Excitation of warm HD and H$_2$ and implications for the interstellar deuterium abundance

Abstract

Rotational emission from HD provides a direct probe of deuterium in warm molecular gas, but the conversion of HD line fluxes to total column densities is sensitive to non-LTE excitation. I present calculations of the non-LTE rotational excitation of HD and H$_2$ and a method for fitting their rotational diagrams simultaneously. Because HD remains subthermally excited to higher densities than H$_2$, the combined analysis constrains the thermal pressure as well as the HD/H$_2$ abundance ratio. I apply this method to JWST/MIRI observations of 26 positions in 10 protostellar outflows from the JOYS survey, published previously by Francis et al. (2025), including 17 positions with secure detections of at least three HD lines. Two-temperature, isobaric models generally provide good fits to the measured H$_2$ and HD level populations. For 11 positions with HD detections, the inferred pressures span $p/k_B=8.4\times10^7$ - $5.1\times10^9\ {\rm K\,cm^{-3}}$; for the remaining six, the HD populations approach LTE and provide only lower limits on the gas pressure. Excluding two positions toward IRAS 4B with comparatively poor fits, the median and mean inferred HD/H$_2$ abundance ratios are $1.42\times10^{-5}$ and $1.47\times10^{-5}$, respectively, only 28$\%$ and 29$\%$ of the value expected if all primordial deuterium were present as HD. These results confirm that HD/H$_2$ is generally low in shocked molecular gas while demonstrating that pressure-dependent non-LTE excitation must be modeled explicitly. Translating the measured HD/H$_2$ ratios into elemental D/H requires additional modeling of shock chemistry.