Prospects for Seismic Probes of Radial Differential Rotation

NHFP Symposium 2022

Joel Ong

15 September 2022 | Slides at http://hyad.es/talks

Motivation

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.

Measuring Rotation

(sun-like star)

Spectroscopic \(v \sin i\):
\(\sim \text{PHz}\) regime \(\to\)

Photometric \(P_\text{rot}\):
\(\ll 1\ \mu\text{Hz}\)
\(\downarrow\)

Power spectra of MDI dopplergrams

Main-sequence stars exhibit p-modes:
pure pressure waves.

Evolved stars dominate our asteroseismic sample!

Kepler Sample (from Yu+ 2020)

Mixed Modes and Rotation

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).

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

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

“pure” g-mode
“pure” p-mode

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.

Mode Coupling and Rotation

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

\[\iff\]

(Ong & Basu 2020)

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

Ong, Bugnet & Basu (in review):
generalisation to include rotational effects

Access to coupling matrices
permits recovery of isolated “pure”
p- and g-mode splittings.

The Inverse Problem

Sketch of (One) Technique

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 (in review):

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

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+, in review)

Further Steps

  • Less evolved stars
  • Larger sample
  • Actual astrophysics??

Summary

Mixed Modes and Rotation

We have derived a semi-analytic description of rotation
in the presence of mode mixing.

This naturally generalises the existing rotational-inversion construction to mixed-mode oscillators.

We are in the midst of testing this new procedure
on an interesting science case.

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

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

Backup Slides II:
Mode Mixing Matrices

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)

Backup Slides III:
The Rotational Inverse Problem

The Rotational Inverse Problem

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

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

Example: Optimally Localised Averages

\[\scriptsize T(x; x_0, \mathbf{\Theta}) \approx \sum_j{\color{olive} {c_j(x_0, \mathbf{\Theta}) K_j(x)}} \implies {\color{orange}{\Omega(x_0)}} \approx \int T(x; x_0, \mathbf{\Theta}) {\color{orange}{\Omega(x)}} \mathrm d x = \sum_j {\color{olive}{c_j(x_0, \mathbf{\Theta})}} {\color{blue}{\delta \omega_j}}\]

Rotational Coupling Matrices

Noncommuting operations!

Backup Slides IV:
Why not \(\gamma\) modes/
core rotation?