The impact of electron temperature and density diagnostics on oxygen abundance determinations in the Orion Nebula

The Orion Nebula (M 42) offers a unique nearby laboratory for testing chemical abundance methods in ionised gas at high spatial resolution. Despite its extensive observational history, uncertainties remain on how electron temperature (Tₑ) and density (nₑ) diagnostics affect abundances derived from collisionally excited lines (CELs). Using VLT/MUSE integral-field observations, we explore several temperature schemes, including classical CEL-based approaches and alternatives combining Tₑ(He I) with updated relations between low- and high-ionisation zones. We also compare density tracers, such as nₑ([S II]), nₑ([Cl III]), nₑ([Fe III]) and hybrid schemes assigning distinct diagnostics to low- and high-ionisation species. These prescriptions reveal systematic variations in the oxygen abundance across the nebula. Abundances based on He I recombination-line temperatures tend to yield higher oxygen abundances in specific regions, in some cases approaching the expected near-solar value, while other areas deviate less from classical results. These findings emphasise the critical role of diagnostic choices in spatially resolved abundance studies and the potential of recombination-line temperatures to refine metallicity estimates in nearby H II regions.