# Mixing-length estimates from binary systems. A theoretical investigation on the estimation errors

## TL;DR

This study investigates the accuracy of recovering the mixing-length parameter in binary star systems through simulations, revealing significant challenges and limitations in constraining the parameter due to observational and methodological factors.

## Contribution

It provides a comprehensive analysis of different fitting approaches for estimating mixing-length in binary stars, highlighting the difficulties and biases involved.

## Key findings

- Full independence assumption yields unconstrained mixing-length estimates.
- Joint fitting with common age and composition improves estimates for secondary star.
- Large uncertainties remain in recovering true mixing-length differences, especially at higher disparities.

## Abstract

We performed a theoretical investigation on the mixing-length parameter recovery from an eclipsing double-lined binary system. We focused on a syntetic system composed by a primary of mass M = 0.95 Msun and a secondary of M = 0.85 Msun. Monte Carlo simulations were conducted at three metallicities, and three 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 reconstructed using 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 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 $\alpha_{ml,1}$ values match the ones recovered under the previous configuration, but $\alpha_{ml,2}$ values are peaked around unbiased estimates. This occurs because the primary star provides a much more tight age constraint in the joint fit than the secondary. Within this second scenario we also explored, for systems sharing a common $\alpha_{ml}$, 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 large standard deviation of 0.3 (15\% of the solar-scaled value). The third technique also imposes the constraint of a common mixing-length value for the two stars, and served as a test for identification of wrong fitting assumptions. In this case the common mixing-length is mainly dictated by the value of $\alpha_{ml,2}$. [...] For $\Delta \alpha_{ml} > 0.4$ less than half of the systems can be recovered and only 20% at $\Delta \alpha_{ml} = 1.0$.