The DESIRED temperature-metallicity relations in star-forming regions: probing the Galactic radial and azimuthal metallicity distributions
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, $T_{\rm e}$([NII]) and $T_{\rm e}$([OIII]), 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 ($t^2 > 0$). Applying these calibrations to 460 radio observations of Galactic HII~regions spanning Galactocentric distances from $\sim0.1$ 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 $t^2 > 0$ 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 onto dust grains or nucleosynthetic processing via the CNO 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used relation in the literature, assuming $t^2 = 0$, produces an excessively steep gradient -- likely due to the use of outdated atomic data and pre-CCD observations.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.