Calibrating convective-core overshooting with eclipsing binary systems. The case of low-mass main-sequence stars

In a robust statistical way, we quantify the uncertainty that affects the calibration of the overshooting efficiency parameter $\beta$ that is owing to the uncertainty on the observational data in double-lined eclipsing binary systems. We also quantify the bias that is caused by the lack of constraints on the initial helium content and on the efficiencies of the superadiabatic convection and microscopic diffusion. We adopted a modified grid-based SCEPtER pipeline using as observational constraints the effective temperatures, [Fe/H], masses, and radii of the two stars. In a reference scenario of mild overshooting $\beta = 0.2$ for the synthetic data, we found both large statistical uncertainties and biases on the estimated $\beta$. For the first 80% of the MS evolution, $\beta$ is biased and practically unconstrained in the whole explored range [0.0; 0.4]. In the last 5% of the MS the bias vanishes and the $1 \sigma$ error is about 0.05. For synthetic data computed with $\beta = 0.0$, the estimated $\beta$ is biased by about 0.12 in the first 80% of the MS evolution, and by 0.05 afterwards. Assuming an uncertainty of $\pm 1$ in the helium-to-metal enrichment ratio $\Delta Y/\Delta Z$, we found that in the terminal part of the MS evolution the error on the estimated $\beta$ values ranges from -0.05 to +0.10, while $\beta$ is basically unconstrained throughout the explored range at earlier evolutionary stages. A uniform variation of $\pm 0.24$ in the mixing-length parameter around the solar-calibrated value causes in last 5% of the MS an uncertainty from -0.09 to +0.15. A complete neglect of diffusion in the stellar evolution computations produces a $1 \sigma$ uncertainty of $\pm 0.08$ in the last 5% of the MS, while $\beta$ is practically unconstrained in the first 80% of the MS. Overall, the calibration appears poorly reliable.