Recent Advances in Gravitoacoustic Mixed-Mode Asteroseismology
Colloquium @ IU Bloomington | Slides at http://hyad.es/talks
January 27, 2023
Whence Asteroseismology?
Power spectra of MDI dopplergrams
\[ \begin{aligned} \Delta\nu &\sim \sqrt{M / R^3} \\ \nu_\text{max} &\sim M / R^2 \sqrt{T_\text{eff}} \end{aligned} \]
Mode frequencies satisfy \(\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_l(\nu)\right) + \mathcal{O}(1/\nu)\)
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\small\color{orange}N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\]
\[\small\color{blue}S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\]
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]
\[{\color{gray} \omega_p > S_l, N}\]
\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]
\[{\color{red} \omega_g < N, S_l}\]
Evolved stars dominate our asteroseismic sample!
Evolved Stars Exhibit Mixed Modes
(proxy for age \(\to\))
Mixed modes exhibit
avoided crossings
between underlying p- and g-modes.
Pure p-modes: \[\nu_{n,l} \sim \Delta\nu \left(n_p + {l \over 2} + \epsilon_{n,l}\right)\]
Pure g-modes: \[{1 \over \nu_{n,l}} \sim \Delta\Pi_l \left(n_g + {l \over 2} + \epsilon_{g, n,l}\right)\]
Over the course of post-main-sequence evolution, \(N^2\) increases dramatically —
an interior g-mode cavity develops.
\[{k_r^2 \sim {\omega^2 \over c_s^2} \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\tiny {\color{gray}\sin \left[\theta_p(\nu)\right] = 0};\ {\color{red}\cos \left[\theta_g(\nu)\right] = 0}\]
\[\tiny {\color{gray}\sin \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right] = 0};\ {\color{red}\cos \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right] = 0}\]
\[\tiny {\color{gray}\tan \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right]} {\color{red}\cot \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right]} = {1 \over 4} \exp \left[-2 \int_{r_{2,g}}^{r_{1,p}} |k_r| \mathrm d r\right]\]
- Requires JWKB approximation — when is this suitable?
- Is there a general analytic construction without Cowling approximation?
- How to incorporate structure perturbations?
(Coupling between oscillators)
Relationship between \((\omega_i, \alpha_i)\) and stellar structure unclear…
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: \[
\begin{bmatrix}
{\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\
\mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}
= -\omega^2 \begin{bmatrix}
\mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1}
\end{bmatrix}
\begin{bmatrix}
{\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma}
\end{bmatrix}.
\] \[
\small
L_{{\color{grey}i}{\color{red}j}} = \langle {\color{grey}\xi_i},
\hat{\mathcal{L}}{\color{red}\xi_j}\rangle;~~~~D_{{\color{grey}i}{\color{red}j}}
= \langle {\color{grey}\xi_i}, {\color{red}\xi_j}\rangle
\]
(Ong & Basu 2020; now also in GYRE v6)
Data: \(y_\text{obs} \in Y\)
Models: \(x_i \in X\);\[F: X \to Y\]
Best-fitting model: \[x = \mathop{\mathrm{argmax}}_{x_j \in X}\ \mathcal{L}\left(x_j\right)\]
\[F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}\]
\[ \color{red} \text{and } \Delta\nu, \nu_{\text{max}}, \left\{\nu_{n,l}\right\} \]
The Forward Problem: Global Parameter Estimation
Hare “Zebedee”, Cunha+ 2021
Precise measurements of field stars: \[ {\sigma_R \over R} \lesssim 2 \%; {\sigma_M \over M} \lesssim 5 \% \]
The Inverse Problem: Model-independent measurements
For the Sun, even standard solar models don’t give the right frequencies!
The asteroseismic “surface term”: a (mostly) smooth function of frequency.
\[\Large \hat{\mathcal{V}}_\text{surf}{\color{red}\xi_\gamma} \to 0\]
\({\color{red}\gamma\text{-modes}}\)
are confined to the stellar interior,
so unaffected by surface
term
How is this compatible with a smooth function of frequency?
Mixing coefficients: customarily defined so that \(\zeta \to 1\) for pure g-modes, and \(\to 0\) for pure p-modes
The Traditional Approximation to Mode Coupling
\[\huge \times\]
Apply a partial correction: \(\delta\nu_\text{surf, mixed} = {\color{red} \delta\nu_\text{surf}} \times {\color{blue}(1 - \zeta)}\)
Order of Operations matters!
\(\delta\nu_\text{mixed} = (1 - \zeta)\delta\nu_\text{surf}\)
These two operations do not necessarily commute.
First-order vs. Full Mode Coupling
First-order vs. Full Mode Coupling
First-order approximation yields
systematically lower stellar masses!
(population-level systematic error)
(two different
correction techniques)
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.
\[\beta_i(r) = \beta_i \int_0^r K_i(r') \mathrm d r'\]
Sensitivity of pure p-modes is distributed
throughout entire stellar structure
(but concentrated at surface)
\[\beta_i(r) = \beta_i \int_0^r K_i(r') \mathrm d r'\]
Mixed modes probe radial differential rotation in two zones (core vs. envelope).
\[\beta_i(r) = \beta_i \int_0^r K_i(r') \mathrm d r'\]
Observed g-mixed modes probe core rotation well
(e.g. Mosser+ 2015; Gehan+ 2018)
Even the most p-dominated mixed modes
are sensitive to core rotation!
If for pure p-modes \(\delta \omega_p \sim
m \beta_p \Omega_\mathrm{env}\)
and for pure g-modes \(\delta \omega_g \sim m
\beta_g \Omega_\mathrm{core}\),
then surely for mixed modes,
\(\delta \omega_\mathrm{mixed} \sim m
\left(\zeta \beta_g \Omega_\mathrm{core} + (1 - \zeta) \beta_p
\Omega_\mathrm{env}\right)\)?
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.
(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):
\[ \small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + 2 m \omega {\color{forestgreen}\mathbf{R}} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = 0. \]
Rotational Coupling Matrices
Rotation matrices are essentially diagonal in \(\pi/\gamma\)-mode basis:
Entire matrix can may be meaningfully constrained observationally.
(this is not the case with mixed modes!)
Rotational Inversion?
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 (2022):
- We can do this with mixed modes under certain conditions
- Explicit numerical procedure based on p/g decomposition
Noncommuting operations again!
\(\pi/\gamma\) basis of kernels has different localisation properties.
Practical Testbed: KIC 9267654 (RGB)
Nonmonotonic rotational profile!
(Tayar+ 2022)
Practical Testbed (TRGB)
traditional approx. OK
perturbative series diverges
???
Ignore mixing altogether? (a la Ball et al. 2018)
All the surface corrections yield similar best-fit models;
main discrepancies arise in the wings of the posterior distribution
EPIC212478598
Single-target Systematics
Joint posterior distributions for TOI 197
(reference values: Huber+ 2019)
(Ong+ 2021c)
Single-target Systematics with Luminosity constraint
