Empirical colour--effective temperature relations in the SDSS system from IRFM temperatures of GALAH and APOGEE stars
Reliable estimates of stellar effective temperature ($T_{\mathrm {eff}}$) 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--$T_{\mathrm {eff}}$ relations essential tools. In this work, we present updated empirical colour--$T_{\mathrm {eff}}$ calibrations based on Sloan Digital Sky Survey (SDSS) $ugriz$ photometry combined with 2MASS $JHK_{\mathrm s}$ 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 $T_{\mathrm {eff}}$ and colour indices constructed from SDSS and 2MASS combinations. We provide both colour--metallicity--$T_{\mathrm {eff}}$ and colour--$T_{\mathrm {eff}}$ relations for dwarfs and giants. The calibrations are derived using low-order polynomial models with iterative $3σ$ clipping. Their performance depends on the adopted colour index, with long-baseline colours such as $(g-K_{\mathrm s})_0$ and $(g-z)_0$ achieving internal precisions of $\sim$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 $T_{\mathrm {eff}}$ from SDSS and 2MASS photometry alone, and are well suited for application to large photometric surveys lacking spectroscopic information.