Empirical colour--effective temperature relations in the SDSS system from IRFM temperatures of GALAH and APOGEE stars

Reliable estimates of stellar effective temperature (T eff) are fundamental to stellar population studies and Galactic astrophysics. However, the majority of stars observed in modern large-scale photometric surveys lack spectroscopic measurements, making empirical colour-- T eff relations essential tools. In this work, we present updated empirical colour-- T eff calibrations based on Sloan Digital Sky Survey (SDSS) ugriz photometry combined with 2MASS JHK s data. Effective temperatures are determined on a homogeneous InfraRed Flux Method (IRFM) scale using a combined sample of 3902 GALAH and 2535 APOGEE stars with high-quality photometry and well-characterised atmospheric parameters. Using this dataset, we establish empirical relations between T eff and colour indices constructed from SDSS and 2MASS combinations. We provide both colour--metallicity-- T eff and colour-- T eff relations for dwarfs and giants. The calibrations are derived using low-order polynomial models with iterative 3 sigma clipping. Their performance depends on the adopted colour index, with long-baseline colours such as (g-K s) 0 and (g-z) 0 achieving internal precisions of about 30--50 K. Comparisons with previous calibrations show general agreement, with differences attributable to sample selection, photometric zero-points, and functional form. The resulting relations provide a homogeneous and internally consistent framework for estimating T eff from SDSS and 2MASS photometry alone, and are well suited for application to large photometric surveys lacking spectroscopic information.