2025MNRAS.543..407R

Astrochemical models can be greatly simplified, with obvious computational advantages, if the reaction networks for key species can be reduced to a bare minimum. In addition, if chemical equilibrium holds, then simple analytical solutions can be formulated. These have particular advantages in the application to complex models evolving over multipoint spatial grids. In this study, the equilibrium solutions to highly simplified chemical networks for CO and H$_2$O have been re-assessed with particular attention to the formulation of the ice desorption rates in the context of 'three-phase' gas-grain astrochemical models. The analytical solutions have also been updated to account for the chemically inert reservoir of molecules below the surface ice layers, and to include the effects of reactive desorption. We find that a very close match is obtained to the results from detailed three-phase models of the time-dependent astrochemistry, and the abundances are typically accurate to within a factor of 2 over the entire range of densities and extinction that are applicable to dense clouds and young star-forming regions. In addition, these solutions give accurate results over most of the range of conditions even for systems undergoing rapid dynamical evolution. Although there are some caveats of applicability, we therefore recommend that these solutions be used in models of cold molecular environments where the rapid calculation of the abundances of CO, H$_2$O, and atomic coolants is helpful.