Analytical Formulas of Molecular Ion Abundances and N2H+ Ring in Protoplanetary Disks

We investigate the chemistry of ion molecules in protoplanetary disks, motivated by the detection of N$_2$H$^+$ ring around TW Hya. While the ring inner radius coincides with the CO snow line, it is not apparent why N$_2$H$^+$ is abundant outside the CO snow line in spite of the similar sublimation temperatures of CO and N$_2$. Using the full gas-grain network model, we reproduced the N$_2$H$^+$ ring in a disk model with millimeter grains. The chemical conversion of CO and N$_2$ to less volatile species (sink effect hereinafter) is found to affect the N$_2$H$^+$ distribution. Since the efficiency of the sink depends on various parameters such as activation barriers of grain surface reactions, which are not well constrained, we also constructed the no-sink model; the total (gas and ice) CO and N$_2$ abundances are set constant, and their gaseous abundances are given by the balance between adsorption and desorption. Abundances of molecular ions in the no-sink model are calculated by analytical formulas, which are derived by analyzing the full-network model. The N$_2$H$^+$ ring is reproduced by the no-sink model, as well. The 2D (R-Z) distribution of N$_2$H$^+$, however, is different among the full-network model and no-sink model. The column density of N$_2$H$^+$ in the no-sink model depends sensitively on the desorption rate of CO and N$_2$, and the flux of cosmic ray. We also found that N$_2$H$^+$ abundance can peak at the temperature slightly below the CO sublimation, even if the desorption energies of CO and N$_2$ are the same.