Statistical errors and systematic biases in the calibration of the convective core overshooting with eclipsing binaries: A case study: TZ Fornacis

Context. Recently published work has made high-precision fundamental parameters available for the binary system TZ Fornacis, making it an ideal target for the calibration of stellar models. Aims. Relying on these observations, we attempt to constrain the initial helium abundance, the age and the efficiency of the convective core overshooting. Our main aim is in pointing out the biases in the results due to not accounting for some sources of uncertainty. Methods. We adopt the SCEPtER pipeline, a maximum likelihood technique based on fine grids of stellar models computed for various values of metallicity, initial helium abundance and overshooting efficiency by means of two independent stellar evolutionary codes, namely FRANEC and MESA. Results. Beside the degeneracy between the estimated age and overshooting efficiency, we found the existence of multiple independent groups of solutions. The best one suggests a system of age 1.10 ± 0.07 Gyr composed of a primary star in the central helium burning stage and a secondary in the sub-giant branch (SGB). The resulting initial helium abundance is consistent with a helium-to-metal enrichment ratio of ΔY/ ΔZ = 1; the core overshooting parameter is β = 0.15 ± 0.01 for FRANEC and fov = 0.013 ± 0.001 for MESA. The second class of solutions, characterised by a worse goodness-of-fit, still suggest a primary star in the central helium-burning stage but a secondary in the overall contraction phase, at the end of the main sequence (MS). In this case, the FRANEC grid provides an age of Gyr and a core overshooting parameter, while the MESA grid gives 1.23 ± 0.03 Gyr and fov = 0.025 ± 0.003. We analyse the impact on the results of a larger, but typical, mass uncertainty and of neglecting the uncertainty in the initial helium content of the system. We show that very precise mass determinations with uncertainty of a few thousandths of solar mass are required to obtain reliable determinations of stellar parameters, as mass errors larger than approximately 1% lead to estimates that are not only less precise but also biased. Moreover, we show that a fit obtained with a grid of models computed at a fixed ΔY/ ΔZ - thus neglecting the current uncertainty in the initial helium content of the system - can provide severely biased age and overshooting estimates. The possibility of independent overshooting efficiencies for the two stars of the system is also explored. Conclusions. The present analysis confirms that to constrain the core overshooting parameter by means of binary systems is a very difficult task that requires an observational precision still rarely achieved and a robust statistical treatment of the error sources.