Mixed Modes
in Theory and Practice

The Good Vibrations Seminar

Joel Ong

1 December 2021 | Slides at http://hyad.es/talks


  1. Introduction
  2. Analytic Construction
  3. Application: the Surface Term
  4. Application: Slow Rotation

I. Background

Power spectra of MDI dopplergrams

Evolved Stars Dominate our Asteroseismic Sample!

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

Mixed Modes

Main Sequence
Red Giant

(proxy for age \(\to\))

(proxy for age \(\to\))

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

(proxy for age \(\to\))

(proxy for age \(\to\))

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

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

II. Understanding
Mixed Modes

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

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

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

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

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

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

\[\tiny \psi(r) \sim {1 \over \sqrt{k_r}} \left(A \sin \left[\int^r k_r\ \mathrm d r\right] + B \cos \left[\int^r k_r\ \mathrm d r\right]\right)\] \[\tiny \boxed{\tan \Theta_p \cot \Theta_g = {1 \over 4} \exp\left[-2 \int_{r_g}^{r_p} \kappa \ \mathrm d r\right]} \]

(Coupling between oscillators)

from Deheuvels and Michel (2011)

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

A Different
Analytic Formulation

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

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

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

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

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

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

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

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

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

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)

III. The Surface Term

Standard Solar Models reproduce the
internal structure of the Sun

From Basu (2020)

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

\(\delta M / M \gtrsim 5\%\):
larger than statistical error!

The asteroseismic “surface term”

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 \[\hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}},\] so that \(\lambda\) interpolates linearly between the two structures.

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

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

Order of Operations matters!

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

These two operations do not necessarily commute.

Evolved red giant: \(\Delta\nu = 3.9\ \mu\text{Hz}\)
Full vs. Traditional mode coupling

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

First-order vs. Full Mode Coupling

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

(two different
correction techniques)

Single-target Systematics

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

(Ong+ 2021c)

Mixed Modes and the Surface Term

Analysis of mixed modes in terms of the decoupled \(\pi\)- and \(\gamma\)-mode basis is necessary to avoid systematic errors in determining
global properties from asteroseismology,
for both single-target measurements and population statistics.

IV. (Slow) Rotation

Spherical Harmonics

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.

Zonal + \(\text{\color{orange}prograde}\) and \(\text{\color{blue}retrograde}\) sectoral
\(\pi\)-modes and \(\gamma\)-modes

\[\text{Asymmetry: }\psi = {(\nu_+ - \nu_0) - (\nu_0 - \nu_-) \over \nu_+ - \nu_-} = 0\]

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

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

Dipole modes (\(l = 1\))

\(\Delta\nu = 17\ \mu\)Hz; \(\Omega_\text{core}/\Omega_\text{surf} = 10\)

Quadrupole modes (\(l = 2\))

Dipole modes (\(l = 1\))

\(\Delta\nu = 5\ \mu\)Hz; \(\Omega_\text{core}/\Omega_\text{surf} = 10\)

Buried B-field: \(\psi = 0.28\)
(from Bugnet+ 2021)

Mixed Modes and Rotation

Avoided crossings must be accounted for
to make credible statements about stellar rotation
and magnetic fields in evolved stars.

The Future

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

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

Rotational kernel for p-dominated mixed mode

\[ \small \begin{aligned} \delta\omega_i = m \beta_i \int_0^R \Omega(r) K_i(r) \mathrm d r &\sim \sum_j A_{ij} \Omega_j \\ \implies \Omega_j &\sim \sum_i A^{+}_{ji} \delta\omega_i? \end{aligned} \]


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 for evolved stars,
to avoid systematic errors in global properties.

Ong, Basu, Roxburgh (2021); Ong, Basu, Lund, Viani, Bieryla, Latham (2021)

Mixed Modes and Rotation:

Radial differential rotation yields
asymmetric rotational splitting on mixed modes.

Symmetrisation in \(\pi/\gamma\) basis is mandatory
for accurate interpretation of observed splitting
(e.g. correct diagnosis of magnetic fields).

Ong, Bugnet & Basu (in prep.)


We analytically decouple mixed modes into \(\pi\)- and \(\gamma\)-like components. This permits the analysis and interpretation of both the surface term, as well as of rotational splitting, in evolved stars.

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

Backup Slides I:
Surface Corrections (see Basu+ 2018, Jørgensen+ 2020 for review)

(Fully) calibrated corrections

\[\nu_\text{model} \mapsto \nu_\text{corr} = \nu_\text{model} + \delta\nu_{nl,\text{surf}},\] with corrections depending only on the model as \(\delta\nu_{nl,\text{surf}} \sim f(\nu_{nl}; x)\).

Parametric corrections

Corrections with free parameters \(\theta \in \Theta\) as \(\delta\nu_{nl,\text{surf}} \sim f(\nu_{nl}; x; \theta)\).

\[\tiny {\text{e.g. Ball \& Gizon (2014) correction: }\delta\nu_{nl,\text{surf}} \sim \left(a_{-1} \left(\nu_{nl}\right)^{-1} + a_{3} \left(\nu_{nl}\right)^{3}\right) / I_{nl}}\]

Alternative Approach I:
Separation Ratios

\[\begin{aligned} \text{e.g. } r_{02, n} = {\nu_{n,0} - \nu_{n-1,2} \over \nu_{n,1} - \nu_{n-1,1}} \end{aligned}\]

Interpretation as surface-independent functions of frequency:
use these to constrain model directly (Otí Floranes et al. 2005)

\[\begin{aligned} r_{02, n} & \sim \delta_2(\nu_{n,0})\\ r_{01, n} \sim \delta_1(\nu_{n,0})&; ~~~ r_{10, n} \sim \delta_1(\nu_{n,1}) \end{aligned}\]

Alternative Approach II:

\[\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_{nl}\right), \implies \epsilon_{nl} = {\nu_{nl} \over \Delta\nu} - n - {l \over 2} \equiv{\epsilon_l(\nu_{nl})}\] \[ \text{Compute } \mathcal{E}_l(\nu^\text{obs}_{nl}) = \epsilon^\text{obs}_{l}(\nu^\text{obs}_{nl}) - \epsilon^\text{model}_{l}(\nu^\text{obs}_{nl}). \]

All of the \(\mathcal{E}_l\) should collapse to a single function if the structural differences are localised to the surface (Roxburgh 2016).

Nonparametric Treatments

Use of transformed variables such that \[O_{nl,\text{surf}} \sim f(\nu_{nl})\]

(where the structure of \(f\) is left underspecified)

Interpolation required to compare
\(f^\text{obs}(\nu_\text{obs})\) vs. \(f^\text{model}(\nu_\text{obs})\) (instead of \(f^\text{model}(\nu_\text{model})\)).

Backup Slides II:
Comparing Surface Corrections

from Basu & Kinnane (2018)

Differences between surface corrections

For each parameter \(P\), we consider normalised differences \[z_{P, \text{BG14 vs. other}} = {\mu_{P, \text{other}} - \mu_{P, \text{BG14}} \over \sqrt{\sigma^2_{P, \text{BG14}} + \sigma^2_{P, \text{other}}}}\]

We examine the distribution of these differences
over a large* sample of stars, ceteris paribus

*Not actually very large


Main-sequence stars

\[\tiny{(\text{Kepler LEGACY sample}: N = 66)}\]

General agreement on the
inferred masses…

\(z\)-score for mass

…and on the inferred radii.

\(z\)-score for radius

Parameter estimates generally
agree quite robustly…

\(z\)-score for age

… but not for all parameters.

\(z\)-score for initial helium abundance


First-ascent red giants

\[\tiny{(\text{NGC 6791}: N = 29; l=0,2\text{ only})}\]

Nonparametric methods appear
to agree with each other…

\(z\)-score for initial helium abundance

…but not with our
fiducial parameterisation…

\(z\)-score for mass

…and these offsets appear
to be systematic.

\(z\)-score for age

What’s going on?

NGC 6791 is an open cluster — what about the
actual distribution of stellar properties?

Stars should be coeval, so red giants should be of similar mass and age.

Disagreements on the age scale?

Disagreements on the mass distribution?

from Jørgensen et al. (2020)

Other parametric methods yield comparable internal scatter!

Different treatments of the surface term exhibit qualitative systematic differences for main-sequence stars vs. red giants.

What happens in between?

Backup Slides III:
Outer Turning Points

Backup Slides IV:
Generalising Classical Parameterisations

The Surface Term as a
Matrix Perturbation

From operator perturbation problem, \(\hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}},\) construct corresponding matrix perturbation problem by evaluating \[V_{ij} = \int \rho\ \vec{\xi}_{\pi,i} \cdot \hat{\mathcal{V}}\vec{\xi}_{\pi,j}\ \mathrm d^3 x.\]

\[\text{``Variational" expression: }\delta\omega_i^2 \sim {\int \rho\ \vec \xi_i\ \delta\hat{\mathcal{L}} (\vec \xi_i)\ \mathrm d^3 x \over \int \rho\ |\vec \xi_i|^2\ \mathrm d^3 x}\]

Matrix Parametrisations
for the Surface Term

So long as the off-diagonal entries of the matrix \(\mathbf{V}\) can be specified, existing parametrisations can adapted as (e.g. for BG14): \[ \large \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} + \mathbf{V}_{(a_{-1}, a_{3})} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \]

Backup Slides V:
\(\zeta\) and Rotation