  Aims: Temperature uncertainties plague our understanding of abundance
variations within the ISM. Using the PHANGS-MUSE large program, we develop and
apply a new technique to model the strong emission lines arising from HII
regions in 19 nearby spiral galaxies at ~50 pc resolution and infer electron
temperatures for the nebulae. Methods: Due to the charge-exchange coupling of
the ionization fraction of the atomic oxygen to that of hydrogen, the
emissivity of the observed [OI]6300/Ha line ratio can be modeled as a function
of gas phase oxygen abundance (O/H), ionization fraction (f_ion) and electron
temperature (T_e). We measure (O/H) using a strong line metallicity
calibration, and identify a correlation between f_ion and
[SIII]9069/[SII]6716,6730, tracing ionization parameter variations. Results: We
solve for T_e, and test the method by reproducing direct measurements of
T_e([NII]5755) based on auroral line detections to within ~600 K. We apply this
charge-exchange method of calculating T_e to 4,129 HII regions across 19
PHANGS-MUSE galaxies. We uncover radial temperature gradients, increased
homogeneity on small scales, and azimuthal temperature variations in the disks
that correspond to established abundance patterns. This new technique for
measuring electron temperatures leverages the growing availability of optical
integral field unit spectroscopic maps across galaxy samples, increasing the
statistics available compared to direct auroral line detections.
