The Metallicity-Electron Temperature Relationship in H <sc>ii</sc> Regions

H ii region heavy-element abundances throughout the Galactic disk provide important constraints to theories of the formation and evolution of the Milky Way. In LTE, radio recombination line (RRL) emission and free-free continuum emission are accurate extinction-free tracers of the H ii region electron temperature. Since metals act as coolants in H ii regions via the emission of collisionally excited lines, the electron temperature is a proxy for metallicity. Shaver et al. found a linear relationship between metallicity and electron temperature with little scatter. Here we use CLOUDY H ii region simulations to (1) investigate the accuracy of using RRLs to measure the electron temperature and (2) explore the metallicity-electron temperature relationship. We model 135 H ii regions with different ionizing radiation fields, densities, and metallicities. We find that electron temperatures derived under the assumption of LTE are about 20% systematically higher owing to non-LTE effects, but overall LTE is a good assumption for centimeter-wavelength RRLs. Our CLOUDY simulations are consistent with the Shaver et al. metallicity-electron temperature relationship, but there is significant scatter since earlier spectral types or higher electron densities yield higher electron temperatures. Using RRLs to derive electron temperatures assuming LTE yields errors in the predicted metallicity as large as 10%. We derive correction factors for log(O/H) + 12 in each CLOUDY simulation. For lower metallicities the correction factor depends primarily on the spectral type of the ionizing star and ranges from 0.95 to 1.10, whereas for higher metallicities the correction factor depends on the density and is between 0.97 and 1.05.