Asteroseismic Diagnostics of
Engulfment and Misalignment
TAPIR Seminar
September 13, 2024
\(\ell = 0\) MDI Doppler velocities
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
Evolved stars dominate our asteroseismic sample.
(e.g. only \(\sim 100\) Kepler main-sequence stars)
(proxy for age \(\to\))
Mixed modes exhibit
avoided crossings
between underlying p- and g-modes.
Pure p-modes: \[\nu_{n,\ell} \sim \Delta\nu \left(n_p + {\ell \over 2} + \epsilon_{n,\ell}\right)\]
Pure g-modes: \[{1 \over \nu_{n,\ell}} \sim \Delta\Pi_\ell \left(n_g + {\ell \over 2} + \epsilon_{g, n,\ell}\right)\]
\[b_i(r) = b_i \int_0^r K_i(r') \mathrm d r'\]
Mixed modes probe radial differential rotation
in two zones (core vs. envelope).
Cumulative sensitivity \(b(r)\)
pure g-mode pure p-mode
Mode mixing yields avoided crossings
between multiplet components of identical \(m\)
(cf. Mosser+ 2012, Ouazzani+ 2013, Deheuvels+ 2017)
Brute-force numerical solution (quantitative)
vs
JWKB approximation (qualitative)
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_\ell}^2 \over \omega^2}\right)\left(1 - {{\color{darkorange}N}^2 \over \omega^2}\right)}\]
\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]
\[{\color{red} \omega_g < N, S_\ell}\]
\[\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\] entropy gradient (\(=0\) in CZ)
\[\small S_\ell^2 = c_s^2 k_h^2 = {\ell(\ell+1) c_s^2 \over r^2}\] wave angular momentum
\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]
\[{\color{gray} \omega_p > S_\ell, N}\]
\[{\color{gray}\sin \left[\int_{r_{1, p}}^{r_{2, p}} k_+ \mathrm d r\right] = 0}\]
\[{\color{red}\cos \left[\int_{r_{1, g}}^{r_{2, g}} k_- \mathrm d r\right] = 0}\]
\[{\color{gray}\tan \left[\int_{r_{1, p}}^{r_{2, p}} k_+ \mathrm d r\right]}{\color{red}\cot \left[\int_{r_{1, g}}^{r_{2, g}} k_- \mathrm d r\right]} = {1\over 4}\exp\left[-2\int_{r_{2,p}}^{r_{1,g}} \kappa \mathrm d r\right]\]
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):
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
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:
- \(\ell = 1\): \(\delta\nu_{\text{rot},1} \sim 0.09\ \mu\mathrm{Hz}\)
- \(\ell = 2\): \(\delta\nu_{\text{rot},2} \sim 0.10\ \mu\mathrm{Hz}\)
\(\implies P_\text{rot} \sim 115\ \mathrm{d}?\)
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?
- Stars tend to have (and interact with) companions: binaries, planetary systems, engulfments…
- Seismic rotational measurements indicate anomalies?
\(\beta = 0\)
\(\beta = {\pi\over10}\)
\(\beta = {\pi\over2}\)
\(\beta = \pi\)
From Huber et al. 2013
\[b_i(r) = b_i \int_0^r K_i(r') \mathrm d r'\]
Mixed modes probe radial differential rotation
in two zones (core vs. envelope).
Cumulative sensitivity \(b(r)\)
\[b_i(r) = b_i \int_0^r K_i(r') \mathrm d r'\]
Even the most p-dominated mixed modes
are sensitive to core rotation!
Cumulative sensitivity \(b(r)\)
From Huber et al. 2013
Caveats:
Model-dependence? Short Cadence?
Amplitudes? Linewidths?
Rotational signal probably got
detrended away
by systematic corrections…
TESS w/ Pixel-Level Decorrelation + Stitching
Very suggestive…
but is this real?
Probably yes!
- Only three known misaligned multiplanet systems (HD 3167
and K2-290A)
- Implications for orbital architectures?
- What about other notionally envelope-counterrotating stars?
- Constraints on realignment \(\mathcal{Q}\) and/or torque mechanisms?
