Stellar Surfaces in the Frequency Domain
Public PhD Defense
Aug 2 2022
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 \(\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_l(\nu)\right) + \mathcal{O}(1/\nu)\)
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\small\color{orange}N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\]
\[\small\color{blue}S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\]
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]
\[{\color{gray} \omega_p > S_l, N}\]
\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]
\[{\color{red} \omega_g < N, S_l}\]
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.
Comparing surface corrections?
Lacunae in literature:
- What is the effect of these methodological choices on population
studies
(i.e. ensemble asteroseismology)? - Parametric vs nonparametric corrections?
- Main Sequence vs. Evolved Stars?
Parameter Estimation
16 Cyg B: from Ong, Basu, McKeever (2021)
For each choice of surface term correction,
we construct a posterior distribution
for some quantities of interest
(e.g. \(M\), \(R\), age) for each star in our samples
General agreement on the
inferred masses…
\(z\)-score for mass
…and on the inferred radii.
\(z\)-score for radius
Parameter estimates generally
agree quite robustly…
\(z\)-score for age
… but not for all parameters. \(\tiny(p \sim 10^{-8})\)
\(z\)-score for initial helium abundance
Nonparametric methods appear
to agree with each other…
\(z\)-score for initial helium abundance
…but not with our
fiducial parameterisation…
\(z\)-score for mass
…and these offsets appear
to be systematic.
\(z\)-score for age
Disagreements on the age scale?
Reference values: McKeever+ 2019 EB’s
Disagreements on the mass distribution?
vs. isochronal mass range
Disagreements on composition?
Reference values: McKeever+ 2019 EB’s
Other parametric methods yield comparable internal scatter!
Evolved Stars Exhibit Mixed Modes
Over the course of post-main-sequence evolution, \(N^2\) increases dramatically —
an interior g-mode cavity develops.
(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: \[{1 \over \nu_{n,l}} \sim \Delta\Pi_l \left(n_g + {l \over 2} + \epsilon_{g, n,l}\right)\]
\[{k_r^2 \sim {\omega^2 \over c_s^2} \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\tiny {\color{gray}\sin \left[\theta_p(\nu)\right] = 0};\ {\color{red}\cos \left[\theta_g(\nu)\right] = 0}\]
\[\tiny {\color{gray}\sin \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right] = 0};\ {\color{red}\cos \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right] = 0}\]
\[\tiny {\color{gray}\tan \left[\int_{r_{1,p}}^{r_{2,p}} k_r \mathrm d r\right]} {\color{red}\cot \left[\int_{r_{1,g}}^{r_{2,g}} k_r \mathrm d r\right]} = {1 \over 4} \exp \left[-2 \int_{r_{2,g}}^{r_{1,p}} |k_r| \mathrm d r\right]\]
- 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
\[\Large \hat H \psi_n = \left(\hat T + \hat V\right)\psi_n = E_n \psi_n\]
\[\large {\color{blue}{\hat H_1}} \psi_n = \left(\hat T + \hat V_1\right)\psi_n = {\color{blue}E_n \psi_n}\]
\[\large {\color{orange}{\hat H_2}} \psi_n = \left(\hat T + \hat V_2\right)\psi_n = {\color{orange}E_n \psi_n}\]
\[\huge \hat H = \hat T + \hat V_1 + \hat V_2 = {\color{blue}{\hat H_1}} + \hat V_2 = \hat V_1 + {\color{orange}{\hat H_2}}\]
\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} + {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 - \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]
\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} - {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 + \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]
\[\Large \hat{\mathcal{L}} \xi_n = -\omega_n^2 \rho_0 \xi_n\]
\[\Large \hat{\mathcal{L}} = {\color{red}\hat{\mathcal{L}}_\gamma} + \hat{\mathcal{R}}_\gamma\]
\[\Large {\color{red}\hat{\mathcal{L}}_\gamma \xi_{\gamma,n} = -\omega^2_{\gamma,n}\xi_{\gamma,n}}\]
\[\Large \hat{\mathcal{L}} = \hat{\mathcal{R}}_\pi + {\color{grey}\hat{\mathcal{L}}_\pi}\]
\[\Large {\color{grey}\hat{\mathcal{L}}_\pi \xi_{\pi,n} = -\omega^2_{\pi,n}\xi_{\pi,n}}\]
\[\Large \hat{\mathcal{L}} = {\color{grey}\hat{\mathcal{L}}_\pi} + \hat{\mathcal{R}}_\pi = \hat{\mathcal{R}}_\gamma + {\color{red}\hat{\mathcal{L}}_\gamma}\]
\[\Large \xi_\text{mixed} \sim {\color{grey}c_\pi \xi_\pi} + {\color{red}c_\gamma \xi_\gamma}\]
\[\Large \hat{\mathcal{V}}{\color{red}\xi_\gamma} \to 0\]
\({\color{red}\gamma\text{-modes}}\)
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.
traditional approx. OK
perturbative series diverges
???
Ignore mixing altogether? (a la Ball et al. 2018)
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)
All the surface corrections yield similar best-fit models;
main discrepancies arise in the wings of the posterior distribution
EPIC212478598
Parametric vs Nonparametric
Parametric vs Nonparametric
Benchmark: differences between grids
Evolved Stars Dominate our Asteroseismic Sample!
e.g. Measuring \(Y_0\) from the TESS CVZs (TASOC WG2 2.12)
Asymmetric Rotational Splitting
Bad: scatter around best-fit function
Better: data points collapse to single function
