Analytical Methods for Muticavity Oscillations

Joel Ong
Hubble Fellow, Univ. of Hawaiʻi at Mānoa

IAStrophysical Waves; January 14, 2025

I.
(Linear) Perturbations

Separation of Variables

\[\left<\mathbf{x}| \xi_{n \ell m}\right> \hat{=} \vec{\xi}_{n \ell m}(r) Y_\ell^m (\theta, \phi)\]

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

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

Under a small perturbation \(\hat{\mathcal{L}} \mapsto \hat{\mathcal{L}} + \lambda \hat{\mathcal{V}}\),

\[\begin{aligned} \omega^2_{n\ell m} &\mapsto \omega^2_{n\ell m} - \lambda \left<\xi_{n\ell m}|\hat{\mathcal{V}}|\xi_{n\ell m}\right> + \mathcal{O}\left(\lambda^2\right)\\ &\approx \omega^2_{n\ell m} - \lambda \int \mathrm d^3 x \cdot \rho \left(\vec{\xi}_{n\ell m}^* \hat{\mathcal{V}} \vec{\xi}_{n\ell m}\right) + \mathcal{O}\left(\lambda^2\right). \end{aligned}\]

e.g. for (slow) rotation in particular

\[ \omega^2 \mapsto \omega^2 + 2 m\ b_{n\ell m} \omega \int \Omega(r) K_{nlm}(r) \mathrm dr \]

Rotational Inversions

For slow rotation, \[\boxed{\delta\omega_{nlm} \sim m b_{nl} \int \Omega(r) K_{nl}(r) \mathrm d r}\]

Variations in \(\delta\omega_\text{rot}\) are

assumed to imply, and
\(\therefore\) used to study,

radial differential rotation.

\[\boxed{{\color{orange}{\delta\omega_{nlm}}} \sim {\color[RGB]{0,100,255}{m b_{nl} \sum_i {\color{black}{\Omega(r_i)}} K_{nl}(r_i) \Delta r_i}}}\] is of the form \({\color[RGB]{0,100,255}\mathbf{A}}\mathbf{x} = {\color{orange}\mathbf{b}}\).

i.e. Inferring the rotational profile
\(\Omega(r)\) is a linear inverse problem.

Rotational Inversions

\[\scriptsize\begin{aligned} \int \Omega(r) {\color[RGB]{0,100,255}\left(\sum_i c_i K_{i}(r)\right)} \mathrm d r &\sim \sum_{i} \left(c_i \over m_i b_{i} \right) \delta\omega_{\mathrm{rot}, i} \\ & \to \boxed{\Omega({\color[RGB]{0,100,255}r_0})}\end{aligned}\]

e.g. OLA (Backus & Gilbert 1968; Gough 1985;
Pijpers & Thompson 1992; Schunker 2016; Ong 2024; etc.)

\(\implies\) we can make measurements of internal rotational structure!

Inversions for Stellar Structure

two standard
solar models

\[\huge {{\color{orange}{\delta\omega_i \over \omega_i}} \approx {\color[RGB]{0,100,255}{\int K_{\rho|c_s^2, i}(r)}} {\delta\rho \over \rho}{\color[RGB]{0,100,255}{\mathrm d r}} + {\color[RGB]{0,100,255}{\int K_{c_s^2|\rho, i}(r)}} {\delta c_s^2 \over c_s^2}{\color[RGB]{0,100,255}{\mathrm d r}}} \]

\[{\Large{\color[RGB]{0,100,255}\mathbf{A}}\mathbf{x} = {\color{orange}\mathbf{b}}}\]

II.
Nonlinearities

Evolved stars dominate our asteroseismic sample.

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

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

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

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

\[\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\] 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

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

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

Eckart-Scuflaire-Osaki Classification Scheme

p-modes:

\[\large\boxed{\nu = \Delta\nu\left(n + {\ell \over 2} + \epsilon_{p,\ell}\right)}\]

\[\small\Delta\nu \sim \left(2 \int {\mathrm d r \over c_s}\right)^{-1}\]

g-modes:

\[\large\boxed{{1 \over \nu} = \Delta\Pi_\ell \left(n + {\ell \over 2} + \epsilon_{g, \ell}\right)}\]

\[\small\Delta\Pi_\ell \sim {2\pi^2 \over \sqrt{\ell(\ell+1)}}\left(\int {N \over r} \mathrm d r\right)^{-1}\]

\[{k_r^2 \sim {\omega^2 \over c_s^2} \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]

\[\small {\color{red}\cos \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right] = 0}\]

\[\small {\color{gray}\sin \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right] = 0}\]

\[\small {\color{gray}\tan \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right]} {\color{red}\cot \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right]} = {1 \over 4} \exp \left[-2 \int_{r_{2,g}}^{r_{1,p}} |k_r| \mathrm d r\right]\]

(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: \[{1 \over \nu_{n,\ell}} \sim \Delta\Pi_\ell \left(n_g + {\ell \over 2} + \epsilon_{g, n,\ell}\right)\]

Mode mixing yields avoided crossings
between multiplet components of identical \(m\)

(cf. Mosser+ 2012, Ouazzani+ 2013, Deheuvels+ 2017)

pure p
pure g
mixed

Structure perturbations
do not reorder mixed modes!
(cf. Ball+ 2018)

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

LCAOs give Molecular Orbitals

\[\Large \hat H \left|n\right> = \left(\hat T + \hat V\right)\left|n\right> = E_n \left|n\right>\]

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

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

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

Energies and Mixing Coefficients
of Molecular Orbitals

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

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; Ong & Gehan 2023)

IV.
Implications

Methodological Modifications for the Inverse Problem

\(\pi/\gamma\) basis of kernels has different localisation properties compared to the natural basis of normal modes.

Observational Well-Posedness

from Ong, Bugnet & Basu 2022

Rotation matrices are essentially diagonal in \(\pi/\gamma\)-mode basis:
Entire matrix can may be meaningfully constrained observationally.

(this is not the case with mixed modes!)

Mixed Modes May Not Even Be Variable-Separable

Pulsation axes may differ between mode cavities if their rotational axes are misaligned.

e.g. Ong et al. 2024; Ong 2025

Analytical Methods for Multicavity Oscillations

New analytical tools are required to render perturbative methods tractable when applied to stars with multiple mode cavities (e.g. gravitoacoustic mixed modes in sub/red giants).

Weakly-coupled mixed modes are linear combinations of isolated modes (just as MOs are LCAOs).

Also enabling new discoveries of star-planet interactions
(planet engulfment and core-envelope misalignment).

\[\mathrm{j}\mathrm{o}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\ \text{@}\ \text{hawaii}.\text{edu}\]

Supplementary Slides

Rotational Signatures of Planetary Engulfment

Very high \(\mathrm{A(Li)}\),
but otherwise innocuous

Mode Identification

\[\ell = 0,2?\]

But Kepler says \(\ell = 0\) have to live here!
(and theory says so too…)

Asteroseismology Happens…

Rotational splittings from seismology:

\(\ell = 1\): \(\delta\nu_{\text{rot},1} \sim 0.09\ \mu\mathrm{Hz}\)
\(\ell = 2\): \(\delta\nu_{\text{rot},2} \sim 0.10\ \mu\mathrm{Hz}\)

\(\implies P_\text{rot} \sim 115\ \mathrm{d}?\)

Rotational signal probably got
detrended away
by systematic corrections…

TESS w/ Pixel-Level Decorrelation + Stitching

Very suggestive…
but is this real?

Probably yes!

Observational Overview

Asteroseismology:

\(M \sim 1.14 \pm 0.04\ M_\odot\)
\(R \sim 23.5 \pm 0.03\ R_\odot\)
\(\boxed{P_\text{rot,bulk} \sim 115 \pm 10\ \text{d}}\)
(Maybe rotational shear?)

Photometry:

\(\boxed{P_\text{rot,surf} \sim 99 \pm 3\ \text{d}}\)

Spectroscopy:

\(\boxed{V\text{sin } i \implies P_\text{rot} \sim 110 \pm 8\ \text{d}}\)
\(\mathrm{A(Li)} = 3.16 \pm 0.08\ \mathrm{dex}\)
\(^{14}\mathrm{N}\)-deficient relative to APOGEE sample

Gaia DR3:

RUWE of 1.06
\(\sigma_V = 0.16\ \mathrm{km/s}\)
over 2.2 years

How did this happen??

Gaia RV scatter rules out
large RV semiamplitudes…

\(P_\text{orb} \gg 99\ \mathrm{d}\) cannot spin star
up to 99-day rotational period…

Remaining permissible orbits are
unstable to tidal dissipation!

\(\implies\) ENGULFMENT?

Misaligned Mixed Modes

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: \(\ell = 1\)

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

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)

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

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

The Misaligned Angular Momentum Operator

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

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

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

\(\implies\) 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.

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

Parameterising Misalignment: Euler Angles

\(\beta = 0\)

\(\beta = {\pi\over10}\)

\(\beta = {\pi\over2}\)

\(\beta = \pi\)

Predicted Phenomenology

Kepler-56

From Huber et al. 2013

\[b_i(r) = b_i \int_0^r K_i(r') \mathrm d r'\]

Mixed modes probe radial differential rotation
in two zones (core vs. envelope).

Cumulative sensitivity \(b(r)\)