[论文解读] A mapping method of age estimation for binary stars: Application to the $α$ Centauri system A and B

F. Thévenin, V. A. Baturin+5 more|arXiv (Cornell University)|Feb 9, 2026

Stellar, planetary, and galactic studies被引用 0

一句话总结

作者开发了一种基于逆映射（基于雅可比矩阵）的方法，用以推断双星系统的共同年龄与初始化学组成，并将其应用于α Centauri A 与 B，对观测不确定性和太阳Z/X混合比的敏感性进行评估。

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.

研究动机与目标

• Motivate accurate stellar age dating for binary systems to reduce parameter degeneracy and improve Galactic population studies.
• Develop an inverse calibration framework that yields age, Y_ini, Z_ini, and mixing-length parameters for binary components.
• Assume both stars share the same age and initial composition, but allow different convective parameters.
• Assess how observational uncertainties in mass, radius, luminosity, and Z/X influence the deduced ages and compositions.
• Evaluate the effect of adopting different solar heavy-element mixtures on the inferred solution.

提出的方法

• Treat stellar evolution as a mapping E(M_j): {alpha_j, Y_ini, Z_ini} -> {R, L, Z/X}_j at age t_s.
• Use inverse mapping E^{-1} to estimate {alpha_j, Y_ini, Z_ini} from observed {R, L, Z/X}_j, allowing alpha_A ≠ alpha_B.
• Compute the Jacobian J_E = d{R,L,Z/X}/d{alpha,Y_ini,Z_ini} and iteratively update initial parameters via P_ini^{i+1} = P_ini^{i} + J_E^{-1} * Delta T.
• Require the Jacobian to be non-singular (det(J_E) ≠ 0) for a unique inverse solution.
• Apply sequential iterations (typical ~12 for star A, ~5 for star B) to converge to consistent age and composition.
• Test observational sensitivity by perturbing inputs and propagating through the inverse Jacobian to quantify impact on derived parameters.

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 obse

实验结果

研究问题

• RQ1 Can a single age and shared initial composition be recovered for α Centauri A and B using observed R, L, and Z/X?
• RQ2 How do uncertainties in mass, radius, luminosity, and Z/X affect the inferred age and initial abundances?
• RQ3 How does the assumed solar Z/X mixture (high-Z vs low-Z) influence the derived age and initial composition?
• RQ4 Is there a common set of initial Y_ini and Z_ini that allows both stars to match their observed properties at the same age within uncertainties?

主要发现

• The method yields ages around 7.8–7.83 Ga for a high-Z solar mixture and ~8.7 Ga for a low-Z solar mixture, with corresponding Y_ini and Z_ini values.
• Mass uncertainties dominate the propagated age uncertainty, with ∆Age ≈ 0.62 Ga for ±0.002 M_sun changes in each star.
• Overshoot in the convective core can increase the inferred age by ~0.6–2.1 Ga depending on overshoot depth.
• Models with higher Z/X and radiative cores tend to better align with asteroseismic frequencies, suggesting a preferred region in parameter space.
• Using the low-Z solar mixture and the given data yields an overall age around 8.7–8.8 Ga with modest changes in α_A, α_B and initial compositions.
• The study demonstrates that a physically consistent, common-age solution exists within a restricted age window (7–9 Ga) depending on input assumptions and data.

Figure 2: Stellar ages as a function of $Y_{\mathrm{ini}}$ for $Z_{\mathrm{ini}}=0.032$ (solid lines) and 0.035 (dashed lines).