Asteroseismic Signatures of
Engulfment and
Misalignment
Seminar @ MSSL | Slides at http://hyad.es/talks
August 29, 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.)
Key Application: Global Parameter Estimation
Hare “Zebedee”, Cunha+ (incl. Ong, 2021)
Precise measurements of field stars: \[ {\sigma_R \over R} \lesssim 2 \%; {\sigma_M \over M} \lesssim 5 \% \]
Very high \(\mathrm{A(Li)}\),
but otherwise innocuous
Mode Identification
\[\ell = 0,2?\]
But Kepler says \(\ell =
0\) have to live here!
(and theory says so too…)
Asteroseismology Happens…
Rotational splittings from seismology:
- \(\ell = 1\): \(\delta\nu_{\text{rot},1} \sim 0.09\ \mu\mathrm{Hz}\)
- \(\ell = 2\): \(\delta\nu_{\text{rot},2} \sim 0.10\ \mu\mathrm{Hz}\)
\(\implies P_\text{rot} \sim 115\ \mathrm{d}?\)
Rotational signal probably got
detrended away
by systematic corrections…
TESS w/ Pixel-Level Decorrelation + Stitching
Very suggestive…
but is this real?
Probably yes!
Gaia RV scatter rules out
large RV semiamplitudes…
\(P_\text{orb} \gg 99\ \mathrm{d}\)
cannot spin star
up to 99-day rotational period…
Remaining permissible orbits are
unstable to tidal dissipation!
\(\implies\) ENGULFMENT?
Albrecht+ 2021:
A Preponderance of Perpendicular Planets
From De+ 2023
Saunders+ (in
prep.)
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)
LCAOs give Molecular Orbitals
\[\Large \hat H \psi_n = \left(\hat T + \hat V\right)\psi_n = E_n \psi_n\]
\[\large {\color{blue}{\hat H_1}} \psi_n = \left(\hat T + \hat V_1\right)\psi_n = {\color{blue}E_n \psi_n}\]
\[\large {\color{orange}{\hat H_2}} \psi_n = \left(\hat T + \hat V_2\right)\psi_n = {\color{orange}E_n \psi_n}\]
\[\huge \hat H = \hat T + \hat V_1 + \hat V_2 = {\color{blue}{\hat H_1}} + \hat V_2 = \hat V_1 + {\color{orange}{\hat H_2}}\]
\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} + {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 - \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]
\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} - {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 + \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]
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: \[
\scriptsize
\begin{bmatrix}
{\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\
\mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}
= -\omega^2 \begin{bmatrix}
\mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}.
\] \[
\small
L_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i},
\hat{\mathcal{L}}{\color{red}\xi_j}\rangle;~~~~D_{{\color{grey}i}{\color{red}j}}
= \langle {\color{grey}\xi_i}, {\color{red}\xi_j}\rangle
\]
(Ong & Basu 2020)
(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):
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
\(\beta = 0\)
\(\beta = {\pi\over10}\)
\(\beta = {\pi\over2}\)
\(\beta = \pi\)
V501 Aur: a red giant accidentally misclassified as
a WTT!
(Vaňko et al. 2017: spectroscopic binary with \(K = 27\ \mathrm{km/s}, P \sim 69\
\mathrm{d}\))
