Revised Best Estimates for the Age and Mass of the Methuselah Star HD 140283 Using MESA and Interferometry and Implications for 1D Convection (2024)

We use the Modules for Experiments in Stellar Astrophysics (MESA v11701; Paxton et al. 2011, 2013, 2015, 2018, 2019) software and an adaptive grid searching method (Joyce & Chaboyer 2018a, 2018b; Murphy et al. 2021) to construct a set of metal-poor stellar evolutionary tracks for HD 140283. We vary as inputs the mass, initial composition, and convective mixing length, α_MLT, which parameterizes the efficiency of energy transport by convection in the low-mass star's outer envelope according to the Mixing Length Theory (MLT) formalism first established by Böhm-Vitense (1960).

Our search covers masses from 0.6 to 2.0 M_⊙, α_MLT values 1.5–2.0× the pressure scale height, , and input metallicities, (Z_in), between 0.00006 and 0.002 ([Fe/H] = −2.46 to −0.94).

Since previously reported ages for HD 140283 approach the age of the universe (e.g., VandenBerg 2000b), we uniformly adopt the helium abundance of the early universe, Y_in = Y_primordial = 0.245.

Models are considered "valid" according to agreement with the effective temperature and luminosity constraints quoted in Karovicova et al. (2020) and shown in Figure 1. A pseudo-χ² ranking scheme assesses relative goodness-of-fit among all classically valid tracks. The pseudo-χ² cost function, equation (1), assigns equal statistical weighting to the modeled radius (R), luminosity (L), and surface metal-to-hydrogen mass fraction Z/X_surf—a metric of the form introduced in Joyce & Chaboyer (2018b). To wit:

where R_obs is the interferometric radius of HD 140283 with uncertainty σ_R,obs, R_mod is the modeled radius, and similarly for L and Z/X.

The model that minimizes Equation (1) has a mass of 0.79 M_⊙, Z_i of 0.0002, and α_MLT = 1.6 H_p . However, as many models yield χ² ≤ 1.0, the optimal parameters are more accurately determined by measuring the frequencies of values among all acceptable models. Because the mass and age estimates do not follow a normal distribution, the confidence interval cannot be calculated directly from the standard deviations. For large data sets, the sampling distribution of the studentized mean is approximately normal (Cook & Weisberg 1975). Consequently, is used to construct the studentized error, representing a confidence interval of 95%.

We apply frequency statistics to parameters from all points of observational intersection: 198 tracks of 1659 total intersect at one or more timesteps (∼12%). This yields a 1σ mass and age for HD 140283 of 0.809 ± 0.001 M_⊙ and 12.01 ± 0.05 Gyr, respectively.

As this analysis uses invariant physical assumptions, the uncertainties do not include contributions from modeling systematics. To provide more realistic estimates of global uncertainty, we replicate the numerical experiment using two different prescriptions for atmospheric surface boundary conditions:

• 1.

  pre-computed photosphere tables based on Hauschildt et al. (1999a, 1999b)s PHOENIX models and Kurucz (2003); and

• 2.

  Eddington t − τ integration (Eddington 1930);

likewise, two convective overshoot prescriptions: MESA's default "step" method and the "exponential" overshooting scheme described by Herwig (2000). These physics, along with heavy element diffusion, were identified as large sources of instrumental variance in the Dartmouth Stellar Evolution Program (DSEP) (Joyce & Chaboyer 2018a), MESA (Joyce et al. 2021, in preparation), and in other stellar evolution codes (Lebreton et al. 2014; Tanner et al. 2014). Presently, we relegate diffusion considerations to future work but strongly emphasize their importance. Results from tests across physical prescriptions revise our mass and age for HD 140283 to 0.81 ± 0.05 M_⊙ and 12 ± 0.5 Gyr, respectively.

Recently, Creevey et al. (2015) reported a best fitting mass and age of 0.780 ± 0.010 M_⊙ and . Joyce & Chaboyer (2018a) found a best-fitting mass range of 0.74–0.79 M_⊙ and a mass-dependent age range of 12.5–14.9 Gyr, with higher masses corresponding to lower ages. Karovicova et al. (2020) reported a best fitting mass of 0.77 ± 0.03 M_⊙. Our median mass and age are slightly higher and lower, respectively, than the results cited above but remain firmly consistent with recent literature and not in conflict with the universe's age.

Importantly, our models show sensitivity to the mixing length parameter, preferring values between 1.6 and 1.8H_p . By comparing the models' preferred α_MLT against solar mixing length calibrations calculated under associated physical conditions, we find that HD 140283 requires a convective mixing length 10%–20% below the solar value. This result contributes to an increasing body of literature showing significantly sub-solar α_MLT values are necessary to reproduce both the observed properties of metal-poor stars in general (e.g., Guenther & Demarque 2000; Tayar et al. 2017; Viani et al. 2018) and of HD 140283 specifically (Creevey et al. 2015; Joyce & Chaboyer 2018a).