How do we know anything?

Solar-like Oscillations

(RHD simulations courtesy of Joel D. Tanner)

Helioseismology: Greatest Hits

• Solar Neutrino Problem (Nobel!)
• How do stars work??
• Rotational Structure
• Solar Abundance Problem
• Far-side imaging (helioseismic holography)

\[\vdots\]

…and not all interesting stars are bright.

3 Main Uses:

Global Properties give Masses and Radii

Modelling with Seismology gives Precision and Ages

• Fundamental Properties
• Dynamics and Activity
• Internal Structure

Multiplet Splittings give Rotation and Orientation

Many Multiplets give Internal Rotation

• Fundamental Properties
• Dynamics and Activity
• Internal Structure

Many Mode Frequencies constrain Structure

Summary

Asteroseismology is a low-investment
yet versatile conceptual instrument
that has revolutionised stellar metrology,
both global and local.

More Data = More Problems

Case Study: Post-Main-Sequence Pulsations

Evolved stars dominate our asteroseismic sample.

(e.g. only \(\sim 100\) Kepler main-sequence stars)

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

Mixed Modes: Evolutionary Diagnostics

from Mosser+ (2014)

A Crash Course in Wave Propagation:
The JWKB Approximation

Simple Wave Equation

More Complicated Wave Equation

Can we apply physical insights in the field?

Energies and Mixing Coefficients
of Molecular Orbitals

Energy eigenvalues and mixing coefficients are solutions to the
Generalised Hermitian Eigenvalue Problem

\[ \mathbf{H} \begin{bmatrix} {\color{blue}\mathbf{c}_1} \\ {\color{orange}\mathbf{c}_2} \end{bmatrix} = E \cdot \mathbf{D} \begin{bmatrix} {\color{blue}\mathbf{c}_1} \\ {\color{orange}\mathbf{c}_2} \end{bmatrix} \] \[ \small H_{{\color{blue}i}{\color{orange}j}} = \langle {\color{blue}\psi_i}|\hat{H}|{\color{orange}\psi_j}\rangle;~~~~D_{{\color{blue}i}{\color{orange}j}} = \langle {\color{blue}\psi_i}|{\color{orange}\psi_j}\rangle \]

Mixed Modes as
Acoustic “Molecular Orbitals”

Mixed mode frequencies and mixing coefficients can likewise be found by solving the
Generalised Hermitian Eigenvalue Problem: \[ \scriptsize \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \] \[ \small L_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i}, \hat{\mathcal{L}}{\color{red}\xi_j}\rangle;~~~~D_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i}, {\color{red}\xi_j}\rangle \]

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

• Fundamental Properties ✔️
• Dynamics and Activity
• Internal Structure

g-modes: Core Magnetism

• Rotational Shear and Magnetic Dynamos
• Spin-Orbit Misalignment
• The Star-Planet Connection

Summary

What I Bring

Data: Synergies on the ground

Asteroseismology in the TESS Era

Convective Boundary Mixing

Theory: g-modes in Massive Stars

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

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

Theory: Turbulence

(RHD simulations courtesy of Joel D. Tanner)

Asteroseismology in is the Future

Stellar oscillations have revolutionised our understanding of
stellar structure, dynamics, and evolution.

The development of theory and technique
unlocks their use as a tool (e.g.: mixed modes \(\to\) RGB science).

My group will combine these developments with new surveys
to make these measurements astrophysically interesting.

Normal modes satisfy an operator eigenvalue problem
\(\hat{\mathcal{L}}\left|\xi_{n\ell m}\right> = -\omega^2_{n\ell m}\left|\xi_{n\ell m}\right>\).

Rotational Inversions

Inversions for Stellar Structure

Mode Identification

Asteroseismology Happens…

TESS w/ Pixel-Level Decorrelation + Stitching

Observational Overview

How did this happen??

Angular Momentum Matrices

\(\forall \ell, \exists (2\ell + 1) \times (2\ell + 1)\) 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
\(\mathbf{J}_z \hat{=} \mathrm{diag}(-\ell, -\ell + 1 \ldots \ell - 1, \ell).\)

Rotation as an Eigenvalue Problem

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

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

Combined Angular Momentum Operator
(Aligned Case)

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

The Misaligned Angular Momentum Operator

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

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

Mixed Modes

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.

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