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Stars with accurate and precise effective temperature (T$_{\rm eff}$)measurements are needed to test stellar atmosphere models and calibrateempirical methods to determine T$_{\rm eff}$. There are few standard starscurrently available to calibrate temperature indicators for dwarf stars. Gaiaparallaxes now make it possible, in principle, to measure T$_{\rm eff}$ formany dwarf stars in eclipsing binaries. We aim to develop a method that useshigh-precision measurements of detached eclipsing binary stars, Gaia parallaxesand multi-wavelength photometry to obtain accurate and precise fundamentaleffective 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 hasvery precise absolute parameters and extensive archival photometry. The methoduses the stellar radii and parallax for stars in eclipsing binaries. We use aBayesian approach to obtain the integrated bolometric fluxes for the two starsfrom observed magnitudes, colours and flux ratios. The fundamental effectivetemperature of two stars in AI Phoenicis are $6199\pm22$ K for the F7Vcomponent and $5094\pm16$ K for the K0IV component. The zero-point error in theflux scale leads to a systematic error of only 0.2% ($\approx$ 11K) in T$_{\rmeff}$. We find that these results are robust against the details of theanalysis, such as the choice of model spectra. Our method can be applied toeclipsing binary stars with radius, parallax and photometric measurementsacross a range of wavelengths. Stars with fundamental effective temperaturesdetermined with this method can be used as benchmarks in future surveys.

Fundamental effective temperature measurements for eclipsing binary stars I. Development of the method and application to AI Phoenicis