Mixing-Length Estimates From Binary Systems. A Theoretical Investigation on the Estimation Errors

We performed a theoretical investigation on the biases and random uncertainties affecting the recovery of the mixing-length parameter αml from an ideal eclipsing double-lined binary system, with well constrained masses and radii. We focused on a test case composed by a primary of mass M = 0.95 M⊙ and a secondary of M = 0.85 M⊙. Synthetic stars were generated coevally and with a common chemical composition by sampling from a dense grid of stellar models. Observational errors were simulated by adding random perturbations to mock data. The αml parameter was then recovered by means of the SCEPtER-binary pipeline. Several Monte Carlo simulations were conducted considering three metallicities, coupled to three different evolutionary stages of the primary. For each configuration, artificial data were sampled assuming an increasing difference between the mixing-length of the two stars. The mixing length values were then reconstructed adopting three alternative set-ups. A first method, which assumes full independence between the two stars, showed a great difficulty to constrain the mixing-length values; the recovered values were nearly unconstrained with a standard deviation of about 0.40. The second technique imposes the constraint of common age and initial chemical composition for the two stars in the fit. We found that αml, 1 values closely match the ones recovered under the previous configuration, but αml, 2 values are much more peaked around unbiased estimates. This occurs because the primary star provides a much tighter age constraint in the joint fit than the secondary, thus leading to the rejection of several extreme solutions for the secondary. Within this second scenario we also explored, for systems sharing a common αml = 2.0, the difference in the mixing-length values of the two stars only due to random fluctuations owing to the observational errors. The posterior distribution of these differences was peaked around zero, with a somewhat large standard deviation of 0.3 (about 15% of the solar-scaled value). Therefore, about 32% of systems with true identical αml are expected to show differences higher than that only owing to random errors. The third technique also imposes the constraint of a common mixing-length value for the two stars. This assumption is generally not true for the sample stars and served as a test for identifying wrong fitting assumptions. In this case, the common mixing-length is mainly dictated by the value of αml, 2. However, an increasing share of systems cannot be fitted by the algorithm as the differences of αml between the two stars in the synthetic systems increases. For Δαml > 0.4, less than half of the systems can be recovered and only 20% at Δαml = 1.0.