How do we know anything?

(RHD simulations courtesy of Joel D. Tanner)

…and not all interesting stars are bright.

Seismology as a Tool

Global Properties give Masses and Radii

Many Mode Frequencies constrain Structure

A Crash Course in Wave Propagation:
The JWKB Approximation

Simple Wave Equation

More Complicated Wave Equation

LCAOs give Molecular Orbitals

Energies and Mixing Coefficients
of Molecular Orbitals

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

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

Mixed Modes as
Acoustic “Molecular Orbitals”

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

(Ong & Basu 2020)

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)

g-modes: Core Magnetism

What about p-modes?

The Small Frequency Separation \(\delta\nu_{02}\)

Integral estimator from asymptotic analysis
(Tassoul 1990, Roxburgh & Vorontsov 1994):

from White+ (2011)

from White+ (2011)

(Ong & Basu 2020)

Where does this asymptotic expression
even come from?

What went wrong?

In deriving the classical estimator, one assumes \[ \begin{aligned} \delta_\ell(\omega,t) \sim \arctan\left(\Theta + \mathcal{O}\left(1\over \omega^3\right)\over 1 - {\ell(\ell+1) \over 2\omega t}\Theta + \mathcal{O}\left(1\over \omega^2\right)\right) &\sim \Theta; \\ \implies \exists t_0 \in (0, T), \text{ s.t. } \Theta(t_0) &\ll 1. \end{aligned} \] How valid is this assumption?

Remediation

The small-angle approximation assumes \[ \begin{aligned} \delta_\ell + {\pi \over 2} &= \arctan\left(\Theta + \mathcal{O}\left(1\over \omega^3\right)\over 1 - {\ell(\ell+1) \over 2\omega t}\Theta + \mathcal{O}\left(1\over \omega^2\right)\right) \\ &= \arctan\left[\Theta - {\ell (\ell + 1) \over 2 \omega t}\right] + \arctan\left[{\ell (\ell + 1) \over 2 \omega t}\right] &\sim \Theta. \end{aligned} \] If this approximation is not made, then one obtains that

Remediation

Inner Phase Shifts

Interpreting Phenomenology

The “knee” in the JCD diagram occurs
when the convective envelope’s boundary
passes a specific distance from the centre of the star:
about 1/3 of a wavelength at \(\nu_\text{max}\).

Observational Confirmation!

(RHD simulations courtesy of Joel D. Tanner)

Sensitivity to Overshoot

TESS GI (funded for Cycle 7!)

Isolation and Coupling of Mixed Modes

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

Resolving an Asteroseismic Catastrophe

We attribute singular behaviour in an integral estimator
for the small separation \(\delta\nu_{02}\) to
a broken small-angle approximation.
(Ong et al., submitted to ApJ)

Understanding Mixed Modes (for Fun and Profit)

Access to g-modes has usually been the main focus
when studying gravitoacoustic mixed modes.

Where does this asymptotic expression
even come from?

1. The Pulsation Equations

\[\scriptsize{1 \over r^2}{\mathrm d \over \mathrm d r}\left(r^2 {\color{red}\xi^r}\right) - {g \over c_s^2} {\color{red}\xi^r} = - {{\color{forestgreen}P'}\over \rho_0 c_s^2}-{\ell(\ell + 1)\over \rho_0\omega^2 r^2}\left({\color{forestgreen}P'} + \rho_0 {\color{orange}\phi'}\right)\]

\[\scriptsize {\mathrm d {\color{forestgreen}P'} \over \mathrm d r} + {g \over c_s^2} {\color{forestgreen}P'}= \rho_0 (\omega^2 - N^2) {\color{red}\xi^r} - \rho_0 {\color{blue}{\mathrm d \phi'\over \mathrm d r}}\]

\[\scriptsize{1 \over r^2}{\mathrm d \over \mathrm d r}\left(r^2 {\color{blue}{\mathrm d \phi'\over \mathrm d r}} \right) = {\ell(\ell + 1) \over r^2} {\color{orange}\phi'} + 4 \pi G \left({{\color{forestgreen}P'} \over c_s^2} + {\rho_0 N^2\over g}{\color{red}\xi^r}\right)\]

\[\scriptsize {\mathrm d {\color{orange}\phi'}\over \mathrm d r} = {\color{blue}{\mathrm d \phi'\over \mathrm d r}}\]

2. Change of Variables

With judicious change of dynamical variables
to \(\xi, \eta, P, S\), one may write as \[ \left({\mathrm{d}^2\over\mathrm{d}r^2} + \mathbf{C}{\mathrm{d}\over\mathrm{d}r} + \mathbf{D}\right)\begin{bmatrix}\eta\\P\end{bmatrix} = 0. \]

Asymptotic Behaviour of the Eigenfunctions

3. Ansatz: Asymptotic Expansion

\[ \begin{bmatrix}\eta\\P\end{bmatrix} \sim \left(\mathbf{Y}_0 + {1\over\omega}\mathbf{Y}_1 + {1\over\omega^2}\mathbf{Y}_2 + \ldots \right)\begin{bmatrix}J_{\ell + {1\over2}}(\omega t)\\J'_{\ell + {1\over2}}(\omega t)\end{bmatrix}. \] One solves for \(Y_{i,jk}\) by inserting this expression back into 2nd-order ODE and exploiting the recurrence relation \[ {\mathrm d \over \mathrm d t}\begin{bmatrix}J_{\ell + {1\over2}}(\omega t)\\J'_{\ell + {1\over2}}(\omega t)\end{bmatrix} = \begin{bmatrix}0 & \omega \\ - \omega + {\left(\ell + {1\over 2}\right)^2\over \omega t^2} & -{1\over t}\end{bmatrix}\begin{bmatrix}J_{\ell + {1\over2}}(\omega t)\\J'_{\ell + {1\over2}}(\omega t)\end{bmatrix}. \]

4. Hankel’s Expansion for Bessel Functions

For \(x \gg 1\), \[ \small \begin{aligned} J_{\ell + {1\over2}}(x) &\sim \sqrt{2\over \pi x} \left[\sin\left(x - {\ell \pi \over 2}\right) + {\ell(\ell+1)\over 2x}\cos\left(x - {\ell \pi \over 2}\right) \right] + \mathcal{O}\left(1\over x^{5/2}\right);\\ J'_{\ell + {1\over2}}(x) &\sim \sqrt{2\over \pi x} \left[\cos\left(x - {\ell \pi \over 2}\right) - {\ell(\ell+1) + 1\over 2x}\sin\left(x - {\ell \pi \over 2}\right) \right] + \mathcal{O}\left(1\over x^{5/2}\right),\label{eq:hankel} \end{aligned} \]

(reminder: phasor addition identity)

\[a \sin \phi + b \cos \phi = \sqrt{a^2 + b^2} \sin \left(\phi + \arctan \left[b \over a \right]\right)\]

5. Frequency Eigenvalues

Defining \(\psi = \xi_r \sqrt{r^2 c_s \rho},\) \[ \begin{aligned} \psi_\text{inner} &\propto \left[{1\over\omega}\left(-{y_{1,12}(t)\over y_{0,11}(t)} + {1 \over 2}{\mathrm d \over \mathrm d t}\log\left[c_s \over h r^2\right] + {1 \over 2t}\right) + \mathcal{O}\left(1\over\omega^3\right)\right]J_{\ell + {1\over2}}(\omega t) \\&+ \left[1 + \mathcal{O}\left(1\over\omega^2\right)\right]J'_{\ell + {1\over2}}(\omega t) \\&\propto \cos\left[\omega t - {\ell \pi \over 2} + \arctan\left(\Theta + \mathcal{O}\left(1\over \omega^3\right)\over 1 - {\ell(\ell+1) \over 2\omega t}\Theta + \mathcal{O}\left(1\over \omega^2\right)\right)\right]; \\\Theta &= {1\over\omega}\left[A_0(t) + \ell(\ell+1)A_\ell(t)\right].\label{eq:arctan} \end{aligned} \]

5. Frequency Eigenvalues (ctd.)

Defining \(\psi = \xi_r \sqrt{r^2 c_s \rho},\) \[ \small\psi_\text{inner} \equiv A(t) \sin \left[\omega t - {\ell \pi \over 2} + \delta_\ell(\omega, t)\right] \]

In the atmosphere, one may show that \[ \small\psi_\text{outer} \sim A(t) \sin \left[\omega (t - T) + \alpha_\ell(\omega, t)\right] \]

If \(\omega\) is an eigenvalue,
the arguments of these sinusoids must match up to sign: \[\small\omega_n T + \delta_\ell(\omega_n, t) - \alpha_\ell(\omega_n, t) = \pi\left(n + {\ell \over 2}\right).\]