Combined model for $\rm ^{15}N$, $\rm ^{13}C$, and spin-state chemistry in molecular clouds

We present a new gas-grain chemical model for the combined isotopic fractionation of carbon and nitrogen in molecular clouds, in which the isotope chemistry of carbon and nitrogen is coupled with a time-dependent description of spin-state chemistry. We updated the rate coefficients of some isotopic exchange reactions considered in the literature, and present here a set of new exchange reactions involving molecules substituted in $\rm ^{13}C$ and $\rm ^{15}N$ simultaneously. We apply the model to a series of zero-dimensional simulations representing a set of physical conditions across a prototypical prestellar core, exploring the deviations of the isotopic abundance ratios in the various molecules from the elemental isotopic ratios as a function of physical conditions and time. We find that the $\rm ^{12}C/^{13}C$ ratio can deviate from the elemental ratio by up to a factor of several depending on the molecule, and that there are highly time-dependent variations in the ratios. The $\rm HCN/H^{13}CN$ ratio, for example, can obtain values of less than 10 depending on the simulation time. The $\rm ^{14}N/^{15}N$ ratios tend to remain close to the assumed elemental ratio within $\sim$ ten per cent, with no clear trends as a function of the physical conditions. Abundance ratios between $\rm ^{13}C$-containing molecules and $\rm ^{13}C$+$\rm ^{15}N$-containing molecules show somewhat increased levels of fractionation due to the newly included exchange reactions, though still remaining within a few tens of per cent of the elemental $\rm ^{14}N/^{15}N$ ratio. Our results imply the existence of gradients in isotopic abundance ratios across prestellar cores, suggesting that detailed simulations are required to interpret observations of isotopically substituted molecules correctly, especially given that the various isotopic forms of a given molecule do not necessarily trace the same gas layers.