Stellar masses and ages in Gaia Data Release 4 from the Final Luminosity Age Mass Estimator algorithm

Technical Summary: Stellar Masses and Ages in Gaia Data Release 4 from the Final Luminosity Age Mass Estimator (FLAME) Algorithm

Problem and Context
The determination of stellar masses and ages is fundamental to understanding the formation and evolution of astrophysical systems, including the history of the Milky Way and the evolutionary context of exoplanetary systems. While the Gaia mission provides high-precision astrometry, photometry, and spectroscopy, deriving intrinsic stellar parameters requires complex modeling. The Gaia Data Processing and Analysis Consortium (DPAC) addresses this through the Apsis pipeline, specifically the Final Luminosity Age Mass Estimator (FLAME) module. This paper details the FLAME methodology, its validation, and its expected performance for Gaia Data Release 4 (GDR4), aiming to provide a large, homogeneous sample of stellar parameters with quantified uncertainties.

Methodology
The FLAME algorithm is structured into two distinct components to separate observational derivation from model inference:

1. Algo1 (Analytical Component): This module derives stellar luminosity ($L$), radius ($R$), and the radial velocity correction due to gravitational redshift ($V_{GR}$). It utilizes Gaia's atmospheric parameters ($T_{\text{eff}}$, $\log g$, $[Fe/H]$, $[\alpha/Fe]$), astrometry (parallax), and photometry ($G$-band magnitude).

   • Luminosity is calculated using the bolometric magnitude equation, incorporating a bolometric correction ($BC$) derived from stellar atmosphere models (MARCS for $2500 < T_{\text{eff}} < 8000$ K; Line-by-Line models for $T_{\text{eff}} > 8000$ K).
   • Radius is derived via the Stefan-Boltzmann relation.
   • $V_{GR}$ is calculated from $\log g$ and $R$.
   • Uncertainties are propagated using a bootstrap method or Markov Chain Monte Carlo (MCMC) samples from the preceding GSP-Phot module.

2. Algo2 (Model Inference Component): This module infers stellar mass ($M$), age ($\tau$), and evolutionary stage ($\epsilon$) by comparing observed properties with theoretical stellar models.

   • Input: The derived $L$, along with $T_{\text{eff}}$, $\log g$, and metallicity ($[M/H]$).
   • Models: The algorithm utilizes the BaSTI (Bag of Stellar Tracks and Isochrones) grid, covering masses from $0.1$ to $15 M_{\odot}$ and metallicities from $-3.197$ to $+0.45$. The grid includes physics such as atomic diffusion and convective-core overshooting.
   • Inference Methods: Two approaches are implemented: a Bayesian framework using the SPInS code (employing MCMC sampling with priors on the Initial Mass Function and Star Formation Rate) and a classical Levenberg-Marquardt minimization approach. The GDR4 results primarily rely on the Bayesian SPInS implementation.
   • Metallicity Correction: To ensure consistency between the MARCS-based observational inputs and the BaSTI models (which use a different solar mixture), a correction is applied to $[M/H]$ based on $\alpha$-element enrichment.

Data Sources
FLAME processes two distinct sets of Gaia data:

• GSP-Phot: Low-resolution XP spectra ($300–1000$ nm) available for the vast majority of sources ($\sim500$ million in GDR4).
• GSP-Spec: High-resolution RVS spectra ($840–870$ nm) for a subset of brighter sources ($\sim34$ million in GDR4).
  The pipeline processes sources individually, without prior knowledge of binarity or cluster membership, which introduces specific challenges for certain parameter spaces (e.g., very low-mass stars or binaries).

Validation and Results
The paper validates FLAME through simulated data, solar parameters, and comparisons with external catalogues:

• Simulated Data: Tests on 372 simulated stars across the HR diagram show that FLAME retrieves input masses and ages with no significant offsets or trends. Mass uncertainties scale with input errors, and age recovery is satisfactory despite the non-linear nature of the age parameter.
• Solar Parameters: Tests on the Sun demonstrate that high-precision inputs yield accurate mass and age. Systematic errors, particularly in $\log g$, have the most significant impact on derived parameters, potentially biasing results if not accounted for.
• Gaia Benchmark Stars (GBS): Comparisons with fundamental parameters of 140 GBS stars show high agreement in radii and masses (median offset $\sim -0.08\sigma$).
• Asteroseismic Data: Comparisons with APOKASC samples (Kepler/APOGEE) reveal that when FLAME uses the same input constraints as the seismic analysis, mass and age agreements are excellent (median offsets $< 1\%$). However, when using independent Gaia atmospheric parameters, discrepancies arise for giant stars due to the degeneracy of evolutionary tracks in the HR diagram.
• External Catalogues (StarHorse): Comparisons with the StarHorse catalogue show good agreement for main-sequence and sub-giant stars (mass offset $\sim 3\%$), but again, giant star results diverge due to input data differences.

Key Contributions and Performance Estimates

• GDR4 Scale: FLAME is expected to provide results for approximately 500 million sources based on GSP-Phot and 34 million based on GSP-Spec.
• Uncertainty Inflation: To address the issue of overly optimistic formal uncertainties in atmospheric parameters (which can bias model inference), GDR4 introduces an inflation of uncertainties (e.g., adding $60$ K to $T_{\text{eff}}$ and $0.1$ dex to $\log g$ for GSP-Spec) before inference.
• Typical Uncertainties:
  • Luminosity: Relative uncertainties range from $\sim1\%$ for bright ($G=11$) solar-type stars to $\sim35\%$ for fainter ($G=17$) stars.
  • Mass: Typical relative uncertainties are between $2\%$ and $6\%$ for main-sequence stars, aligning with analytical mass-luminosity predictions but improved by model constraints.
  • Age: Relative uncertainties vary significantly by mass and evolutionary stage. For solar-like stars, uncertainties range from $20\%$ to $40\%$. For giants, uncertainties are typically $10\%$ for lower masses, increasing to $15–20\%$ for higher masses. Low-mass main-sequence stars ($<0.8 M_{\odot}$) exhibit large relative uncertainties ($\sim70\%$) due to slow surface property evolution.

Significance and Limitations
The paper concludes that FLAME produces valid and statistically consistent results for Gaia Data Releases, providing a massive dataset for galactic archaeology and exoplanet characterization. The authors emphasize that the quality of the output is strictly dependent on the quality of the input atmospheric parameters.

A critical limitation highlighted is the difficulty in validating masses and ages for giant stars. Because evolved models occupy a narrow region in the HR diagram, small variations in input data lead to large variations in derived parameters. Consequently, agreement with external catalogues for giants is only achieved when the exact same input data is used. The paper advises users to consult Gaia documentation and flags, particularly for samples with decreased data quality or for specific evolutionary stages where parameter degeneracies are high.

Applications Presented
The authors apply FLAME to derive new parameters for:

1. High-velocity, metal-poor stars (Katz et al. 2025), finding ages peaking around $11$ Ga.
2. Solar-like stars with very low-mass companions (Maxted 2025), constraining companion properties via primary star characterization.
3. A subset of the PLATO Input Catalog (PIC), providing homogeneous parameters for future mission validation.