Stars with accurate and precise effective temperature ( T _ { eff } ) measurements are needed to test stellar atmosphere models and calibrate empirical methods to determine T _ { eff } .
There are few standard stars currently available to calibrate temperature indicators for dwarf stars .
Gaia parallaxes now make it possible , in principle , to measure T _ { eff } for many dwarf stars in eclipsing binaries .
We aim to develop a method that uses high-precision measurements of detached eclipsing binary stars , Gaia parallaxes and multi-wavelength photometry to obtain accurate and precise fundamental effective temperatures that can be used to establish a set of benchmark stars .
We select the well-studied binary AI Phoenicis to test our method , since it has very precise absolute parameters and extensive archival photometry .
The method uses the stellar radii and parallax for stars in eclipsing binaries .
We use a Bayesian approach to obtain the integrated bolometric fluxes for the two stars from observed magnitudes , colours and flux ratios .
The fundamental effective temperature of two stars in AI Phoenicis are 6193 \pm 24 K for the F7 V component and 5090 \pm 17 K for the K0 IV component .
The zero-point error in the flux scale leads to a systematic error of only 0.2 % ( \approx 11 K ) in T _ { eff } .
We find that these results are robust against the details of the analysis , such as the choice of model spectra .
Our method can be applied to eclipsing binary stars with radius , parallax and photometric measurements across a range of wavelengths .
Stars with fundamental effective temperatures determined with this method can be used as benchmarks in future surveys .