The DESIRED temperature–metallicity relations in star-forming regions: probing the Galactic radial and azimuthal metallicity distributions

ABSTRACT We analyse a sample of 225 star-forming regions from the DESIRED-E project, each with simultaneous determinations of the electron temperature from ionized nitrogen and oxygen, $T_{\rm e}$([N ii]) and $T_{\rm e}$([O iii]), respectively. We derive new empirical relations connecting the gas-phase metallicity to the global electron temperature, $T_{\rm e}$(H$^+$), as determined via radio observations. We establish two calibrations: one assuming a homogeneous temperature distribution ($t^2 = 0$, the ‘direct method’), and another accounting for internal temperature fluctuations ($t^2 \,\gt\, 0$). Applying these calibrations to 460 radio observations of Galactic H ii regions spanning Galactocentric distances from ${\sim} 0.1$ to 16 kpc, we determine the radial O/H gradient in the Milky Way under both assumptions. We further compare these nebular gradients to independent metallicity estimates from young O- and B-type stars and Cepheid variables. We find that the $t^2 \,\gt\, 0$ calibration yields a gradient in excellent agreement with stellar-based determinations, whereas the $t^2 = 0$ method underestimates metallicities by up to $\sim$0.3 dex. This discrepancy cannot be reconciled by invoking oxygen depletion on to dust grains or nucleosynthetic processing via the C-N-O cycle in massive stars. We also find that one widely used relation in the literature, assuming $t^2 = 0$, produces an excessively steep gradient – likely due to the use of outdated atomic data and pre-CCD observations. Finally, we explore potential azimuthal variations in the Galactic metallicity distribution driven by the presence of the spiral arms, finding no evidence for variations larger than $\sim$0.1 dex with respect to the general radial gradient.