Despite the substantial refinements of the accuracy and reliability of stellar evolutionary predictions, several mechanisms involved in the evolution of stars are still poorly understood. A major and long-standing problem affecting stellar model computations is the treatment of superadiabatic convection. This lack prevents a firm and reliable prediction of the extension of the external convective regions.

A precise treatment of external convection would require 3D hydrodynamical calculations, greatly improved in recent years (see e.g., Tanner et al., 2013; Trampedach et al., 2014; Magic et al., 2015); however, they still cannot cover the wide range of input physics needed to model stellar computations. Moreover, defining alpha in one-dimensional codes from the results of 3D simulations is quite ambiguous due to the differences between the mixing length representation of convection and the convection behavior in 2D/3D simulations, even if there have been relevant attempts in the literature (Lydon et al., 1992; Ludwig et al., 1999; Tanner et al., 2013; Magic et al., 2015; Mosumgaard et al., 2017). Therefore, the current generation of stellar evolution codes still addresses this problem by relying, almost universally, on the mixing-length theory (Böhm-Vitense, 1958). In this framework, the efficiency of the convective transport and the stellar structure in the superadiabatic transition layers depends on the mixing-length l, which is supposed to be proportional to the pressure scale height H_p, i.e., l = α_mlH_p, where α_ml is a non-dimensional free parameter to be somehow calibrated.

As a result of this freedom, neither the effective temperature nor the radius of stars with a thick outer convective envelope can be firmly predicted by the current generation of 1D stellar models since they strongly depend on the calibrated value of α_ml. This obviously influences the stellar characteristics recovered by fit techniques that exploit these observables.

The classical target for mixing-length calibration is the Sun, but the generalization of this calibration to different evolutionary phases, metallicity, and mass ranges has been questioned both on theoretical and observational grounds. In particular, a growing amount of observations suggests that the adoption of the solar calibrated α_ml does not allow proper modeling of all kinds of stars (see e.g., Guenther and Demarque, 2000; Yıldız et al., 2006; Yıldız, 2007; Clausen et al., 2009; Deheuvels and Michel, 2011; Bonaca et al., 2012; Mathur et al., 2012; Wu et al., 2015; Joyce and Chaboyer, 2018a,b; Li et al., 2018).

Several of the above mentioned studies consider stars in double-lined detached eclipsing binary systems. In fact, in this case it is possible to obtain an accurate measurement of both masses and radii of the two stars. The availability of these two fundamental quantities allows for stringent tests of the stellar models (see e.g., Claret, 2007; Stancliffe et al., 2015; Claret and Torres, 2016, 2017; Gallenne et al., 2016; Valle et al., 2017) and thus also on the possible variations of the mixing-length parameter in different mass and metallicity ranges.

A recent theoretical analysis (Valle et al., 2019) pointed out many limitations for a mixing-length calibration from field stars, so it is interesting to investigate the question of the reliability of this calibration in binary systems. The aim of this work is to perform such an investigation for a system composed of two low mass stars, thus avoiding the supplementary complication of dealing with the concurrent calibration of the convective core overshooting parameter.

The investigation is focused on a synthetic binary system composed by a primary of M₁ = 0.95 M_⊙ and a secondary of M₂ = 0.85 M_⊙, resulting in a mass ratio of about 1.1. This configuration, quite common for real systems, was chosen also to allow the stars to be sampled in different evolutionary phases, because this provides the most stringent constraints for the recovery (Claret and Torres, 2016; Valle et al., 2017). The focus of our analysis is in quantifying the errors in the recovered mixing-length values arising only from observational errors. Therefore, no systematic discrepancies between the grid of models adopted in the recovery and the artificial stars are assumed. Although this is a far too optimistic assumption when dealing with real-world binary systems, it is nevertheless a mandatory step that allows one to assess the very minimum amount of errors that can be expected in the calibration process. Moreover, only a theoretical investigation can highlight the presence of hidden biases and dependencies (see e.g., Valle et al., 2019, for an analysis of spurious metallicity dependencies of the recovered mixing-length on field stars) that would otherwise be neglected.

The analysis was performed at three different metallicity values, centered around the solar value (Z = 0.0074, 0.0129, 0.0221) and at three different evolutionary stages of the primary star. More precisely, we define a relative age r with respect to the main sequence (MS) lifetime, and we selected models at r = 60, 90, and 120% of the central hydrogen exhaustion time. The first two points correspond to a primary at the middle and nearly at the end of the MS, while the third one corresponds to a primary in the red giant branch (RGB) phase. Correspondingly, the secondary star is always in the MS, at about 40, 55, and 75% of its central hydrogen exhaustion time. The synthetic stars' initial helium abundances were obtained by the linear relation Y=Yp+ΔYΔZZ, with the primordial abundance Y_p = 0.2485 from WMAP (Peimbert et al., 2007a,b) and with a helium-to-metal enrichment ratio of ΔY/ΔZ = 2.0 (see e.g., Gennaro et al., 2010).

We considered different possible values of α_ml for the two stars. The reference scenario has the two stars with a common mixing-length parameter α_{ml, 1} = α_{ml, 2} = 2.0. The other five cases adopted a systematically increasing difference between the two mixing-length values at a step of 0.2; while the mixing length of the primary star was increased at a step of 0.1 (α_{ml, 1} = 2.1, 2.2, 2.3, 2.4, and 2.5), the values of α_{ml, 2} were correspondingly decreased by 0.1 (α_{ml, 2} = 1.9, 1.8, 1.7, 1.6, 1.5). Thus, the extreme scenario has a difference in mixing-length value α_{ml, 1} − α_{ml, 2} = 1.0. The considered simulation parameters are summarized in Table 1.

For each possible mixing-length combination, metallicity and evolutionary stage artificial stars were sampled from the grid described in section 2.2, and their observables were perturbed by means of a Monte Carlo procedure assuming Gaussian errors. The procedure was repeated N = 5, 000 times for each artificial system. The adopted observable constraints were the effective temperature T_eff, metallicity [Fe/H], mass M, and radius R of both stars. We assumed uncertainties of 100 K in T_eff, 0.1 dex in [Fe/H], 0.5% in M and 0.25% in R. Similar precisions on the mass and radii are achievable for a small subset of binary systems, but they are mandatory for calibration purposes (see e.g., Valle et al., 2017).

The analysis of the fit results for the considered scenarios revealed some expected behaviors and also some peculiar effects. The following subsections explore in detail the outcomes in the three considered fitting configurations.

We performed a theoretical investigation on the biases and random uncertainties affecting the calibration of the mixing-length value from a mock eclipsing double-lined binary system, composed of a primary artificial star of mass M₁ = 0.95 M_⊙ and a secondary of mass M₂ = 0.85 M_⊙. We used the SCEPtER-binary pipeline (Valle et al., 2015a) to estimate the mixing-length of the mock stars, adopting as observational constraint the effective temperature, the metallicity [Fe/H], the radius, and the mass of the two stars. The comparison between the true and the estimated mixing-length values allows to evaluate the calibration reliability.

More in detail, several Monte Carlo simulations were conducted considering nine different scenarios, consisting of three metallicities, coupled to three different evolutionary stages of the primary (0.6, 0.9 and 1.2 of the central hydrogen exhaustion time). For each configuration, data were sampled assuming an increasing difference between the mixing-length of the two stars, from perfect agreement at α_{ml, 1} = α_{ml, 2} = 2.0 to a maximum difference of 1.0 (α_{ml, 1} = 2.5, α_{ml, 2} = 1.5).

The mixing-length values were then estimated adopting a different hypothesis in the recovery procedure. In the first case we assumed full independence between the two stars and reconstructed them without imposing any constraint in age and chemical composition between the stars. Great difficulty to estimate the mixing-length value for the two stars resulted under these hypotheses. The standard deviation of the recovered values was about 0.40, confirming the difficulties pointed out for field stars in Valle et al. (2019). Thus, even from stars with exceptionally well constrained masses and radii, the calibration of the mixing-length parameter seems unreliable.

In the second case, we imposed the constraint of common age and initial chemical composition for the two stars in the recovery. While the fitted α_{ml, 1} values closely match those recovered under full independence, the α_{ml, 2} values are much more peaked around unbiased estimates of the values adopted in the sampling. The standard deviations of the recovered α_{ml, 2} values range from one half to one third of those of α_{ml, 1}. This occurs because the primary star provides a much tighter age constraint in the joint fit than the secondary. This leads to the rejection of several extreme solutions for the secondary, while the solutions of the primary are unaffected. In this scenario, we also explored the difference in the mixing-length values of the two stars due to random fluctuations owing to observational errors. We considered stars sampled at common α_ml = 2.0 and focused the analysis on the distribution of the differences α_{ml, 1} − α_{ml, 2}. We found that the posterior distribution of these differences 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 α_ml are expected to show differences higher than that caused only by random errors. These results should be carefully considered when obtaining a fit from a real binary system, because a difference lower than this has a great chance to be only a random fluctuation.

In the third case we also imposed the constraint of common mixing-length value for the two stars, beside those on chemical composition and age. Two interesting effects were detected. First, the estimated common mixing-length is mainly dictated by the value of α_{ml, 2}. This happens because, as discussed above, the posterior distribution of the mixing-length of the secondary star under partial independence is much more peaked than that of the primary, thus dominating in the joint estimate process. Second, an increasing share of systems cannot be fitted by the algorithm as the differences between the true α_ml increase. For Δα_ml > 0.4 less than half of the systems can be recovered; at Δα_ml = 1.0 the values decreases at 20%. Therefore, it seems that moderate differences in the mixing-length value between the two stars are difficult to detect and in these cases the solution is biased toward α_{ml, 2}.

While most of the results presented in this paper can be considered general, such as the effect of the shrink of the estimated mixing-length value around the value of the secondary star, nonetheless this work deals only with a specific binary system with fixed masses. Therefore, the quantitative results cannot be expected to be valid without modifications for other binary systems with different masses and in different evolutionary phases. Although checking the robustness of the results presented here for different ranges of mass or by adopting a different mass ratio would be highly desirable, this possibility is actually limited by the very large computational burden required to compute the stellar models for the recovery at the required level of accuracy. For these reasons, our study should be considered as a first step in this exploration, adopting a quite common value of masses, and mass ratio. More theoretical investigations are required to fully address this open topic.

GV and MD computed the stellar models grid and performed the recovery. GV, MD, PP, and SD wrote the manuscript.