Chronos: Towards a self-consistent and absolute stellar age scale. I. A Bayesian hierarchical lithium-age model: Validation on the Pleiades cluster

Technical Summary: Chronos: I. A Bayesian hierarchical lithium-age model: Validation on the Pleiades cluster

Problem Statement
Establishing a self-consistent and absolute stellar age scale remains a central challenge in astrophysics, particularly due to the lack of systematic cross-calibration between different dating techniques, which often yield discrepancies of up to a factor of two. While the lithium depletion boundary (LDB) provides a robust, physically grounded chronometer for low-mass stellar populations, its application is currently limited by heterogeneous temperature scales and astrophysical dispersion. Specifically, the transition between fully convective ultra-cool dwarfs (UCDs) and partially radiative FGK stars is often treated with sharp temperature cuts, failing to propagate uncertainties consistently across the regime transition. Furthermore, the FGK population exhibits significant lithium dispersion driven by stellar rotation, where fast rotators display enhanced lithium abundances and increased scatter compared to slow rotators. Standard methods often rely on single-valued prescriptions or empirical fits that do not fully capture this intrinsic diversity or the non-linear physics of lithium depletion.

Methodology
The authors present Chronos, a Bayesian hierarchical model (BHM) designed to infer stellar population ages by jointly modeling effective temperatures ($T_{\text{eff}}$), lithium abundances ($A_{\text{Li}}$), and global cluster age. The framework integrates three core components:

1. Hierarchical Structure: The model treats observed data ($T_{\text{eff,obs}}, A_{\text{Li,obs}}$) as noisy realizations of latent true stellar parameters. It avoids hard boundaries between stellar regimes, instead using a smooth logistic function to model the transition between fully convective and partially radiative stars.
2. Neural Network Surrogate: A multilayer perceptron (MLP) trained on BT-Settl evolutionary models serves as a surrogate for theoretical lithium depletion curves. This network predicts the mean theoretical lithium abundance ($\mu_{A_{\text{Li}}}$) as a function of age and $T_{\text{eff}}$, providing a fast and continuous approximation of the physical models.
3. Rotation-Aware Mixture Model: To account for rotation-induced lithium enhancement in FGK stars, the model employs a two-component mixture distribution for the true lithium abundance. This consists of a "slow rotator" component (modeled as a Normal distribution centered on the theoretical prediction) and a "fast rotator" component (modeled as a Skew-Normal distribution with an amplitude offset $\Delta A_{\text{Li,rot}}$ and additional dispersion). The probability of a star belonging to the fast rotator population is treated as a function of its spectral regime and a global rotation fraction parameter ($p_{\text{rot}}$).

The inference is performed using Markov Chain Monte Carlo (MCMC) with the No-U-Turn Sampler (NUTS) implemented in PyMC. The model was validated using synthetic datasets generated from the same hierarchical assumptions and applied to a homogeneous dataset of 122 Pleiades cluster members.

Key Contributions

• Unified Framework: Chronos recasts lithium-based stellar chronology as a coherent hierarchical inference problem, allowing for the simultaneous estimation of cluster age, intrinsic dispersion, and rotation parameters without imposing arbitrary temperature thresholds.
• Probabilistic Regime Transition: The model explicitly handles the transition between UCDs and FGK stars via a continuous probabilistic function, ensuring consistent uncertainty propagation across the LDB.
• Rotation Modeling: By incorporating a mixture model for rotation effects, the framework accounts for the observed asymmetric lithium dispersion in FGK stars, distinguishing between intrinsic astrophysical variance and rotation-induced enhancement.
• Surrogate Modeling: The use of a neural network trained on evolutionary models allows for the efficient evaluation of lithium depletion physics within the Bayesian inference loop, bridging the gap between complex theoretical grids and statistical modeling.

Results
Applied to the Pleiades cluster, Chronos yields a posterior age distribution centered at Age = 124.53$^{+3.34}_{-2.70}$ Myr. This result is consistent with classical LDB estimates and recent isochrone fitting results but features a tighter uncertainty constraint. The model successfully recovers the input age in synthetic tests across a range of 20–400 Myr, demonstrating robustness even with limited numbers of objects in the LDB-sensitive regime ($T_{\text{eff}} < 3700$ K).

Diagnostic checks, including convergence statistics ($\hat{R} \approx 1$), residual analysis, and posterior predictive checks (PPCs), confirm the model's stability. The analysis reveals that the inferred age is primarily driven by stars near the steep LDB transition ($T_{\text{eff}} \sim 2800$ K) and remains largely uncorrelated with the rotation parameters, indicating that the age determination is robust against the specific parametrization of rotation-induced lithium enhancement. The model also successfully constrains the fraction of fast rotators ($p_{\text{rot}} \approx 0.32$) and the magnitude of lithium enhancement ($\Delta A_{\text{Li,rot}} \approx 1.78$ dex).

Significance
The paper positions Chronos as a methodologically robust step toward a self-consistent and absolute stellar age scale. By demonstrating that lithium-based chronology can be formulated as a hierarchical inference problem, the work provides a flexible and statistically rigorous framework for age determination in young stellar populations (1–600 Myr). The authors emphasize that this approach allows for the self-consistent assessment of the relative constraining power of multiple age diagnostics. While the current validation is limited to the Pleiades and solar-metallicity models, the framework is designed to be extensible to include metallicity as a dimension and to incorporate additional age-sensitive observables in future iterations. The work does not claim to solve all age-dating discrepancies but establishes a physically interpretable, statistically sound foundation for integrating lithium diagnostics into a broader, unified age scale.