(RHD simulations courtesy of Joel D. Tanner)

LCAOs give Molecular Orbitals

Energies and Mixing Coefficients
of Molecular Orbitals

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

$$\mathbf{H}\left[\begin{array}{c}{\mathbf{c}}_1\\ {\mathbf{c}}_2\end{array}\right]=E\cdot \mathbf{D}\left[\begin{array}{c}{\mathbf{c}}_1\\ {\mathbf{c}}_2\end{array}\right]$$ $$H_{ij}=⟨{\psi }_i|\hat{H}|{\psi }_j⟩;D_{ij}=⟨{\psi }_i|{\psi }_j⟩$$

Mixed Modes as
Acoustic “Molecular Orbitals”

Mixed mode frequencies and mixing coefficients can likewise be found by solving the
Generalised Hermitian Eigenvalue Problem: $$\left[\begin{array}{cc}{\mathbf{L}}_{\pi \pi }& {\mathbf{L}}_{\pi \gamma }\\ {\mathbf{L}}_{\pi \gamma }^T& {\mathbf{L}}_{\gamma \gamma }\end{array}\right]\left[\begin{array}{c}{\mathbf{c}}_{\pi }\\ {\mathbf{c}}_{\gamma }\end{array}\right]=-{\omega }^2\left[\begin{array}{cc}\mathbb{1}& \mathbf{D}\\ {\mathbf{D}}^T& \mathbb{1}\end{array}\right]\left[\begin{array}{c}{\mathbf{c}}_{\pi }\\ {\mathbf{c}}_{\gamma }\end{array}\right].$$ $$L_{ij}=⟨{\xi }_i,\hat{\mathcal{L}}{\xi }_j⟩;D_{ij}=⟨{\xi }_i,{\xi }_j⟩$$

(Ong & Basu 2020)

(Ong & Basu 2020)

Mode Identification

Asteroseismology Happens…

How did this happen??

Why???

Parameterising Misalignment

Euler Angles

Isolating and Decoupling Mixed Modes

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

A Rapidly-Rotating Engulfment Candidate

We find rapid rotation in Zvrk with seismology
(confirmed independently with TESS, ASAS-SN, $V\sin i$)

Core-Envelope Misalignment

We extend this formalism to the tensor product
of the radial and horizontal problem.

Asteroseismic Diagnostics of
Engulfment & Misalignment

A better understanding of gravitoacoustic
mixed modes, and how they respond to rotation,
has permitted us to diagnose signatures of
planetary engulfment and core-envelope misalignment.

Angular Momentum Matrices

$\mathrm{\forall }\ell ,\mathrm{\exists }(2\ell +1)\times (2\ell +1)$ matrices
${\mathbf{J}}_x^{\ell }$, ${\mathbf{J}}_y^{\ell }$, ${\mathbf{J}}_z^{\ell }$ satisfying commutation relations
$[{\mathbf{J}}_i,{\mathbf{J}}_j]=-i{ϵ}_{ijk}{\mathbf{J}}_k$, with
${\mathbf{J}}_z\widehat{=}\mathrm{diag}(-\ell ,-\ell +1\dots \ell -1,\ell ).$

Rotation as an Eigenvalue Problem

For fixed $m$ (to leading order):

$$(-{\mathbf{\Omega }}_0^2+2\omega m\mathbf{R}+{\omega }^2\mathbf{\Delta })\mathbf{c}=0$$

Combined Angular Momentum Operator
(Aligned Case)

$${\overrightarrow{\xi }}_{n\ell m}(r,\theta ,\phi )={\overrightarrow{\xi }}_{n\ell }(r)Y_{\ell }^m(\theta ,\varphi )⟺\underset{\text{this is a }\mathbf{\text{tensor product}}!}{\underbrace{{\tilde{R}}_{n,n^{\prime },m,m^{\prime }}=R_{n,n^{\prime }}{J}_{m,m^{\prime }}}}$$

The Misaligned Angular Momentum Operator

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

$⟹$ 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.

$$(\left[\begin{array}{cc}{\mathbf{L}}_{\pi \pi }& {\mathbf{L}}_{\pi \gamma }\\ {\mathbf{L}}_{\pi \gamma }^T& {\mathbf{L}}_{\gamma \gamma }\end{array}\right]\otimes {\mathbb{1}}_{2\ell +1}+2\omega \left[\begin{array}{cc}{\tilde{\mathbf{R}}}_{\pi }& 0\\ 0& {\tilde{\mathbf{R}}}_{\gamma }\end{array}\right]+{\omega }^2\left[\begin{array}{cc}\mathbb{1}& \mathbf{D}\\ {\mathbf{D}}^T& \mathbb{1}\end{array}\right]\otimes {\mathbb{1}}_{2\ell +1})\mathbf{x}=0.$$

TESS w/ Pixel-Level Decorrelation + Stitching

Observational Overview