Asteroseismic Modelling Techniques Overview

• Asteroseismic modelling techniques are a set of methods that extract fundamental stellar parameters from oscillation data using statistical, computational, and observational tools.
• They employ seismic observable extraction, forward grid-based modelling, and Bayesian frameworks to infer quantities like mass, radius, and age with robust uncertainty quantification.
• Recent advances include automated pipelines, seismic inversions, and correlated noise modelling to mitigate systematic errors and enhance the accuracy of stellar interior characterizations.

Asteroseismic modelling techniques comprise a collection of statistical, computational, and observational methodologies enabling the inference of fundamental stellar parameters (mass, radius, age, chemical composition, core structure, mixing properties, and more) from stellar oscillation data. The field has progressed rapidly with the advent of high-precision space photometry (e.g., Kepler, CoRoT, TESS, PLATO), shifting from classical mode-by-mode identification to ensemble statistical and probabilistic modelling frameworks. This article presents a technical overview of the principal methodologies, advances, and critical considerations in asteroseismic modelling.

1. Extraction and Characterization of Seismic Observables

Asteroseismic analysis begins with the detection and precise measurement of oscillation mode frequencies from time-series data. For stars such as δ Scuti, Kepler or CoRoT photometry yields dense oscillation spectra containing hundreds to thousands of observed modes. The amplitude distribution and the challenges of unknown degree ℓ in these stars make classical mode identification intractable. Instead, the entire observed frequency spectrum $f(\nu) = \sum_{k} \delta(\nu - \nu_k)$ is often analyzed statistically.

Key global seismic observables include:

• Large frequency separation: $\Delta\nu = \nu_{n+1, \ell} - \nu_{n, \ell}$, correlating closely with the stellar mean density via the scaling $\Delta\nu \propto \sqrt{M/R^3}$.
• Frequency of maximum power: $\nu_{\max}$, often used with scaling relations: $$\nu_{\max}/\nu_{\max,\odot} \simeq (M/M_{\odot})/[(R/R_{\odot})^2 (T_{\rm eff}/T_{\rm eff,\odot})^{1/2}]$$.

In evolved red giants and subgiants, echelle diagrams tailored for mixed modes are constructed, and Lorentzian profile fitting in the power density spectrum (peakbagging) is employed to extract individual frequencies, especially dipole and quadrupole mixed modes characterized by strong coupling between pressure and gravity wave cavities (Larsen et al., 29 Mar 2025).

High-order gravity mode (g-mode) pulsators (e.g., γ Doradus and Slowly Pulsating B stars) are modeled by extracting period spacings and constructing frequency ratios that probe the deep interior. Mode identification in these stars leverages frequency ratios and the Brunt–Väisälä integral, facilitating statistical identification schemes in stars with limited angular resolution (Grigahcene et al., 2010).

2. Forward Modelling: Grids, Statistical Matching, and Optimization

The core of asteroseismic inference is forward modelling: stellar structure and evolution models are computed over a multidimensional grid spanning relevant physical parameters. For each model, theoretical oscillation spectra are computed (using, e.g., ADIPLS, GYRE, GraCo, or Guenther's code), producing predictions for the observed quantities.

Classical, grid-based methodologies (Gai et al., 2010, Tang et al., 2010, Nsamba et al., 2016, Bowman et al., 2 Oct 2024):

• A precomputed grid (e.g., with MESA, YREC, CESAM) is compared to observed frequencies or global observables (e.g., $\Delta\nu$, $\nu_{\max}$).
• The fit is performed using a likelihood function, often in the form $$\mathcal{D} = \prod_{i=1}^n [1/(\sqrt{2\pi} \sigma_i)] \exp(-\chi^2/2)$$, with $$\chi^2 = \sum_{i=1}^n [(q_i^{\text{obs}} - q_i^{\text{model}})/\sigma_i]^2$$.
• Uncertainties are quantified via Monte Carlo sampling, bootstrapping, or Markov Chain Monte Carlo (MCMC) (1711.01896), with both fully Bayesian and frequentist approaches in use.

Model selection and improvement:

• Observational constraints (e.g., effective temperature, luminosity, metallicity) reduce the parameter space by discarding inconsistent models.
• Nonadiabatic stability calculations (e.g., with GraCo) evaluate mode excitation, retaining only models with physically driven pulsational modes.
• Recent techniques include iterative coupling of global and local minimization (FICO procedure (Buldgen et al., 25 Aug 2025)), and local Levenberg–Marquardt refinement after global grid convergence.

3. Statistical and Probabilistic Advances: Bayesian and Correlated Noise Modelling

The increasing quality of observed data necessitates robust statistical frameworks:

• Bayesian grid fitting (Gruberbauer et al., 2012, 1711.01896): Posterior distributions for stellar parameters are obtained by marginalizing over all sources of uncertainty, including:
  • Uncertain mode identification: summing likelihoods over possible frequency assignments.
  • Discrete grid resolution: integrating over interpolations between grid points (using, e.g., Delaunay triangulation and barycentric interpolation).
  • Surface effects: systematic offsets are introduced as free (nuisance) parameters per frequency, assigned prior distributions and marginalized analytically or numerically.
• Correlated noise models (Li et al., 2023): Recognizing that systematic error in computed frequencies is highly frequency-correlated, Gaussian Process (GP) kernels are used to construct a covariance matrix

$$k(\nu, \nu') = \sigma\, \exp\left[-\frac{(\nu - \nu')^2}{2l^2}\right],$$

allowing the likelihood to properly account for smooth, structured systematics. This approach yields more stable parameter posteriors and mitigates spurious, overly precise solutions arising from grid undersampling.

4. Seismic Inversions, Ratios, and Model-Independent Constraints

Several quasi–model-independent techniques supplement forward grid modelling:

• Seismic inversion (Bétrisey et al., 2023, Bétrisey et al., 2023):
  • Mean density inversion computes $⟨\rho⟩$ directly from the observed mode set using inversion coefficients $c_i$, minimizing a cost function for the reproduction of a target kernel (e.g., $T_{\bar{\rho}}(x) = 4\pi x^2\, (\rho/\rho_R)$).
  • Acoustic radius and central entropy inversions target additional internal structure indicators. By-products from the inversion (averaging kernels, coefficient distributions) are used to automate quality assessment (K-flag: kernel fit, R-flag: randomness).
  • For pipeline application in large surveys (PLATO), robust flagging and thresholding ensure reliable statistical error control and automation (Bétrisey et al., 2023).
• Frequency ratio fitting and glitch analysis (Verma et al., 2022, Bétrisey et al., 2023):
  • Frequency ratios (e.g., $r_{02}$, $r_{01}$), formed from combinations of individual mode frequencies, are nearly insensitive to the outermost layers and hence robust against surface-effect modeling errors. He glitch signatures provide envelope helium content independent of near-surface physics.
  • Simultaneous fitting of ratios and glitch parameters, with full covariance accounted, enables unique determination of mass and initial helium abundance, breaking traditional anti-correlations and degeneracies in stellar model grids.

5. Physical Effects, Systematics, and Precision

Asteroseismic modelling must contend with various sources of theoretical and observational systematics:

• Surface correction methodologies: Polynomial power-law, mode inertia–dependent (Ball & Gizon), or Lorentzian (Sonoi et al.) approaches regularize frequency residuals. However, direct frequency fitting is extremely sensitive to these corrections and may yield over-optimistic parameter uncertainties or bias (Bétrisey et al., 2023).
• Internal systematics (Nsamba et al., 2018): Uncertainties in diffusion, solar abundance mixture, treatment of mixing, and undershooting are directly propagated by considering models both with and without these physics, quantifying the mean density, radius, mass, and age biases (e.g., systematics in age exceeding 15% when comparing diffusion vs. no diffusion).
• Cluster membership and coeval priors: For open cluster stars, external constraints from membership dramatically reduce the model space and increase parameter precision (Grigahcene et al., 2010, Gai et al., 2010).

The table below summarizes selected modelling techniques and their main purposes:

Technique	Key Purpose	Robustness to Systematics
Direct frequency fitting	Mass, radius, age from individual modes	Sensitive to surface effects
Frequency ratio + mean density	Model-independent core constraints	Surface-insensitive
Bayesian/MCMC grid search	Full-posterior parameter estimation	Incorporates full error budget
Seismic inversion (e.g., mean ρ)	Direct internal structure measurement	High if input kernel is robust
Correlated noise/GP kernel	Statistical error regularization	Corrects for frequency covariance

6. Incorporation of Additional Physics: Mixing, Rotation, and Chemistry

Advanced models now integrate additional transport and mixing processes, especially as asteroseismic constraints grow more precise:

• Lithium depletion (Buldgen et al., 25 Aug 2025): Constrained through simultaneous fitting of turbulence ($D_{\rm turb} = D_T \times (\rho_{BCZ}/\rho)^n$) and convective penetration ($d_{\rm under} = \alpha_{\rm under} H_P$). F-type stars require large penetration depths to match Li depletion but may then be incompatible with observed low-degree frequency separation ratios, indicating missing or poorly modeled physics in mixing and convection implementation.
• Rotation: Treated differentially via rotational splitting in oscillation spectra, with gravity-mode modeling pipelines such as FOAM (Michielsen, 10 Jun 2024) leveraging the traditional approximation and Mahalanobis distance merit functions to robustly sample over rotation rates and their impact on g-mode frequency patterns.
• Helium and heavy element diffusion, core overshoot, and mixing-length parameter calibration are now routinely varied within statistical frameworks and their effect on the final parameter distributions is explicitly assessed (Li et al., 2018, Nsamba et al., 2018).

7. Large-Scale Pipelines, Automation, and the Era of Ensemble Seismology

The proliferation of high-cadence, long-duration time series from missions such as Kepler, TESS, and the anticipated PLATO mission necessitates highly automated, efficient, and statistically rigorous pipelines:

• Open-source tools and packages: AIMS (Asteroseismic Inference on a Massive Scale) leverages Bayesian MCMC sampling on top of precomputed model grids with advanced interpolation (age and parameter), modular design, and user-defined merit functions (1711.01896).
• Automation of inversion quality assessment (Bétrisey et al., 2023): Pipeline-ready methods for kernel fitting and inversion coefficient randomness are benchmarked and shown to perform at the level of expert manual assessment, enabling application to thousands of targets.
• Statistical frameworks: Nested grid methodologies, model selection via information criteria (AIC), and explicit penalization of added free parameters are implemented in packages like FOAM (Michielsen, 10 Jun 2024).

Application to unique populations (e.g., very metal-poor red giants (Larsen et al., 29 Mar 2025), binary systems (Nsamba et al., 2016)) or ensemble settings (open clusters (Grigahcene et al., 2010, Gai et al., 2010)) leverages these frameworks to address inherent parameter degeneracies, systematic uncertainties, and the challenges of limited or complex frequency spectra.

---

Asteroseismic modelling techniques represent an overview of high-precision observational data, sophisticated grid-based and inversion methodologies, statistical/machine-learning frameworks, and continual advances in stellar physics inputs. The ongoing development toward robust, automated, and physically complete models underpins the current and future capability to reliably extract fundamental interior properties for large, diverse stellar samples across the Hertzsprung–Russell diagram.