Liberating Asteroseismic    Inversions from the Tyranny of Stellar Modelling

Joel Ong
University of Sydney
with Nicholas Rui, Claudia Reyes, Sarbani Basu, Willem Hoogendam, Vincent Vanlaer

ISSI-BJ, April 16 2026

I.
What are Inversions?

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}

All models are wrong,
but some are useful.
— George E. P. Box

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,\ell}\right\}

f-modes: \omega_f \sim \sqrt{\left(\ell + {1\over 2}\right){GM\over R^3}}
↘

↑
p-modes: \omega_p \sim {\pi \over T}\left(n + {\ell \over 2} + \epsilon_p\right)

\begin{array}{c} {\scriptsize\text{Power}}\\ \big\uparrow \end{array}

\longrightarrow {\scriptsize\text{Frequency}}

Inversions in a nutshell

Frequencies constrain.

Frequency differences constrain differentially.

Rotational inversions constrain differential rotation

(e.g. Yoshiki Hatta’s talk)

from Basu (2020)

two standard
solar models

Structure Differences give Frequency Differences

S_{ij}[{\color{blue}{\rho}}, {\color{orange}c_s}] \equiv \int \bm\xi_i^*\cdot \left(\hat{\mathcal L} \bm \xi_j\right) \mathrm d^3 x \equiv \int L_{ij}\ \mathrm d r\ \ (= -\delta_{ij} \omega_i^2)

\delta S_{ij} = \int {\delta L_{ij} \over \delta \rho} {\color{blue}\delta\rho(r)}\mathrm d r + \int \underbrace{\delta L_{ij} \over \delta c_s}_{\mathclap{{\partial L_{ij} \over \partial c_s} - {\mathrm d \over \mathrm d r}{\partial L_{ij} \over \partial c_s'} + \ldots}} {\color{orange}\delta c_s(r)}\mathrm d r = V_{ij}

{\delta\omega_i \over \omega_i} = -{V_{ii} \over 2\omega_i^2} = \int K_{c_s,\rho,i}{\delta c_s \over c_s}\ \mathrm d r + \int K_{\rho,c_s,i}{\delta \rho \over \rho}\ \mathrm d r

The Inverse Problem

{\delta\omega_i \over \omega_i} = -{V_{ii} \over 2\omega_i^2} = \int K_{c_s,\rho,i}{\delta c_s \over c_s}\ \mathrm d r + \int K_{\rho,c_s,i}{\delta \rho \over \rho}\ \mathrm d r

{\delta\omega_i \over \omega_i} \approx \sum_j {\color{green}\Delta r K_{c_s,i}(r_j)}{\color{purple}{\delta c_s(r_j) \over c_s}} + \sum_j {\color{green}\Delta r K_{\rho,i}(r_j)}{\color{purple}{\delta \rho(r_j) \over \rho}}

\huge \mathbf{b} = {\color{green}\mathbf{A}}{\color{purple}\mathbf{x}}

We can* do this for ~30 other (cool MS) stars

Recipe:

  1. Model star well (expensive)
  2. Linearise perturbations around best-fit model (fragile)
  3. Invert for structure
    (the easy part)

(e.g. Bellinger+ 2017, 2019; Pedersen+ 2018;
Vanlaer+ 2023; Buchele+ 2024)

Bellinger+ 2019

Buchele+ 2024

Relative difference in isothermal sound speed

* it’s hard

II.
Why Are Inversions Hard?

Evolved stars and classical pulsators dominate our asteroseismic sample.

(avoided crossings can be decoupled:
Ong and Basu 2020)

Kepler Sample (from Yu+ 2020)

We can only meaningfully compare dimensionless
quantities, scaled by combinations
of R and \omega_0 = \sqrt{G M \over R^3}.

Problem: We don’t (usually) know the mass and radius of a star in advance.

We needdon’t usually have accurate radii for this

κ Cyg has TESS asteroseismology…

…and state-of-the-art CHARA interferometry.

Asteroseismic and Interferometric
radius estimates
do not agree.

(Chowhan et al, 2026)

Problem: Kernel shapes themselves depend on structure!
(and are nonuniformly distributed)

III.
Asteroseismic Inversions

From asymptotic analysis of p-modes, \xi_r \sqrt{\rho c_s r^2} \sim s_{\ell}(\omega t + \phi), where t(r) = \int_0^r \mathrm d r'/c_s is the acoustic radius and s_\ell(x) = \sqrt{\pi x \over 2} J_{\ell + {1\over2}}.

Proposal: Let’s choose t as the integration coordinate

(kernels for differences at matching t rather than r)

(similar analysis possible for g-modes)

(use for rotational inversions: Ong et al. 2024)

Highly similar kernels for very different radiistellar structures!

Blue = ; Orange =

Inversions now viable with kernels from models that are not exact matches to a star’s internal structure!

Perturbation should remain linear over relatively large
evolutionary/mass range.

Summary

Liberating Asteroseismic Inversions from
the Tyranny of Stellar Modelling

Asteroseismic inversions have historically been expensive because stellar modelling is hard.
By performing inversions in carefully chosen coordinates,

  • We eliminate the need for the stellar mass and radius to be simultaneously accurately measured.
  • Accuracy of results become much less sensitive to an accurate model of internal structure

Star-vs.-star inversions (e.g. in clusters) may now be possible.

\mathrm{j}\mathrm{o}\mathrm{e}\mathrm{l}\cdot\mathrm{o}\mathrm{n}\mathrm{g}\ \text{@}\ \text{sydney}.\text{edu}.\text{au}

Backup slides

What are Normal Modes?

\newcommand{\rr}{{\color{blue}{\rho}}} \newcommand{\gv}{{\color{blue}{\mathbf{g}}}} \begin{aligned} \Large -\omega^2 \rr \bm{\xi} &= \small \nabla \left(\rr {\color{orange}c_s^2} \nabla \cdot \bm\xi\right) + \nabla \left(\rr \bm\xi \cdot \gv\right) - \gv \nabla \cdot (\rr \bm\xi) - \rr G \nabla\left(\int \mathrm d^3 x' {\nabla' \cdot(\rr \bm\xi) \over |\mathbf{x - x}'|}\right)\\ &\Large \equiv \hat{\mathcal{L}}\bm{\xi}. \end{aligned}

Consider two stars with identical M and R,
but with different internal structures: \hat{\mathcal{L}}_0\xi_{0,n} = -\omega_{0,n}^2 \xi_{0,n};~~~~\hat{\mathcal{L}}\xi_{n} = -\omega_{n}^2 \xi_{n}. We write \hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}}, so that \lambda \in [0,1] interpolates linearly between the two structures.