The First Spatially-resolved Detection of $^{13}$CN in a Protoplanetary Disk and Evidence for Complex Carbon Isotope Fractionation

Recent measurements of carbon isotope ratios in both protoplanetary disks and exoplanet atmospheres have suggested a possible transfer of significant carbon isotope fractionation from disks to planets. For a clearer understanding of the isotopic link between disks and planets, it is important to measure the carbon isotope ratios in various species. In this paper, we present a detection of the $^{13}$CN $N=2-1$ hyperfine lines in the TW Hya disk with the Atacama Large Millimeter/submillimeter Array. This is the first spatially-resolved detection of $^{13}$CN in disks, which enables us to measure the spatially resolved $^{12}$CN/$^{13}$CN ratio for the first time. We conducted non-local thermal equilibrium modeling of the $^{13}$CN lines in conjunction with previously observed $^{12}$CN lines to derive the kinetic temperature, ${\rm H_2}$ volume density, and column densities of $^{12}$CN and $^{13}$CN. The ${\rm H_2}$ volume density is found to range between $ (4 - 10)\times10^7 \ {\rm cm^{-3}}$, suggesting that CN molecules mainly reside in the disk upper layer. The $^{12}$CN/$^{13}$CN ratio is measured to be $ 70^{+9}_{-6}$ at $30 < r < 80$ au from the central star, which is similar to the $\rm ^{12}C/^{13}C$ ratio in the interstellar medium. However, this value differs from the previously reported values found for other carbon-bearing molecules (CO and HCN) in the TW Hya disk. This could be self-consistently explained by different emission layer heights for different molecules combined with preferential sequestration of $\rm ^{12}C$ into the solid phase towards the disk midplane. This study reveals the complexity of the carbon isotope fractionation operating in disks.