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 (T$_e$) and density (n$_e$) 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 T$_e$(He I) with updated relations between low- and high-ionisation zones. We also compare density tracers, such as n$_e$([S II]), n$_e$([Cl III]), n$_e$([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.