Seismic Signatures of
Core-Envelope
Misalignment
MIAPbP 2023 | Slides at http://hyad.es/talks
August 8, 2023
Albrecht+ 2021:
A Preponderance of Perpendicular Planets
Saunders+ (in
prep.)
From De+ 2023
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
\(\implies\) equipartition
between different \(m\)
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
- used to study,
radial differential rotation.
e.g. Rotational Inversions
\[\scriptsize\begin{aligned} \int \Omega(r) {\color[RGB]{0,100,255}\left(\sum_i c_i K_{i}(r)\right)} \mathrm d r &\sim \sum_{i} \left(c_i \over m_i b_{i} \right) \delta\omega_{\mathrm{rot}, i} \\ & \to \boxed{\Omega({\color[RGB]{0,100,255}r_0})}\end{aligned}\]
OLA
(Backus & Gilbert 1968; Gough 1985;
Pijpers & Thompson 1992; Schunker 2016; etc.)
How will this change,
accounting for misalignment?
Two Misaligned Zones
Let’s say \[\mathbf{\Omega}(r) = \Omega_0 \times \left\{\begin{matrix}\mathbf{e}_z, & r < r_0 \\ \hat{\mathbf{n}}, & r > r_0\end{matrix}\right.,\] s.t. \(\cos \beta = \mathbf{e}_z \cdot \hat{\mathbf{n}}\).
\[\text{Define}\ \alpha = \int_{r_0}^R
K(r)\ \mathrm d r\] (i.e. \(\alpha\) varies
from multiplet to multiplet).
Kepler-56: Red giant
with multiple planets,
misaligned axis, and
outer companion.
(TTV and RV measurements: Huber+ 2013)
\(\star\)
Kepler-56: Rotational Splittings
adapted from Huber+ 2013
\(\delta\nu_\text{rot}/\mu\mathrm{Hz}\)
\(i/^\circ\)
adapted from Huber+ 2013
\[i_g = 43 \pm 4 \ ^\circ\]
\[i_p = 51 \pm 4 \ ^\circ\]
Potential misalignment?
If for pure p-modes \(\delta \omega_p \sim
m b_p \Omega_\mathrm{env}\)
and for pure g-modes \(\delta \omega_g \sim m
b_g \Omega_\mathrm{core}\),
then surely for mixed modes,
\(\delta \omega_\mathrm{mixed} \sim m
\left(\zeta b_g \Omega_\mathrm{core} + (1 - \zeta) b_p
\Omega_\mathrm{env}\right)\)?
Mode mixing yields avoided crossings
between multiplet components of identical \(m\)
(cf. Mosser+ 2012, Ouazzani+ 2013, Deheuvels+ 2017)
How to incorporate misalignment into nonlinear treatment of mode coupling?
JWKB Analysis?
\[{k_r^2 \sim {\omega^2 \over c_s^2} \left(1 - {{\color{blue} S_\ell}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[ {\color{gray}\tan \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right]} {\color{red}\cot \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right]} = {1 \over 4} \exp \left[-2 \int_{r_{2,g}}^{r_{1,p}} |k_r| \mathrm d r\right]\]
Asymptotic analysis performed after separation of variables. (i.e. a priori alignment throughout star)
(Ong & Basu 2020)
\[\small\psi_\text{mol} = {\color{blue}\sum_i c_{1,i} \psi_{1,i}} + {\color{orange}\sum_j c_{2,j} \psi_{2,j}}\]
\[\iff\]
\[\small\vec{\xi}_\text{mixed} \sim {\color{grey}
\sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j}
\vec{\xi}_{\gamma,j}}\]
Rotation gives a Quadratic Hermitian Eigenvalue
Problem
(Ong et al. 2022):
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
\(\beta = 0\)
\(\beta = {\pi\over10}\)
\(\beta = {\pi\over2}\)
\(\beta = \pi\)