Methods for Asteroseismic Modelling of Solar-Like Oscillators
ANYE, Dec. 1 2025
\ell = 0 MDI Doppler velocities
Power spectra of MDI dopplergrams
\begin{aligned} {\Delta\nu_\odot} &\sim 135\ \mathrm{\mu Hz} \\ {\nu_{\text{max},\odot}} &\sim 3090\ \mathrm{\mu Hz} \end{aligned}
(roughly 5-minute oscillations)
p-mode frequencies satisfy \nu_{n\ell} \sim \Delta\nu\left(n + {\ell \over 2} + \epsilon_\ell(\nu)\right) + \mathcal{O}(1/\nu)
Stochastic,
broad-band
excitation
Telescopes can only point at one star at time…
Required photometric stability not achievable from ground \Longrightarrow Space photometry!
\begin{aligned} {M \over M_\odot} &\sim \left(\nu_{\text{max}} \over \nu_{\text{max},\odot}\right)^{3}\left(\Delta\nu \over \Delta\nu_\odot\right)^{-4} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{3/2} \\ {R \over R_\odot} &\sim \left(\nu_{\text{max}} \over \nu_{\text{max},\odot}\right)\left(\Delta\nu \over \Delta\nu_\odot\right)^{-2} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{1/2} \end{aligned}
Data: y_\text{obs} \in Y
Models: x_i \in X;F: X \to Y
Best-fitting model: x = \mathop{\mathrm{argmax}}_{x_j \in X}\mathcal{L}\left(x_j\right)
F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}
, 
, GARSTEC,
…
\color{darkorange} \to \Delta\nu, \nu_{\text{max}}, \left\{\nu_{n,l}\right\}
Modelling with Seismology gives Precision and Ages
Hare “Zebedee”, Cunha+ (2021)
Using \Delta\nu and \nu_\mathrm{max} only
Precise measurements of field stars: {\sigma_R \over R} \lesssim 2 \%;\ {\sigma_M \over M} \lesssim 5 \%;\ \sigma_\text{Age} \lesssim 0.4\ \mathrm{Gyr}
Machine-learning emulation
of mode frequencies
for rapid inference
(Scutt et al. 2025)
\to
Nonradial modes constrain internal structure
\delta\nu_{02} probes
mixing processes
near convective boundaries
(Ong, Lindsay, Reyes, et al. 2025)
Knee feature is
bourne out observationally by \delta\nu_{02} measurements of the open
cluster M67.
(Reyes et al. 2025, Nature)
Many Mode Frequencies constrain lots of internal structure
Probes of the internal states of stars … now return constraints on stellar structure previously only theorized. —Astro 2020
Decadal Survey
(e.g. Bellinger+ 2017, 2019; Pedersen+ 2018;
Vanlaer+ 2023; Buchele+ 2024)
Bellinger+ 2019
Buchele+ 2024
Relative difference in isothermal sound speed
Multiplets constrain Rotation and Orientation
Further applications:
calibrating gyrochronology with
ensemble asteroseismology
(e.g. Hall+ 2021).
Evolved stars dominate our asteroseismic sample.
(facultative with Kepler, obligate with TESS)
(proxy for age \to)
Mixed modes exhibit
avoided crossings
between underlying p- and g-modes.
Pure p-modes: \nu_{n,\ell} \sim \Delta\nu \left(n_p + {\ell \over 2} + \epsilon_{n,\ell}\right)
Pure g-modes: {1 \over \nu_{n,\ell}} \sim \Delta\Pi_\ell \left(n_g + {\ell \over 2} + \epsilon_{g, n,\ell}\right)
period differences \Delta P give asymptotic g-mode period spacings \Delta\Pi_\ell, and mixing fractions \zeta
Ground truth
Mixed Modes: Evolutionary Diagnostics
g-mode Period Spacing \Delta\Pi_1/\mathrm{s}
p-mode Frequency Spacing \Delta\nu/\mu\mathrm{Hz}
from Mosser+ (2014)
clump stars
first-ascent RGs
↖
Single-star electron degeneracy sequence
stars found in forbidden region → merger
remnants?
(Rui and Fuller 2021, Deheuvels+ 2021)
g-modes: Core Rotation
e.g. Mosser+ 2012; Gehan+ 2018
\delta P_{\text{rot}, g, \ell=1} \sim - {m \Omega_\text{core} \over 4\pi \nu^2}
(\leftarrow proxy for age)
g-modes: Core Magnetism
Li et al. Nat. 2022:
Asymmetric splittings probe
core magnetic fields
(Li+ 2023, Deheuvels+ 2023;
Rui+ 2024)
Population studies
of rotation vs. magnetism
(Hatt+ 2024)
\scriptsize \delta \nu_{\text{mag}, g, \ell=1} \sim {m^2 \over \nu^3}
1D Modelling Challenges Limit Inference
1D numerical models encounter significant difficulty
reproducing seismology of
core-helium-burning stars
(Schimak et al., in review)
Mixing near convective boundaries is still essentially unsolved
Asteroseismic constraints on convective core
sizes
constrain 1D theories of
convective-boundary mixing
(Lindsay et al., 2024, 2025,…)
from Blouin+ 2023
Comparing methods on astrophysical benchmarks
κ Cyg has TESS asteroseismology…
…and state-of-the-art CHARA interferometry.
Asteroseismic and Interferometric
radius estimates
might not agree?
(Chowhan et al, in prep.)
🇪🇺 PLATO: Launch January 2027
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