A mapping method of age estimation for binary stars: Application to the $α$ Centauri system A and B

Given the wealth of data provided by Gaia and the upcoming PLATO mission, it is essential to improve stellar models to obtain accurate stellar ages. Our objective is to apply a mapping technique to estimate the age of a system and the initial chemical composition. We also evaluate the influence of observational uncertainties in mass and heavy-element mixtures on results. We applied an inverse calibration method to the evolution of a multiple stellar system, assuming that the stars share the same age and initial chemical composition. This approach determines age, the initial mass fractions of helium ($Y_{ini}$) and heavy elements ($Z_{ini}$), as well as the convective mixing-length parameters ($α_A $ and $α_B$). It uses the observed luminosities ($L_A$ and $L_B$), radii ($R_A$ and $R_B$), and surface chemical compositions ($Z/X_A$ and $Z/X_B$). We used the most recent observational data for $M$, $R$, $L$, and $[Fe/H]$ of $α$ Centauri A and B as input data for our method. We compared two assumptions for the $Z/X$ ratio, following the results for the solar composition. For an assumed high solar $Z/X_\odot =0.0245$, we obtain an age of $7.8 \pm 0.6$ Ga, $Y_{ini} = 0.284 \pm 0.004$, and $Z_{ini} = 0.0335 \pm 0.0015$. For a low solar $Z/X_\odot = 0.0181$, the derived age is $8.7 \pm 0.6$ Ga, $Y_{ini} = 0.267 \pm 0.008$, and $Z_{ini} = 0.025 \pm 0.002$. Observational errors in the stellar masses of $\pm$0.002 lead to an age error of 0.6 Ga. Overshooting of $0.05-0.20H_p$ at the boundary of the convective core increases the age by $0.6-2.1$ Ga. Models with higher $Z/X$ and radiative cores, with ages of $7.2-7.8$ Ga, appear preferable and show better agreement with the observed asteroseismic frequencies.

💡 Research Summary

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This paper presents a novel inverse‑mapping technique for simultaneously determining the age and initial chemical composition of binary (or higher‑order multiple) star systems. The method exploits the fact that the two components of a binary are assumed to have formed at the same time from the same protostellar cloud, and therefore share a common age, initial helium mass fraction (Y_ini) and initial heavy‑element mass fraction (Z_ini). The only parameters allowed to differ between the two stars are their convective mixing‑length parameters (α_A and α_B).

The authors formulate stellar evolution as a three‑dimensional mapping
E(M): {α, Y_ini, Z_ini} → {R, L, Z/X} at a given age t_s, where R is radius, L is luminosity and Z/X is the surface heavy‑element‑to‑hydrogen ratio. By computing the Jacobian matrix J_E = ∂(R, L, Z/X)/∂(α, Y_ini, Z_ini) for a grid of ages, they obtain its inverse J_E⁻¹. The inverse Jacobian provides a linear correction ΔP_ini = J_E⁻¹·ΔT that updates the initial parameters in response to the difference ΔT between observed quantities and the current model prediction. An iterative scheme rapidly converges (≈12 iterations for the primary, ≈5 for the secondary) to a set of initial parameters that reproduces the observed R, L and Z/X for both stars at a common age.

The method is implemented with the 1‑D stellar evolution code CESAM2k, which includes mixing‑length convection, microscopic diffusion (Burgers equations), OPAL EOS and opacities, NACRE nuclear rates, and a simple radiative atmosphere. The authors adopt two solar heavy‑element mixtures to explore the impact of the solar Z/X controversy: a “high‑Z” case (Z/X_⊙ = 0.0245, Grevesse & Noels 1993) and a “low‑Z” case (Z/X_⊙ = 0.0181, Asplund et al. 2009).

The test case is the nearby α Centauri system. Recent interferometric, astrometric and spectroscopic measurements provide the following input values (adopted from Akeson et al. 2021 and Morel 2018): M_A = 1.0788 M⊙, M_B = 0.9092 M⊙, R_A = 1.2234 R⊙, R_B = 1.2175 R⊙, L_A = 1.521 L⊙, L_B = 1.506 L⊙, and surface metallicities