Asteroseismology
in the Era of PLATO

Joel Ong
Hubble Fellow, Univ. of Hawaiʻi at Mānoa

CEA Paris-Saclay;
June 30, 2025

I.

How do we know anything?

xkcd #2347: Munroe (2020)

physics of stellar interiors

quantitative

astronomy & astrophysics

Asteroseismology is our
only direct probe of
stellar interiors
(in the electromagnetic spectrum)

(illustration courtesy of Conny Aerts)

PLATO: Launch December 2026

The PLATO Sample

“Bright” sample:
16.000 FGK dwarfs
(\(V < 11\))

“Statistical” sample:
245.000 FGK stars
(dwarfs and subgiants)

(Entire existing sample not visible on this diagram)

More Data = More Problems

The Wrong Trousers: Park et al. 1993

big data deluge

physical interpretation

analytic theory
computational technique
statistical methodology

I am a methodologist.

My work sits at the interface
between analytic theory and
observational data analysis.

As such, I uniquely possess
expertise in both.

 

(illustration courtesy of Conny Aerts)

Blackman, Ong, Fischer 2019 Petersburg, Ong, et al. 2020 Li, Huber, Ong, et al. 2025 Ong: California Planet Search

Ong+ 2022; Ong & Gehan 2023; Ong 2024;
Rui†, Ong, Mathis 2024; Hatt†, Ong, et al. 2024…

Lindsay†, Hon, Ong et al.
2025; 3+ coauthor
publications

Hey, Li, Ong 2025

Ong et al. 2024a; Ong 2025;
\(15+\) coauthor publications

Ong et al. 2021;
Reyes†, Stello, Ong,
et al., Nature, 2025

Ong et al. 2024b;
Nielsen, Ong et al. 2025
+3 coauthor publications

†: mentoree coauthor

Ong:
PLATO WP 120, 128;
PLATO CS WP 162

from Jeffery & Saio (2016)
 
 

Asteroseismology is rich:
each of these patches is
a different class of variable star!

The PLATO main mission
focuses specifically on
cool dwarfs and subgiants.

II.
Solar-like Oscillations
From the Sun to the Stars

SOHO EIT Image (2016)

HMI Dopplergram (2017)

(RHD simulations courtesy of Joel D. Tanner)

Convection excites
pressure waves (p-modes).

\(\ell = 0\) 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

Solar-like oscillators from 1995 onwards: n = 15; from Arentoft+ (2008)

Asteroseismology went from tens of stars…

…to thousands.

MOST (2003-2014)
CoRoT (2006-2013)
Kepler & K2 (2009-2016)
TESS (2018—)

III.
The Asteroseismic Revolution

\[ \begin{aligned} {\Delta\nu} & \sim 1/t_\text{cross} \sim \sqrt{M/R^3}\\ {\nu_{\text{max}}} &\sim{g/c_s} \sim {M/R^2\sqrt{T_\text{eff}}} \end{aligned} \]

\[V_\text{osc} \sim L / M\]

Henrietta Swan Leavitt
δ Cep: from Elver+ 2014

\[\Huge P \sim L^\alpha\]

\(\Delta\nu\)  and  \(\nu_\mathrm{max}\)  give Masses and Radii

\[ \begin{aligned} {\Delta\nu} &\sim \sqrt{M/R^3}\\ {\nu_{\text{max}}} &\sim {M/R^2\sqrt{T_\text{eff}}} \end{aligned} \]

\[ \begin{aligned} {M \over M_\odot} &\sim \left(\nu_\text{max} \over \nu_{\text{max},\odot}\right)^{3}\left(\Delta\nu \over \Delta\nu_\odot\right)^{-4} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{3/2} \\ {R \over R_\odot} &\sim \left(\nu_\text{max} \over \nu_{\text{max},\odot}\right)\left(\Delta\nu \over \Delta\nu_\odot\right)^{-2} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{1/2} \end{aligned} \]

(better scaling relations: Ong & Basu 2019a,b)

All-Sky Mass Mapping: Hon+ 2021

Asteroseismology du-museau-à-la-queue

Asteroseismology du-museau-à-la-queue

  • What are stars like?
  • How do stars work?
  • How do stars behave?
  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

Modelling with Seismology gives Precision and Ages

Hare “Zebedee”, Cunha+ incl. Ong (2021);
HPC & pipeline: Ong+ 2021a

Using \(\Delta\nu\) and \(\nu_\mathrm{max}\) only

Precise measurements of field stars: \[ {\sigma_R \over R} \lesssim 2 \%;\ {\sigma_M \over M} \lesssim 5 \%;\ \sigma_\text{Age} \lesssim 0.4\ \mathrm{Gyr} \]

PLATO DP5:
Masses, Radii, Ages

Asteroseismology as a Tool

  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

from White+ (2011)

Dashed lines = isochrones,
spaced by 1 Gyr

(n.b. typical Kepler uncertainty of \(<1\ \mu\)Hz)

Main-sequence ages (\(\Delta\nu \gtrsim 50\ \mu\)Hz)
can be read off \(r_{02}\) diagrams directly.

Many Mode Frequencies constrain Structure

In particular, Asteroseismology…
has led to the discovery of new avenues of research in our understanding of stellar interiors across the whole HR diagram. —astronet Roadmap
2022-2035

(e.g. Bellinger+ 2017,2019;Ong & Basu 2019a,b;
Lindsay, Ong, Basu 2022†, 2023†, 2024†;
Vanlaer+ 2023; Ong+ in prep†)

†: mentoree paper

Bellinger+ 2019

Relative difference in isothermal sound speed

Asteroseismology as a Tool

  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

Multiplet Splittings give Rotation and Orientation

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

\[\longrightarrow {\scriptsize\text{Frequency}}\]

  • Rapid Rotation \(\to\) Planet Engulfments
  • Spin-Orbit Misalignment
  • Rotational Shear \(\to\) Magnetic Dynamos

(e.g. Ong et al. 2024a, Ong 2024, Ong 2025)

PLATO DP4:
Rotation and Activity

Asteroseismology as a Tool

Each independent aspect of asteroseismic phenomenology
gives us a different observational tool.
(they are PLATO DPs for a reason!)

Space missions like CoRoT and Kepler
revolutionised stellar astrophysics
by giving us opportunities to apply these tools.

IV.
Unknown Unknowns

Case Study: Post-Main-Sequence Pulsations

Evolved stars dominate any asteroseismic sample, because \[\large V_\text{osc} \sim L/M\]

(from Yu+ 2020)
(from Schofield+ 2019)

\(T_\text{eff}/\mathrm{K}\)

\(R/R_\odot\)

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

\[\longrightarrow {\scriptsize\text{Frequency}}\]

(proxy for age \(\to\))

Mixed modes exhibit avoided crossings
between underlying p- and g-modes.

(adapted from Ong & Basu 2020)

Unresolvable from the ground — but visible from space!

Hypothesis
(Unno+ 1989)

Data Set
(CoRoT: 2009)

Analysis
(de Ridder et al. 2009)

Interpretation
(Dupret et al. 2009)

Mixed modes propagate separately in the core
vs
in the envelope.

\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]

\[{\color{red} \omega_g < N, S_\ell}\]

\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]

\[{\color{gray} \omega_p > S_\ell, N}\]

\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]

?

(Ong & Basu 2020)

\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]

\[\iff\]

\[\small\psi_\text{mol} = {\color{blue}\sum_i c_{1,i} \psi_{1,i}} + {\color{darkorange}\sum_j c_{2,j} \psi_{2,j}}\]

State of the art for determining (sub)giant structure and
properties (Ong+ 2021a, b, c), and internal rotation
(Ong+ 2022, 2023; Ong 2024, 2025).

We know more about giant cores
than about the core of our own Sun!

We know more about giant cores
than about the core of our own Sun!

  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

Pressure waves (p-modes)
propagate isotropically.

p-modes:
Characteristic overtone frequency spacing \(\Delta\nu\)

Buoyancy waves (g-modes)
propagate anisotropically.

g-modes:
Characteristic overtone period spacing \(\Delta\Pi_\ell\)

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

(Ong & Basu 2020)

\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]

\[\longrightarrow\]

PBJam/reggae:
Nielsen, Ong et al. 2025; Ong et al. 2024b

from Lindsay, Hon, Ong, et al. 2025

Unknown Unknowns: Highlights

  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

from White+ (2011)

Main-sequence ages (\(\Delta\nu \gtrsim 50\ \mu\)Hz)
can be read off JCD or \(r_{02}\) diagrams directly.
(\(r_{02} = \delta\nu_{02}/\Delta\nu_1\))

What’s going on here?? (A: avoided crossings!)

(Ong & Basu 2020)

\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]

\[\longrightarrow\]

Fast numerical calculations of pure quadrupole p-modes,
and therefore \(\delta\nu_{02}\) or \(r_{02}\), are now possible
in sub- and red giants.
(Ong et al. 2025)

\[\text{Old theory: }\delta\nu_{02} \sim \int {1 \over r}{\mathrm d c_s\over \mathrm d r} \mathrm d r\]

?????
Catastrophe!

Observational Confirmation!

\(\delta\nu_{02}\) 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†, Stello, Ong, et al. 2025, Nature)

†: mentoree paper authors

Theory
(Tassoul 1990)

Data Set
(Kepler/K2: 2013-2015)

New Theory
(Ong et al. 2025)

Analysis and
(Re)interpretation
(Reyes, Stello, Ong+ 2025)

Unknown Unknowns: Highlights

  • Fundamental Properties
  • Internal Structure
  • Dynamics and Activity

g-modes: Core Rotation

e.g. Mosser+ 2012; Gehan+ 2018; Ong & Gehan 2023; Ahlborn, Ong, et al. in review

\[\delta P_{\text{rot}, g, \ell=1} \sim - {m \Omega_\text{core} \over 4\pi \nu^2}\]

g-modes: Core Magnetism

Li et al. Nat. 2022:
Asymmetric splittings probe
core magnetic fields

(Li+ 2023, Deheuvels+ 2023;
Rui, Ong, Mathis, 2024†;
Rui, Fuller, Ong 2025)

Population studies
of rotation vs. magnetism

(Hatt, Ong, et al., 2024†)

\[\scriptsize \delta \nu_{\text{mag}, g, \ell=1} \sim {m^2 \over \nu^3}\]

†: mentoree paper

Frequency/\(\mu\mathrm{Hz}\)

g-mode phase

RMS Radial Magnetic Field Strength

Hypothesis
(Unno+ 1989)

Data Set
(Kepler: 2013)

Analysis
(Li et al. 2022)

Interpretation
(Rui, Ong, Mathis 2024;
Hatt, Ong, et al. 2024)

Star-Planet Interactions

from Albrecht et al. (2021)
  • Stars tend to have (and interact with) companions: binaries, planetary systems, engulfments…
  • Seismic rotational measurements indicate anomalies?

Saunders et al. 2024

  1. \(\forall \ell, \exists (2\ell + 1) \times (2\ell + 1)\) matrices
    \(\mathbf{J}_x^\ell\), \(\mathbf{J}_y^\ell\), \(\mathbf{J}_z^\ell\) satisfying commutation relations
    \(\left[\mathbf{J}_i, \mathbf{J}_j\right] = -i\epsilon_{ijk}\mathbf{J}_k\), with
    \(\mathbf{J}_z \hat{=} \mathrm{diag}(-\ell, -\ell + 1 \ldots \ell - 1, \ell).\)

for \(\ell = 1,\)

\[\small\mathbf{J}_x \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}; \mathbf{J}_y \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & i & 0 \\ -i & 0 & i \\ 0 & -i & 0\end{bmatrix}; \mathbf{J}_z \hat{=} \begin{bmatrix}-1 & 0 & 0 \\ 0 &0 &0 \\ 0& 0 &1\end{bmatrix}.\]

  1. For fixed \(m\) (to leading order):

\[\left(-\mathbf{\Omega}_0^2 + 2 \omega m \mathbf{R} + \omega^2 \mathbf{\Delta}\right)\mathbf{c} = 0\]

For fixed \(n\) (to leading order):

\[(-\omega_0^2 \mathbb{1}_{2\ell+1} + 2 \omega \mathbf{J}_z R_{n,n} + \omega^2 \mathbb{1}_{2\ell+1})\mathbf{y} = 0\]

  1. Under separation of variables, \[\vec{\xi}_{n\ell m}(r, \theta, \varphi) = \vec{\xi}_{n\ell}(r) Y_\ell^m(\theta, \phi) \iff \underbrace{\tilde{R}_{n, n', m, m'} = R_{n,n'} J_{m,m'}}_{\text{this is a }\textbf{tensor product}!}\]

\[\implies \left(-\mathbf{\Omega_0}^2 \otimes \mathbb{1}_{2\ell+1} + 2 \omega \underbrace{\mathbf{R} \otimes \mathbf{J}_z}_{\tilde{\mathbf{R}}} + \omega^2 \mathbf{\Delta} \otimes \mathbb{1}_{2\ell+1}\right)\mathbf{x} = 0\]

  1. Let’s associate with each mass shell at \(r\) both \(\Omega(r)\)
    (as is customary), and also a rotational axis
    \(\hat{\mathbf{n}}(r) =\sum_i n_i \mathbf{e}_i\).

\[\small \begin{aligned} \mathbf{R}_{n\ell, n\ell} &= b_{n\ell}\int {\mathbf{d}^\ell}^\dagger(\beta(r)) \Omega(r) \mathbf{J}_z {\mathbf{d}^\ell}(\beta(r))\ K(r)\ \mathrm{d} r\\ &= b_{n\ell}\int \Omega(r) (\hat{\mathbf{n}} \cdot \vec{\mathbf{J}})\ K(r)\ \mathrm{d} r\\ &= \boxed{b_{n\ell}\left(\int \vec{\mathbf{\Omega}}(r) K(r)\ \mathrm{d} r\right) \cdot \vec{\mathbf{J}}}. \end{aligned} \]

\(\implies\) For each mode, AM matrix is
specified by usual vector addition.

  1. We only assume that the pure p- and g-mode solutions
    are separately amenable to separation of variables;
    the mixed-mode eigenfunctions need not be.

\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} + 2 \omega \begin{bmatrix} {\color{forestgreen}\tilde{\mathbf{R}}_\pi} & 0 \\ 0 & {\color{forestgreen}\tilde{\mathbf{R}}_\gamma}\end{bmatrix} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} \right)\mathbf{x} = 0. \]

Mode visibilities are specified by \(\mathbf{x}^\dagger(\mathbb{1} \otimes \mathbf{P})\mathbf{x}\), where \(\mathbf{P}\) is the projection matrix onto \(m=0\) in the observer’s coordinate frame.

\[ \vdots \]

New Theory
(Ong 2025)

Ong 2025

Existing Data,
New Techniques
(Ong 2025)

From Huber et al. 2013

Ong 2025

Technique-driven Scientific Discovery!

Hypothesis
(Winn+ 2010)

Data Set
(Kepler: 2013)

Analysis and Interpretation
(Ong 2025)

Unknown Unknowns: Highlights

  • Generative Models for Mixed Modes (e.g. Ong+ 2024b; Nielsen, Ong+ 2025)
  • Mixed Modes \(\to\) Age-Dating (e.g. Ong+ 2021a,b; Lindsay, Ong, Basu 2022,2025)
  • K-dwarf Asteroseismology \(\to\) Radius Inflation (e.g. Li, Huber, Ong+ 2025)
  • Small Separations \(\to\) Convective Evolution (e.g. Ong+ 2025; Reyes+ 2025)
  • Mixed Modes \(\to\) Convective-Boundary Mixing (e.g. Lindsay, Ong, Basu 2023,2024)
  • Rapid Rotation \(\to\) Planet Engulfment (e.g. Ong+ 2024a, “Zvrk”)
  • Rotational Shear \(\to\) Magnetic Dynamos (e.g. Rui, Ong, Mathis 2024; Hatt, Ong+ 2024)
  • Asteroseismology of Black-Hole Binaries (e.g. Hey, Li, Ong 2025, Gaia BH2* & BH3*)
  • Spin-Orbit Misalignments \(\to\) Orbital Architectures (e.g. Ong 2025)

Unknown Unknowns

In the data-rich régime,
technique development and
interpretation are
rate-limiting steps to discovery.

V.
The Coming
Asteroseismic Revolution

Data: The PLATO Mission

\(\sim100 \to 10,000++\) stars:

How do we cope with
a deluge of new data?

e.g. Nielsen, Ong, et al., 2025;
Ong et al. 2024b

(incl. Ong: Cunha+ 2021; Nielsen+ 2021, 2023; Campante+ 2023)

How do we search for
and interpret
unknown unknowns?

PLATO DP3: Mode Frequencies

?

Data: PLATO and Inversions

Qualitative new capability:
Ensemble Inversions for Structure and Rotation
(Ong, Hoogendam†, et al. in prep. — TASC Poster I)

Bellinger+ 2019

Buchele+ 2024

PLATO will bring us from tens to thousands.

?

PLATO DP3,4,5;
Complementary Science

Rotational Inversions Core-Envelope Misalignment RRRGs

Asteroseismic Age-Dating

Constraints on CBM from p-modes The Asteroseismic Surface Term

EPRV Asteroseismology Asteroseismology
+ Doppler Imaging

Binary Asteroseismology

Constraints on CBM from g-modes

Generalised Structure Inversions

Galactic Archaeology w/ LFEMI

Internal Mixing and Transport Processes (w/ S. Mathis)

Stellar Rotation and Magnetism (w/ R. Garcia; A. Strugarek)

Star-Planet Interactions (w/ A. Garcia; E. Ducrot; B. Perri)

Non-canonical Evolution

Accurate Stellar Ages

?

Data: Synergies on the ground

Combining with Photometric surveys — e.g. Ong et al. (2024a); Hart+ incl. Ong (2023)

Seismology from Extreme Precision Radial Velocities (Li, Huber, Ong, et al. 2025; Hon+ incl. Ong 2024)

Ong: ASAS-SN; SONG WG1 & WG2; Keck Planet Finder via CPS

Cool Dwarfs
with ESPRESSO

Long-Period Variables
with Rubin/LSST

EPRV Asteroseismology
with the E-ELT

Data-processing techniques for gapped data sets: Ong, Li, Hey (in prep. — TASC Poster II)

?

🇺🇸 TESS (Ongoing)
🇨🇳 Earth 2.0: 2028 (Planned)
🇪🇺 PLATO Mission: 2026
🇺🇸 Roman: 2026 (Planned)

Statistical sample (Ong: WP 120, 128; CS WP 162)

\(\sim 10^6\) red giants in galactic bulge & globular clusters

Long temporal baselines
(Ong: collaboration member)

All-sky shallow search
(Ong: TASOC WG1, WG2, WG7)

The Coming Asteroseismic Revolution

Asteroseismology has revolutionised our understanding of
stellar structure, dynamics, and evolution.

Large-scale surveys now dominate our scientific landscape.

My group will combine PLATO data with new techniques
to make these measurements astrophysically interesting.

Supplementary Slides

Why Space?

Solar-like oscillators from 1995 onwards: n = 15; from Arentoft+ (2008)

Telescopes can only point at one star at time…

…and not all interesting stars are bright.

Required photometric stability not achievable from ground

Why Not Space?

The Sun as seen by SOHO:
Bedding & Kjeldsen 2005

Procyon from MOST vs. RVs:
Huber et al. 2011

The Star-Planet Connection

Probing the star-planet connection, and non-canonical evolution

e.g. Ong 2025; Ong et al. 2024a;
Hon et al. (incl Ong) Nat. 2023

(also incl. Ong: Huber+ 2019, 2022; …)

Kepler-56’s core and envelope
rotate around different axes.

(Ong 2025)

Zvrk rotates too fast to
not have eaten something recently.

(Ong et al. 2024a)

8 UMi b should have
been consumed, but wasn’t?

(Hon+ 2023 incl. Ong)

PLATO DP4: Rotation and Activity

?

Constraints on Interior Mixing from p-modes

Probes of mixing processes
near convective boundaries

(Lindsay, Ong, Basu 2022†, and 2024†;
Ong, Lindsay, Reyes et al. 2025;
Reyes, Stello, Ong, et al. 2025†, Nature)

†: mentoree paper

Ong: TESS Guest Investigator Cycle 7, PI

?

Convective Turbulence

Theory: The Surface Term

?

Theory: Turbulence

(RHD simulations courtesy of Joel D. Tanner)

Mode amplitudes are usually ignored,
but are entirely determined by turbulent convective driving.

How do we predict mode amplitudes and lifetimes?
&
Why is there a \(\nu_\text{max}\)???
&
How do mode frequencies depend on turbulent stresses?

(Ong+ 2021a,b,c; Li+ 2023; Zhou+ 2020, 2021)

?

Constraints on Interior Mixing from g-modes

from Jeffery & Saio (2016)
 
 

Convective Boundary Mixing

from Blouin et al. 2023a, b

Theory: Analytical Methods for g-modes

Nonlinear evolution is
the primary obstacle to
g-mode inversions.

Will understanding this
(Ong, Hoogendam†, et al. in prep.) permit further technique development?

from Vanlaer et al. 2023

proxy for age \(\to\)

†: mentoree paper

?

Stellar Modelling

Data: \(y_\text{obs} \in Y\)

Models: \(x_i \in X\);\[F: X \to Y\]

Best-fitting model: \[x = \mathop{\mathrm{argmax}}_{x_j \in X}\ \mathcal{L}\left(x_j\right)\]

\[F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}\]

\[F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \textbf{input physics}\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}\]

\[ \color{darkorange} \to \Delta\nu, \nu_{\text{max}}, \left\{\nu_{n,l}\right\} \]

PLATO DP5: Masses, Radii, Ages

?