Equilibrium \(\mathrm{H}_2^+/\mathrm{H}^-\) Abundance Ratios in MARCS Atmosphere Models

Abstract

We investigate the equilibrium abundance ratio of the molecular hydrogen ion \(\mathrm{H}_2^+\) and the negative hydrogen ion \(\mathrm{H}^-\) in stellar atmospheres using compact Saha-type equilibrium expressions and layer-by-layer partial pressures from MARCS model atmospheres. The analysis distinguishes the temperature-dependent coefficient ratio \(K_{\mathrm{H}_2^+}(T)/K_{\mathrm{H}^-}(T)\) from the full abundance ratio, which also depends on the local ionisation factor \(N_{\mathrm{H}^+}/N_e\). The internal partition function \(Z_{\mathrm{H}_2^+}(T)\) is included in the \(\mathrm{H}_2^+\) equilibrium coefficient. We processed \(51{,}994\) MARCS models, corresponding to \(2{,}911{,}664\) atmospheric layers, and computed \(N(\mathrm{H}_2^+)/N(\mathrm{H}^-)\) directly from the tabulated equilibrium partial pressures, constructing the first systematic map of this ratio across the MARCS grid. \(\mathrm{H}_2^+\) exceeds \(\mathrm{H}^-\) in \(827{,}350\) layers, corresponding to \(F=0.284\), or \(28.4\%\) of all analysed layers. Although \(\mathrm{H}^-\) remains dominant in the global logarithmic mean, with \(\langle \log_{10}[N(\mathrm{H}_2^+)/N(\mathrm{H}^-)] \rangle=-4.229\), \(\mathrm{H}_2^+\) becomes comparable to or more abundant than \(\mathrm{H}^-\) in a substantial subset of layers. Its relative significance increases toward higher \(T_{\mathrm{eff}}\), lower \([\mathrm{Fe}/\mathrm{H}]\), and deeper layers. The balance is therefore controlled by the combined local thermodynamic and ionisation structure, not by temperature alone. These ratios are equilibrium abundance diagnostics and do not imply opacity dominance; assessing opacity requires wavelength-dependent cross-sections and radiative-transfer calculations.