MIAPbP 2023 | Slides at http://hyad.es/talks
Joel Ong
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
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?
\(\forall \ell, \exists (2\ell + 1) \times
(2\ell + 1)\) matrices
\(\mathbf{J}_x^\ell\), \(\mathbf{J}_y^\ell\), \(\mathbf{J}_z^\ell\) satisfying commutation
relations
\(\left[\mathbf{J}_i, \mathbf{J}_j\right] =
-i\epsilon_{ijk}\mathbf{J}_k\), with
\(\mathbf{J}_z \hat{=} \mathrm{diag}(-\ell,
-\ell + 1 \ldots \ell - 1, \ell).\)
\[\mathbf{J}_x \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}; \mathbf{J}_y \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & i & 0 \\ -i & 0 & i \\ 0 & -i & 0\end{bmatrix}; \mathbf{J}_z \hat{=} \begin{bmatrix}-1 & 0 & 0 \\ 0 &0 &0 \\ 0& 0 &1\end{bmatrix}.\]
Let’s associate with each mass shell at \(r\) both \(\Omega(r)\)
(as is customary), and also an axis \(\hat{\mathbf{n}}(r) =\sum_i n_i
\mathbf{e}_i\).
\[ \begin{aligned} \mathbf{R}_{n\ell} &= b_{n\ell}\int {\mathbf{d}^\ell}^\dagger(\beta(r)) \Omega(r) \mathbf{J}_z {\mathbf{d}^\ell}(\beta(r))\ K(r)\ \mathrm{d} r\\ &= b_{n\ell}\int \Omega(r) (\hat{\mathbf{n}} \cdot \vec{\mathbf{J}})\ K(r)\ \mathrm{d} r\\ &= \boxed{b_{n\ell}\left(\int \vec{\mathbf{\Omega}}(r) K(r)\ \mathrm{d} r\right) \cdot \vec{\mathbf{J}}}. \end{aligned} \]
For each mode, AM matrix is specified by usual vector addition.
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).
Misalignment \(\implies\) Different apparent \(i\) per multiplet.
Geometrical degeneracy owing to
unconstrained orientation angle \(\gamma\).
Kepler-56: Red giant
with multiple planets,
misaligned axis, and
outer companion.
(TTV and RV measurements: Huber+ 2013)
\(\star\)
Linear approximation:
\[\huge\delta\omega_\text{rot} \sim \zeta \cdot {1\over2}\left<\Omega_\text{core}\right> + \underbrace{(1-\zeta) \left<\Omega_\text{env}\right>}_{\scriptsize\mathrm{usually~ignored}}\]
\[\Large\implies \alpha = {\zeta \over 2 - \zeta}\]
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?
R-M measurements might break geometrical degeneracy.
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?
\[{k_r^2 \sim {\omega^2 \over c_s^2} \left(1 - {{\color{blue} S_\ell}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[ {\color{gray}\tan \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right]} {\color{red}\cot \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right]} = {1 \over 4} \exp \left[-2 \int_{r_{2,g}}^{r_{1,p}} |k_r| \mathrm d r\right]\]
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):
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
With the core aligned along \(\mathbf{e}_z\),
and the envelope misaligned along \(\hat{\mathbf{n}}\)
under a rotation described by \(\mathbf{d}^\ell(\beta)\),
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
\[ \scriptsize \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} + 2 \omega \begin{bmatrix} {\color{forestgreen}\mathbf{R}_\pi} \otimes \mathbf{J}_z & 0 \\ 0 & {\color{forestgreen}\mathbf{R}_\gamma} \otimes \mathbf{J}_z \end{bmatrix} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
\[ \scriptsize \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} + 2 \omega \begin{bmatrix} {\color{forestgreen}\mathbf{R}_\pi} \otimes {\mathbf{d}^\ell}^\dagger\mathbf{J}_z\mathbf{d}^\ell & 0 \\ 0 & {\color{forestgreen}\mathbf{R}_\gamma} \otimes \mathbf{J}_z \end{bmatrix} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
\(\beta = 0\)
\(\beta = {\pi\over10}\)
\(\beta = {\pi\over2}\)
\(\beta = \pi\)
Stratified misalignment systematically reduces rotational splittings, modifies the shape of rotational avoided crossings, and may be diagnosed by per-multiplet variability of apparent \(i_\star\). Inferences of \(\beta\) are geometrically degenerate without further constraints from e.g. Rossiter-McLaughlin measurements.
\[\mathrm{j}\mathrm{o}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\ \text{@}\ \text{hawaii}.\text{edu}\]
\(\forall \ell, \exists (2\ell + 1) \times
(2\ell + 1)\) matrices
\(\mathbf{J}_x^\ell\), \(\mathbf{J}_y^\ell\), \(\mathbf{J}_z^\ell\) satisfying commutation
relations
\(\left[\mathbf{J}_i, \mathbf{J}_j\right] =
-i\epsilon_{ijk}\mathbf{J}_k\), with
\(\mathbf{J}_z \hat{=} \mathrm{diag}(-\ell,
-\ell + 1 \ldots \ell - 1, \ell).\)
\[\mathbf{J}_x \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}; \mathbf{J}_y \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & i & 0 \\ -i & 0 & i \\ 0 & -i & 0\end{bmatrix}; \mathbf{J}_z \hat{=} \begin{bmatrix}-1 & 0 & 0 \\ 0 &0 &0 \\ 0& 0 &1\end{bmatrix}.\]
\(\forall\) coordinate rotations by
angle \(\beta\) sending \(\mathbf{e}_z\) to \(\hat{\mathbf{n}}\),
\(\exists\) unitary \(\mathbf{d}^\ell\) of rank \(2\ell + 1\) such that \[Y^\ell_{m}(\theta_{\mathbf{e}_z},
\varphi_{\mathbf{e}_z}) = \sum_{m'} d^\ell_{m', m}(\beta)
Y^\ell_{m'}(\theta_\hat{\mathbf{n}},
\varphi_\hat{\mathbf{n}}).\]
Under this change of basis, \(\mathbf{A} \mapsto {\mathbf{d}^\ell}^\dagger \mathbf{A} {\mathbf{d}^\ell}\).
e.g. rotations around \(\mathbf{e}_y\) generate
\(\mathbf{d}^\ell(\beta) = \exp\left[ i \beta
\mathbf{J}_y \right]\). In this case
\(\mathbf{e}_z \mapsto \mathbf{e}_z \cos \beta
- \mathbf{e}_x \sin \beta\).
Under the change of basis generated by \(\mathbf{d}^\ell\),
\({\mathbf{d}^\ell}^\dagger \mathbf{J}_z
{\mathbf{d}^\ell} = \mathbf{J}_z \cos \beta - \mathbf{J}_x \sin \beta
\equiv \boxed{\hat{\mathbf{n}}(\beta) \cdot
\vec{\mathbf{J}}}\).
\(\mathbf{d}^\ell\) is unitary \(\implies \hat{\mathbf{n}} \cdot
\vec{\mathbf{J}}\) has
same eigenvalues as \(\mathbf{J}_z\),
with basis functions aligned wrt \(\hat{\mathbf{n}}\).