0.1$) using the vertical separations of the disk surfaces in the channel maps and based on the known 2D gas structure of the HD 163296 disk combined with our excitation analysis.