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

TL;DR

This study addresses the challenge of precisely dating binary stars by introducing an inverse-mapping framework that infers a system's common age $t_s$ and initial composition $(Y_{\mathrm{ini}},Z_{\mathrm{ini}})$ from observables $(R,L,Z/X)$ while allowing component-specific mixing-length parameters. The inverse Jacobian method solves for $(\alpha,Y_{\mathrm{ini}},Z_{\mathrm{ini}})$ via iterative updates, enforcing $t_A=t_B$ and identical initial composition for the pair. Applied to the α Centauri A/B system with CESAM2k, the authors show that two solar mixtures yield ages of $7.8\pm0.6$ Ga (high-$Z$) and $8.7\pm0.6$ Ga (low-$Z$), with $Y_{\mathrm{ini}}$ and $Z_{\mathrm{ini}}$ around the 0.27–0.28 and 0.032–0.035 ranges, respectively; mass uncertainties and convective-core overshoot can shift ages by up to $0.6$–$2.1$ Ga. Models with higher $Z/X$ and radiative cores provide better asteroseismic agreement, suggesting a radiative-core, high-$Z$ solution is preferred under current data. The work lays a framework for robust age dating of solar-type binaries in Gaia/PLATO-era surveys, with future incorporation of seismic frequencies expected to further reduce ambiguities.

Abstract

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.

Figures (7)

• Figure 1: Inverse mapping results. Initial conditions for stars A (circles) and B (stars) reproduce the observed parameters $(R,L,Z/X)_{\mathrm{obs}}$ at given ages. Colors indicate the ages, as shown in the legend. Colored lines show the initial chemical compositions obtained by accounting for observation errors $\pm\Delta(Z/X)$. The black line shows the common initial conditions $(Y_{\mathrm{ini}},Z_{\mathrm{ini}})$ for which the stars achieve the observed values $(R,L,Z/X\pm\Delta(Z/X))_{\mathrm{obs}}$ at ages = 7.801--7.826 Ga.
• Figure 2: Stellar ages as a function of $Y_{\mathrm{ini}}$ for $Z_{\mathrm{ini}}=0.032$ (solid lines) and 0.035 (dashed lines).
• Figure 3: Examples of evolutionary tracks of stars A and B for luminosity (upper panel), radius (middle panel), and $Z/X$ (lower panel), all with $Z_{\mathrm{ini}} = 0.0320$. The age of the pair is 7.826 Ga, corresponding to the intersection point in Fig. \ref{['fig:age_yini']}.
• Figure 4: Squared sound speed profile (left) and hydrogen mass fraction (right) for models of $\alpha$ Centauri A. Models with a radiative core are shown by solid curves, and those with convective core are shown by dashed curves.
• Figure 5: Ratio $r_{02}(n)$ of small and large separations as function of radial order $n$ for different models of $\alpha$ Centauri A (circles). Colors indicate the stellar age and diamonds show observational data.