Recent Advances in Gravitoacoustic Mixed-Mode Asteroseismology

Colloquium @ IU Bloomington | Slides at

Joel Ong

January 27, 2023

Background Asteroseismology Per Se

Whence Asteroseismology?

Some Observational Facts

from Jeffery & Saio (2016)

Animation: NASA

Power spectra of MDI dopplergrams

\[ \begin{aligned} \Delta\nu &\sim \sqrt{M / R^3} \\ \nu_\text{max} &\sim M / R^2 \sqrt{T_\text{eff}} \end{aligned} \]

Mode frequencies satisfy \(\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_l(\nu)\right) + \mathcal{O}(1/\nu)\)

How does this relate
to stellar structure?

Simple Wave Equation

\[\huge {\partial^2 \over \partial t^2}f = c_s^2 \nabla^2 f\]

\[\huge -\omega^2 f = c_s^2 \nabla^2 f\]

\[\large \nabla^2 f = -{\omega^2 \over c_s^2} f \equiv -|\mathbf{k}|^2 f\]

Dispersion relation: \[\omega^2 = c_s^2 |\mathbf{k}|^2\]

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 \(k^2(r, \omega) < 0\).

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

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

\[\small\color{blue}S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\]

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

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

\[{\color{gray} \omega_p > S_l, N}\]

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

\[ \begin{aligned} \oint \mathbf{k} \cdot \mathrm d \mathbf{r} &= 2 \pi \left(n' - {1 \over 2}\right) \\ 2 \omega \int_0^R {\mathrm d r \over c_s} &= 2 \pi \left(n + {l \over 2} + \epsilon_l\right) \end{aligned} \]

\[\large\nu = {1 \over 2T}\left(n + {l \over 2} + \epsilon\right)\]

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

\[{\color{red} \omega_g < N, S_l}\]

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

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

Evolved stars dominate our asteroseismic sample!

Kepler Sample (from Yu+ 2020)
TESS ATL (from Schofield+ 2019)

Evolved Stars Exhibit Mixed Modes

Main Sequence
Red Giant

(proxy for age \(\to\))

Mixed modes exhibit avoided crossings
between underlying p- and g-modes.

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

Pure g-modes: \[{1 \over \nu_{n,l}} \sim \Delta\Pi_l \left(n_g + {l \over 2} + \epsilon_{g, n,l}\right)\]

Over the course of post-main-sequence evolution, \(N^2\) increases dramatically —
an interior g-mode cavity develops.

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

\[\tiny {\color{gray}\sin \left[\theta_p(\nu)\right] = 0};\ {\color{red}\cos \left[\theta_g(\nu)\right] = 0}\]

\[\tiny {\color{gray}\sin \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right] = 0};\ {\color{red}\cos \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right] = 0}\]

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

  • Requires JWKB approximation — when is this suitable?
  • Is there a general analytic construction without Cowling approximation?
  • How to incorporate structure perturbations?

(Coupling between oscillators)

adapted from Deheuvels and Michel (2011)

Relationship between \((\omega_i, \alpha_i)\) and stellar structure unclear…

A New Analytic Treatment of Mixed Modes (Ong & Basu 2020; ApJ, 898, 127)

LCAOs give Molecular Orbitals

I stole this from a chemistry textbook

\[\Large \hat H \psi_n = \left(\hat T + \hat V\right)\psi_n = E_n \psi_n\]

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

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

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

\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} + {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 - \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]

\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} - {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 + \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]

Energies and Mixing Coefficients
of Molecular Orbitals

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

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:
\[ \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \] \[ \small L_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i}, \hat{\mathcal{L}}{\color{red}\xi_j}\rangle;~~~~D_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i}, {\color{red}\xi_j}\rangle \]

(Ong & Basu 2020; now also in GYRE v6)

Operationalising Asteroseismology Asteroseismology as an Instrument, or, “How do we use this?”

Data: \(y_\text{obs} \in Y\)

Models: \(x_i \in X\);\[F: X \to Y\]

Best-fitting model: \[x = \mathop{\mathrm{argmax}}_{x_j \in X}\ \mathcal{L}\left(x_j\right)\]

\[F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}\]

\[ \color{red} \text{and } \Delta\nu, \nu_{\text{max}}, \left\{\nu_{n,l}\right\} \]

The Forward Problem: Global Parameter Estimation

Hare “Zebedee”, Cunha+ 2021

Precise measurements of field stars: \[ {\sigma_R \over R} \lesssim 2 \%; {\sigma_M \over M} \lesssim 5 \% \]

The Inverse Problem: Model-independent measurements

From Basu (2020)

Stellar Surfaces
are Complicated.

For the Sun, even standard solar models don’t give the right frequencies!

The asteroseismic “surface term”: a (mostly) smooth function of frequency.

Mixed Modes and
the Surface Term (Ong, Basu & Roxburgh 2021, ApJ 920, 8; Ong, Basu, Lund et al. 2021, ApJ 922, 18)

The Surface Term as a
Structural Perturbation

Consider two stars with identical \(M\) and \(R\),
differing only in the near surface layers: \[\hat{\mathcal{L}}_0\xi_{0,n} = -\omega_{0,n}^2 \xi_{0,n};~~~~\hat{\mathcal{L}}\xi_{n} = -\omega_{n}^2 \xi_{n}.\] We write \[\hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}_\text{surf}},\] so that \(\lambda\) interpolates linearly between the two structures.

\[\Large \hat{\mathcal{V}}_\text{surf}{\color{red}\xi_\gamma} \to 0\]

\({\color{red}\gamma\text{-modes}}\) are confined to the stellar interior,
so unaffected by surface term

How is this compatible with a smooth function of frequency?

Mixing coefficients: customarily defined so that \(\zeta \to 1\) for pure g-modes, and \(\to 0\) for pure p-modes

The Traditional Approximation to Mode Coupling

\[\huge \times\]

Apply a partial correction: \(\delta\nu_\text{surf, mixed} = {\color{red} \delta\nu_\text{surf}} \times {\color{blue}(1 - \zeta)}\)

Order of Operations matters!

\(\delta\nu_\text{mixed} = (1 - \zeta)\delta\nu_\text{surf}\)

These two operations do not necessarily commute.

What does this do to inference?

\[\tiny{\text{Numerical Experiment: Various Kepler/K2/TESS subgiants}; N = 47;\ \ell=0, 1, 2}\]

First-order vs. Full Mode Coupling

First-order vs. Full Mode Coupling

First-order approximation yields
systematically lower stellar masses!
(population-level systematic error)

(two different
correction techniques)

Stellar Rotation (Ong, Bugnet & Basu 2022; Ong & Gehan, in review; Ahlborn, Ong, et al., in prep.)

Why Rotation?

  • Gyrochronology: rotation gives ages (e.g. Skumanich 1972).
  • Rotation tells us about stellar activity (e.g. Noyes+ 1984)
  • Rotation needed to fully explain stellar evolution (e.g. Pinsonneault+ 1992)

Almost all rotational measurements are at the stellar surface.
Interior rotation is much more poorly constrained.

Three quantum numbers \(n, l, m\): \[ \begin{aligned} l &= 0, 1, 2, \ldots \\ m &= -l, -l+1, \ldots, l-1, l \end{aligned} \]

Zonal (\(m = 0\))

Prograde sectoral
(\(m = +l\))

Retrograde sectoral
(\(m = -l\))

Rotational Splitting

For slow rotators (\(\Omega \ll 2\pi\Delta\nu\)),

\[\boxed{\delta\omega_{nlm} \sim m \beta_{nl} \int \Omega(r) K_{nl}(r) \mathrm d r}\]

(\(\equiv m \Omega \beta_{nl}\) for solid-body rotation).

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

Sensitivity of pure p-modes is distributed
throughout entire stellar structure
(but concentrated at surface)

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

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

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

Observed g-mixed modes probe core rotation well
(e.g. Mosser+ 2015; Gehan+ 2018)

Even the most p-dominated mixed modes
are sensitive to core rotation!

If for pure p-modes \(\delta \omega_p \sim m \beta_p \Omega_\mathrm{env}\)
and for pure g-modes \(\delta \omega_g \sim m \beta_g \Omega_\mathrm{core}\),
then surely for mixed modes,
\(\delta \omega_\mathrm{mixed} \sim m \left(\zeta \beta_g \Omega_\mathrm{core} + (1 - \zeta) \beta_p \Omega_\mathrm{env}\right)\)?

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

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

Existing seismic rotational measurement techniques
have considerable difficulty
simultaneously returning core and envelope rotation rates
in evolved stars.

(Ong & Basu 2020)

\[\small\psi_\text{mol} = {\color{blue}\sum_i c_{1,i} \psi_{1,i}} + {\color{orange}\sum_j c_{2,j} \psi_{2,j}}\]


\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]

Rotation gives a Quadratic Hermitian Eigenvalue Problem
(Ong et al. 2022):

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

Rotational Coupling Matrices

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!)

The Rotational Inverse Problem

\[ {\color{blue}\delta\omega_i} = {\color{olive}m \beta_i \int_0^R} {\color{orange}\Omega(r)} {\color{olive}K_i(r) \mathrm d r} + {\color{red}{\mathcal{O}(\Omega^2)}} \sim \sum_j {\color{olive}A_{ij}} {\color{orange}\Omega_j}. \]

\[ \text{(linear problem of the form $\mathbf{{\color{olive}A}{\color{orange}x}} = \mathbf{{\color{blue}b}}$)} \iff {\color{orange}\Omega_j} \sim \sum_i {\color{olive}A^{+}_{ji}} {\color{blue}\delta\omega_i}? \]

Rotational Inversion?

OLA (Backus & Gilbert 1968; Gough 1985;
Pijpers & Thompson 1992; Schunker 2016; etc.):

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

Ong, Bugnet, Basu (2022):

  • We can do this with mixed modes under certain conditions
  • Explicit numerical procedure based on p/g decomposition

Noncommuting operations again!

\(\pi/\gamma\) basis of kernels has different localisation properties.

The Way Forward

  1. Decompose mixed modes into \(\pi/\gamma\) modes (✔️: Ong & Basu 2020)
  2. Constrain structure from mixed modes (✔️: Ong+ 2021a, b)
  3. Infer splittings & kernels for \(\pi\)/\(\gamma\)-modes (✔️: Ong, Bugnet & Basu)
  4. Perform standard linear inversion in modified basis (*WIP)

Practical Testbed: KIC 9267654 (RGB)

Nonmonotonic rotational profile!
(Tayar+ 2022)

Practical Testbed (TRGB)


Decoupling of Mixed Modes:

Decomposition of wave operator into purely acoustic/buoyant propagation terms and their remainder operators
permits closed-form evaluation of coupling matrix elements.

Ong and Basu (2020)

Mixed Modes and the Surface Term:

Traditional surface term corrections handle mode coupling
only to first order, if at all.

More sophisticated techniques are required
as stars get more evolved.

Inference of stellar masses and compositions depends on
how mode mixing is treated
when correcting for the surface term.

Ong, Basu, Roxburgh (2021); Ong, Basu, Lund, et al. (2021)

Mode Mixing and Rotation:

Rotational characterisation techniques which
do not work on mixed modes may still be applied
if they are decomposed into \(\pi/\gamma\) basis.

We describe how this is to be done.

Ong, Bugnet, Basu (2022)


We generalise conventional surface corrections to apply to mixed modes, by analytically decoupling mixed modes into p- and g-like components. We apply these techniques to the problem of correcting the asteroseismic surface term, and to the task of internal rotational measurement.

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

Supplementary Slides

Noncommuting Operations in Surface Term Corrections

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Full matrix coupling
\(\lambda\) from \(0 \to 1\)

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Traditional approximation to coupling

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Full vs. Traditional mode coupling

Young red giant: \(\Delta\nu = 17.2\ \mu\text{Hz}\)

Evolved red giant: \(\Delta\nu = 3.9\ \mu\text{Hz}\)

Surface-term corrections
should not reorder mixed modes!
(cf. Ball+ 2018)

traditional approx. OK

perturbative series diverges


Ignore mixing altogether? (a la Ball et al. 2018)

Correcting the Surface Term
for Mixed Modes

  1. Determine an appropriate correction for p-modes
  2. Construct modified coupling matrices
  3. Solve the perturbed GHEP:

\[ \left( \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + \lambda \begin{bmatrix} {\color{blue}\mathbf{V}} & 0 \\ 0 & 0 \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \]

Does this actually change our results?

TOI 197

All the surface corrections yield similar best-fit models;
main discrepancies arise in the wings of the posterior distribution


Single-target Systematics

Joint posterior distributions for TOI 197
(reference values: Huber+ 2019)

(Ong+ 2021c)

Single-target Systematics with Luminosity constraint

Avoided Crossings and Rotation