Asteroseismic Adventures Around Azimuthal Asymmetry and: an assay at astrophysical alliteration

Joel Ong
University of Sydney

ANU Astronomy Colloquium, 5 May 2026

Overview

Introduction
Avoided Crossings (Old)
Avoided Crossings (New)
Core-Envelope Misalignment
Magnetic Red Giants

I.
Solar-like Oscillations

How do we know anything?

physics of stellar interiors

quantitative

astronomy & astrophysics

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

\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

(RHD simulations courtesy of Joel D. Tanner)

Convection excites
pressure waves (p-modes).

Most stars are
solar-like oscillators.

\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

Seismology as a Tool

Global Properties 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}

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

Many Mode Frequencies constrain 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, Ong+ 2026…)

Bellinger+ 2019

Relative difference in isothermal sound speed

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

\longrightarrow {\scriptsize\text{Frequency}}

II.
Mixed Modes

Evolved stars dominate our asteroseismic sample.

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

Kepler Sample (from Yu+ 2020) — *Kepler* Sample (from Yu+ 2020)

Pure p-modes: \boxed{\nu_{n,\ell} \sim \Delta\nu \left(n_p + {\ell \over 2} + \epsilon_{n,\ell}\right)}

Pure g-modes: \boxed{{1 \over \nu_{n,\ell}} \sim \Delta\Pi_\ell \left(n_g + {\ell \over 2} + \epsilon_{g, n,\ell}\right)}

\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.

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)

A Crash Course in Wave Propagation:
The JWKB Approximation

Simple Wave Equation

\huge {\partial^2 \over \partial t^2}f(t, \mathbf{x}) = c_s^2 \nabla^2 f(t, \mathbf{x})

\huge -\omega^2 \hat{f}(\omega, \mathbf{x}) = c_s^2 \nabla^2 \hat{f}(\omega, \mathbf{x})

\huge -\omega^2 \hat{f}(\omega, \mathbf{k}) = -c_s^2 |\mathbf{k}|^2 \hat{f}(\omega, \mathbf{k})

Dispersion relation: \boxed{\huge \omega^2 = c_s^2 |\mathbf{k}|^2}

More Complicated Wave Equation

\huge {\partial^2 \over \partial t^2}f = \left(c_s^2 \nabla^2 - \omega^2_\text{ac}(\mathbf{x})\right) f

\huge -\omega^2 f = \left(c_s^2 \nabla^2 - \omega^2_\text{ac}(\mathbf{x})\right) f

\large \nabla^2 f = -\left({\omega^2 - \omega_\text{ac}^2\over c_s^2}\right) f \equiv -k^2 f

Wave propagates where k^2(r, \omega) > 0,
and decays where k^2(r, \omega) < 0.

JWKB approximation (qualitative)
vs
Brute-force numerical solutions (quantitative)

{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_\ell}^2 \over \omega^2}\right)\left(1 - {{\color{darkorange}N}^2 \over \omega^2}\right)}

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

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

\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r} entropy gradient (=0 in CZ)

\small S_\ell^2 = c_s^2 k_h^2 = {\ell(\ell+1) c_s^2 \over r^2} wave angular momentum

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

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

III.
Isolating Mixed Modes “Therefore what God has joined together, let not man separate.” — Mark 10:9

LCAOs give Molecular Orbitals

\Large \hat H \left|n\right\rangle = \left(\hat T + \hat V\right)\left|n\right\rangle = E_n \left|n\right\rangle

\large {\color{blue}{\hat H_1}} \left|n\right\rangle = \left(\hat T + \hat V_1\right)\left|n\right\rangle = {\color{blue}E_n \left|n\right\rangle}

\large {\color{orange}{\hat H_2}} \left|n\right\rangle = \left(\hat T + \hat V_2\right)\left|n\right\rangle = {\color{orange}E_n \left|n\right\rangle}

\huge \hat H = \hat T + \hat V_1 + \hat V_2 = {\color{blue}{\hat H_1}} + \hat V_2 = \hat V_1 + {\color{orange}{\hat H_2}}

Energies and Mixing Coefficients
of Molecular Orbitals

\left|\psi_\text{mol}\right\rangle = {\color{blue}\sum_i c_{1,i}\left|\psi_{1,i}\right\rangle} + {\color{orange}\sum_j c_{2,j}\left|\psi_{2,j}\right\rangle}.

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

\Large \hat{\mathcal{L}} \xi_n = -\omega_n^2 \rho_0 \xi_n

\Large \hat{\mathcal{L}} = {\color{red}\hat{\mathcal{L}}_\gamma} + \hat{\mathcal{R}}_\gamma

\Large {\color{red}\hat{\mathcal{L}}_\gamma \xi_{\gamma,n} = -\omega^2_{\gamma,n}\xi_{\gamma,n}}

\Large \hat{\mathcal{L}} = \hat{\mathcal{R}}_\pi + {\color{grey}\hat{\mathcal{L}}_\pi}

\Large {\color{grey}\hat{\mathcal{L}}_\pi \xi_{\pi,n} = -\omega^2_{\pi,n}\xi_{\pi,n}}

\Large \hat{\mathcal{L}} = {\color{grey}\hat{\mathcal{L}}_\pi} + \hat{\mathcal{R}}_\pi = \hat{\mathcal{R}}_\gamma + {\color{red}\hat{\mathcal{L}}_\gamma}

\Large \xi_\text{mixed} \sim {\color{grey}c_\pi \xi_\pi} + {\color{red}c_\gamma \xi_\gamma}

Mixed Modes as
Acoustic “Molecular Orbitals”

\xi_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \xi_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \xi_{\gamma,j}}

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

(Ong & Basu 2020)

OK, so what?

g-modes: Core Rotation

e.g. Mosser+ 2012; Gehan+ 2018; Ong & Gehan 2023

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

(\leftarrow proxy for age)

IV.
Differential Rotation

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

\longrightarrow {\scriptsize\text{Frequency}}

Neglecting avoided-crossing effects means many estimates of envelope rotation have likely been systematically misestimated.

Systematic measurement error (sigmas)

(proxy for age \rightarrow)

from Ahlborn, Ong, et al. 2025

from Li et al. 2024

Counter-rotating cores vs envelopes?

Geometrical differential rotation

e.g. Anomalous Envelope Rotation: Li Gang et al. (2024) — e.g. Anomalous Envelope Rotation:
Li Gang et al. (2024)

Seismic rotational measurements indicate anomalies?
Stars tend to have (and interact with) companions: binaries, planetary systems, engulfments…

\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}.

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

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

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.

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

(some assembly required)

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

\longrightarrow {\scriptsize\text{Frequency}}

Kepler-56

From Huber et al. 2013

Rapidly rotating: P_\text{rot} \sim 74\ \mathrm{d}

Multitransiting, Spin-Orbit Misaligned.

T_\text{eff} \sim 4840\ \mathrm{K}: Coolward of Kraft Break

\implies should not be (rapidly) rotating

\implies should not be spin-orbit misaligned

+ spin-orbit misalignment vs mutually aligned planets is exceedingly rare

From Wang, Wang & Ong (2026)

What does this mean???

(most likely scenario given rapid obliquity damping)

Problems:

Rapidly rotating despite being cool
Spin-orbit misaligned despite being cool
Spin-orbit misaligned with multiple planets

Problems:

Rapidly rotating despite being cool
~~Spin-orbit misaligned despite being cool~~
~~Spin-orbit misaligned with multiple planets~~
VERY WEIRD internal rotation profile??

Most obvious explanation:
what if spun up by tidal inspiral?

Even unphysically aggressive (low-Q) inspiral
requires rapid rotation at TAMS:
incompatible with magnetic braking!

\implies tidal torques insufficient to supply present AM

(cf. Tokuno 2025 for alternative argument)

TAMS

Today

Engulfment?

V.
What About Magnetic Fields?

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

\longrightarrow {\scriptsize\text{Frequency}}

g-modes: Core Magnetism

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

(Li+ 2023, Deheuvels+ 2023)

Nonperturbative description
for rotation interacting with magnetic dipole:

\ell(\ell+1) \to \lambda\left({\color{blue}q}={2{\color{blue}\Omega}\over\omega}, {\color{orange}b} = {k_r {\color{orange}v_{A,r}}\over \omega}\right)

(Rui, Ong, Mathis, 2024)

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}

The TAR + M

\hat{\mathcal{L}}y = {\mathrm d \over \mathrm d x}\left(p(x) {\mathrm dy \over \mathrm d x}\right) + q(x) y= -\lambda w(x) y

\hat{\mathcal{L}}P_\ell^m = {\mathrm d \over \mathrm d \mu}\left(\left(1 - \mu^2\right) {\mathrm d P_\ell^m \over \mathrm d \mu}\right) - {m^2 \over 1 - \mu^2}P_\ell^m

\begin{aligned} \hat{\mathcal{L}}^{m,{\color{orange}b},{\color{blue}q}}[f(\mu)] &= \frac{\mathrm{d}}{\mathrm{d}\mu}\left(\frac{(1-\mu^2)(1-{\color{orange}b^2}\mu^2)}{(1-{\color{orange}b^2}\mu^2)^2-{\color{blue}q^2}\mu^2}\frac{\mathrm{d}f(\mu)}{\mathrm{d}\mu}\right) \\ &- \frac{m^2}{1-\mu^2}\frac{1-{\color{orange}b^2}\mu^2}{(1-{\color{orange}b^2}\mu^2)^2-{\color{blue}q^2}\mu^2}f(\mu) \\ &- m{\color{blue}q}\left(\frac{4{\color{orange}b^2}\mu^2(1-{\color{orange}b^2}\mu^2)+2{\color{blue}q^2}\mu^2}{\left[(1-{\color{orange}b^2}\mu^2)^2-{\color{blue}q^2}\mu^2\right]^2} + \frac{1}{(1-{\color{orange}b^2}\mu^2)^2-{\color{blue}q^2}\mu^2}\right)f(\mu)\mathrm{.} \end{aligned}

How do we identify modes with so many complicated moving parts?

It’s not just a phase

Normalised Complex
Cross-correlation

Standard time-series analysis uses the
Fourier power spectrum P(\omega) = |\hat{f}(\omega)|^2, but this discards phase information
encoded in the complex amplitudes.

Power spectrum (SNR)

(Rui, Fuller, Ong 2025)

Complex cross-correlation (including phase information) identifies modes across multiplets!

Time-dependent Mode Coupling

(Ma, Ong, Rui in prep.)

Why stop at a magnetic dipole?

(Rui, Ong et al. in prep.)

Wrapping Up

Isolating and Decoupling Mixed Modes

Mixed modes may be described as
“acoustic molecular orbitals”:
Ong & Basu 2020, ApJ, 898, 127

Why is Kepler-56?

As previously understood, Kepler-56
— rapidly rotating, misaligned, multitransiting —
should not exist.

Revisited asteroseismology resolves tension with obliquity damping,
but implied configuration still
cannot be explained by canonical stellar evolution.

Planetary engulfment is most likely explanation.
(so far)

Ong 2025, ApJ, 980, 40

Magnetism and Oblique Pulsations

Existing techniques have to be generalised to describe the intersection of strong magnetism, rapid rotation,
and p/g mode coupling.

Qualitative consequences and morphology
of this theory are still being explored.

Asteroseismic Adventures Around Azimuthal Asymmetry

A better understanding of gravitoacoustic
mixed modes permits us to learn new things about
stellar rotation, magnetism, and
star-planet interactions.

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

\text{Spon}\text{sored}\ \text{Messages}

I. Visit Tokyo

II. A PhD in Sydney

Why Stellar Physics?

All modern astronomy and astrophysics
is built on stellar astrophysics —
whose foundations remain poorly defined.

Meaningful progress will require advances
in both analytic theory and
observational data analysis,
combined.

Dear colleagues,

I am pleased to announce a 3-year PhD scholarship at the University of Sydney to work with me (Dr. J. M. Joel Ong) on the topic of the asteroseismology and astrophysics of core-envelope misalignment.

This opportunity is a part of an ARC-funded project, led by Dr. Ong, which aims to address the following key questions: a) how prevalent is differential rotational misalignment? b) what are its astrophysical consequences? and c) how can we use it to better understand the astrophysics of stellar rotation and magnetism?

Application deadline: TBD (expected June 2026)

Keep an eye out for a job opening on the AAS Job Register (~May 2026).

Additional information

Expected start date: October 2026
Possibility of an up to 6-month extension of the duration of the PhD scholarship.
Applicants should have a background in mathematics, physics, astronomy, or equivalent.
Prior experience with asteroseismology desirable but not required.

Please forward this information to any interested students.

With thanks and best regards,
Joel Ong

Supplementary Material

Spin-Orbit Misalignment

Reminder: 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).

Example: \ell = 1

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

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

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.

\boxed{\tilde{\mathbf{R}} = \int \mathbf{K}\otimes\left(\Omega_\text{rot}(r)\ \hat{\mathbf{n}}\cdot \vec{\mathbf{J}}\right)\ \mathrm{d}r}

\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.

Asteroseismic Adventures Around Azimuthal Asymmetry and: an assay at astrophysical alliteration

Overview

I. Solar-like Oscillations

How do we know anything?

Seismology as a Tool

Global Properties give Masses and Radii

Modelling with Seismology gives Precision and Ages

Many Mode Frequencies constrain Structure

II. Mixed Modes

A Crash Course in Wave Propagation: The JWKB Approximation

Simple Wave Equation

More Complicated Wave Equation

III.Isolating Mixed Modes “Therefore what God has joined together, let not man separate.” — Mark 10:9

LCAOs give Molecular Orbitals

Energies and Mixing Coefficients of Molecular Orbitals

Mixed Modes as Acoustic “Molecular Orbitals”

OK, so what?

g-modes: Core Rotation

IV. Differential Rotation

Geometrical differential rotation

Kepler-56

What does this mean???

V. What About Magnetic Fields?

g-modes: Core Magnetism

The TAR + M

It’s not just a phase

Time-dependent Mode Coupling

Why stop at a magnetic dipole?

Wrapping Up

Isolating and Decoupling Mixed Modes

Why is Kepler-56?

Magnetism and Oblique Pulsations

Asteroseismic Adventures Around Azimuthal Asymmetry

\text{Spon}\text{sored}\ \text{Messages}

I. Visit Tokyo

II. A PhD in Sydney

Why Stellar Physics?

Supplementary Material

Spin-Orbit Misalignment

Reminder: Angular Momentum Matrices

Example: \ell = 1

Rotation as an Eigenvalue Problem

Combined Angular Momentum Operator (Aligned Case)

The Misaligned Angular Momentum Operator

Mixed Modes

I.
Solar-like Oscillations

II.
Mixed Modes

A Crash Course in Wave Propagation:
The JWKB Approximation

III.
Isolating Mixed Modes “Therefore what God has joined together, let not man separate.” — Mark 10:9

Energies and Mixing Coefficients
of Molecular Orbitals

Mixed Modes as
Acoustic “Molecular Orbitals”

IV.
Differential Rotation

V.
What About Magnetic Fields?

Combined Angular Momentum Operator
(Aligned Case)