OH mid-infrared emission as a diagnostic of H$_2$O UV photodissociation. I. Model and application to the HH 211 shock

Water is an important molecule in interstellar and circumstellar environments. Previous observations of mid-infrared rotational lines of OH suggest that these lines may be used to probe the photodissociation of water. In this work, cross sections for the photodissociation of H$_2$O resolving the state of the OH fragment are collected and incorporated in a new molecular excitation code called GROSBETA, which includes radiative pumping, collisional (de)excitation, and prompt emission (i.e., following the production of OH in excited states). We find that the OH rotational line intensities in the range 9-16$\mu$m, covering rotational transitions with $N_{up}=18$ to 45, are proportional to the column density of H$_2$O photodissociated per second by photons in the range 114-143nm and do not depend on other local properties. Provided an independent measurement of the column density of water is available, the strength of the local UV radiation field can be deduced with good accuracy. In contrast, the OH lines at longer far-infrared wavelengths are primarily produced by IR radiative pumping and collisions. Our model successfully reproduces the OH mid-IR lines in the $10-16\mu$m range observed by Spitzer toward the tip of the HH 211 bow-shock and shows that the jet shock irradiates its surroundings, exposing H$_2$O to a UV photon flux that is about $5 \times 10^3$ times larger than the standard interstellar radiation field. We also find that chemical pumping by the reaction H$_2$ + O may supplement the excitation of lines in the range $16-30\mu$m. The mid-infrared lines of OH constitute a powerful diagnostic for inferring the photodissociation rate of water and thus the UV field water is exposed to. Future JWST-MIRI observations will be able to map the photodestruction rate of H$_2$O in various dense and irradiated environments and provide robust estimates of the local UV radiation field.