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

Technical Summary: Empirical colour–effective temperature relations in the SDSS system from IRFM temperatures of GALAH and APOGEE stars

Problem Statement
Accurate stellar effective temperatures ($T_{\text{eff}}$) are fundamental to stellar population studies and Galactic astrophysics, underpinning the determination of radii, luminosities, chemical abundances, ages, and distances. While high-resolution spectroscopy provides precise atmospheric parameters, it remains unavailable for the vast majority of stars in modern large-scale photometric surveys. Consequently, empirical colour–$T_{\text{eff}}$ relations derived from broadband photometry are essential tools. However, existing relations often suffer from inconsistencies due to differences in adopted temperature scales, photometric zero-points, extinction treatments, and underlying stellar samples. Furthermore, small zero-point offsets in the Sloan Digital Sky Survey (SDSS) photometry can propagate directly into temperature estimates, limiting the achievable accuracy of photometric $T_{\text{eff}}$ determinations.

Methodology
This work establishes updated empirical colour–$T_{\text{eff}}$ calibrations for the SDSS $ugriz$ system combined with 2MASS $JHK_s$ data. The methodology relies on a homogeneous temperature scale derived using the InfraRed Flux Method (IRFM).

• Reference Samples: The study utilizes a combined sample of 3,902 stars from the GALAH DR3 survey and 2,535 stars from the APOGEE survey. These samples were selected for high-quality spectroscopic parameters and reliable photometry. The APOGEE metallicity scale was calibrated onto the GALAH scale to ensure homogeneity.
• Temperature Determination: Effective temperatures were determined using the IRFM, which compares observed bolometric fluxes to monochromatic infrared fluxes. This method yields an absolute temperature scale weakly dependent on stellar atmosphere models and relatively insensitive to metallicity and surface gravity. Crucially, the SDSS $ugriz$ photometry was corrected for zero-point offsets (as detailed in a companion study, Zhou et al. 2026) to place them on the AB magnitude scale before being used in the IRFM.
• Data Selection: Interstellar reddening corrections were applied using the Bayestar2019 dust map, with a strict cut of $E(B-V) \le 0.10$ to minimize extinction uncertainties. The final calibration sample spans $4000 \lesssim T_{\text{eff}} \lesssim 7000$ K.
• Calibration Strategy: To account for surface gravity effects, separate calibrations were derived for dwarfs and giants using a temperature-dependent boundary in the $T_{\text{eff}}$–$\log g$ plane. A buffer region ($\Delta \log g = 0.10$ dex) was excluded from fitting to prevent cross-contamination.
• Functional Form: The relations are modeled using low-order polynomials of the form:
  $$T_{\text{eff}} = a_0 + a_1X + a_2X^2 + a_3X^3 + a_4[\text{Fe/H}] + a_5X[\text{Fe/H}] + a_6[\text{Fe/H}]^2$$
  where $X$ is the dereddened colour index. The polynomial order (first to third) was selected based on the curvature of the specific colour index. Coefficients were derived using iterative $3\sigma$ clipping to remove outliers.

Key Contributions and Results
The paper provides a self-consistent set of colour–metallicity–$T_{\text{eff}}$ relations for FGK-type stars.

• Precision and Baseline Dependence: The precision of the relations is strongly dependent on the wavelength baseline. Long-baseline colours, particularly optical–infrared combinations such as $(g-K_s)_0$, $(g-H)_0$, and $(g-J)_0$, achieve the highest internal precisions, with RMS scatters of $\sim 20\text{--}40$ K. Purely optical colours (e.g., $(g-r)_0$, $(r-i)_0$) exhibit larger dispersions ($\sim 60\text{--}140$ K) and stronger sensitivity to metallicity.
• Metallicity Dependence: Including metallicity terms significantly improves the calibration, particularly for blue optical colours where line blanketing effects are prominent. The relations are valid for metallicities roughly between $-2.6$ and $+0.7$ dex, though the authors recommend adopting metallicity floors of $[\text{Fe/H}] = -1$ for dwarfs and $[\text{Fe/H}] = -2$ for giants when applying the calibrations to stars with lower measured metallicities, as the colour dependence on metallicity weakens at low values.
• Dwarf vs. Giant Consistency: The transition region between dwarfs and giants was validated using stars excluded from the fitting process. Long-baseline colours show excellent agreement between the two calibrations (mean offsets of $\sim 15\text{--}20$ K), whereas purely optical colours show larger offsets, reflecting their higher sensitivity to surface gravity.
• Comparison with Previous Work:
  • Pinsonneault et al. (2012): The new calibrations show good agreement, with differences typically within $\sim 50$ K.
  • Huang et al. (2015): Systematic offsets of $\sim 50\text{--}100$ K (with the new relations being hotter) are observed, particularly at lower temperatures. The authors attribute these discrepancies to differences in the adopted temperature scales (specifically, earlier interferometric measurements used by Huang et al. that may have overestimated stellar diameters) and the use of colour transformations from the Johnson system rather than direct SDSS/2MASS photometry.

Significance
The primary significance of this work lies in providing a homogeneous and internally consistent framework for estimating stellar effective temperatures directly from SDSS and 2MASS photometry alone. By anchoring the calibrations to an IRFM temperature scale derived from corrected SDSS zero-points, the study addresses systematic uncertainties inherent in previous calibrations. The resulting relations are well-suited for application to large photometric surveys lacking spectroscopic information, enabling the robust characterisation of millions of stars across the Milky Way. The authors emphasize that these relations are most reliable when used within the specified colour and metallicity ranges and with the appropriate dwarf/giant classification.