Revisiting the statistical equilibrium of H$^-$ in stellar atmospheres

The negative hydrogen ion H$^-$ is, almost without exception, treated in local thermodynamic equilibrium (LTE) in the modelling of F, G, and K stars, where it is the dominant opacity source in the visual spectral region. This assumption rests in practice on a study from the 1960s. Since that work, knowledge of relevant atomic processes and theoretical calculations of stellar atmospheres and their spectra have advanced significantly, but this question has not been reexamined. We present calculations based on a slightly modified analytical model that includes H, H$_2$, and H$^-$, together with modern atomic data and a grid of 1D LTE theoretical stellar atmosphere models with stellar parameters ranging from T$_\mathrm{eff} = 4000$ to 7000~K, $\log{g} = 1$ to 5 cm/s$^2$, and [Fe/H]$=-3$ to 0. We find direct non-LTE effects on populations in spectrum-forming regions, continua, and spectral lines of about 1-2% in stars with higher T$_\mathrm{eff}$ and/or lower $\log g$. Effects in models for solar parameters are smaller by a factor of 10, about 0.1-0.2%, and are practically absent in models with lower T$_\mathrm{eff}$ and/or higher $\log g$. These departures from LTE found in our calculations originate from the radiative recombination of electrons with hydrogen to form H$^-$ exceeding photodetachment, that is, overrecombination. Modern atomic data are not a source of significant differences compared to the previous work, although detailed data for processes on H$_2$ resolved with vibrational and rotational states provide a more complete and complex picture of the role of H$_2$ in the equilibrium of H$^-$. In the context of modern studies of stellar spectra at the percent level, our results suggest that this question requires further attention, including a more extensive reaction network, and indirect effects due to non-LTE electron populations.