Stellar Surfaces in the Frequency Domain

Public PhD Defense

Joel Ong

Aug 2 2022

Background Asteroseismology Per Se

Whence Asteroseismology?

Some Observational Facts

from Jeffery & Saio (2016)

Animation: NASA

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 \(\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_l(\nu)\right) + \mathcal{O}(1/\nu)\)

Three quantum numbers \(n, l, m\): \[ \begin{aligned} l &= 0, 1, 2, \ldots \\ m &= -l, -l+1, \ldots, l-1, l \end{aligned} \]

Zonal (\(m = 0\))

Prograde sectoral
(\(m = +l\))

Retrograde sectoral
(\(m = -l\))

How does this relate
to stellar structure?

Simple Wave Equation

\[\huge {\partial^2 \over \partial t^2}f = c_s^2 \nabla^2 f\]

\[\huge -\omega^2 f = c_s^2 \nabla^2 f\]

\[\large \nabla^2 f = -{\omega^2 \over c_s^2} f \equiv -|\mathbf{k}|^2 f\]

Dispersion relation: \[\omega^2 = c_s^2 |\mathbf{k}|^2\]

More Complicated Wave Equation

\[\huge {\partial^2 \over \partial t^2}f = \left(c_s^2 \nabla^2 - \omega^2_\text{ac}(\mathbf{x})\right) f\]

\[\huge -\omega^2 f = \left(c_s^2 \nabla^2 - \omega^2_\text{ac}(\mathbf{x})\right) f\]

\[\large \nabla^2 f = -\left({\omega^2 - \omega_\text{ac}^2\over c_s^2}\right) f \equiv -k^2 f\]

Wave propagates where \(k^2(r, \omega) > 0\) and decays where \(k^2(r, \omega) < 0\).

\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]

\[\small\color{orange}N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\]

\[\small\color{blue}S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\]

\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]

\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]

\[{\color{gray} \omega_p > S_l, N}\]

\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]

\[\large\mathbf{v}_g = {\partial \omega \over \partial \mathbf{k}} = c_s {\mathbf{k} \over |\mathbf{k}|}\]

\[ \begin{aligned} \oint \mathbf{k} \cdot \mathrm d \mathbf{r} &= 2 \pi \left(n' - {1 \over 2}\right) \\ 2 \omega \int_0^R {\mathrm d r \over c_s} &= 2 \pi \left(n + {l \over 2} + \epsilon_l\right) \end{aligned} \]

\[\large\nu = {1 \over 2T}\left(n + {l \over 2} + \epsilon\right)\]

\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]

\[{\color{red} \omega_g < N, S_l}\]

\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]

\[ \begin{aligned} \oint \mathbf{k} \cdot \mathrm d \mathbf{r} &= 2 \pi n' \\ \implies {2 \sqrt{l(l+1)} \over \omega} &\int_{N^2 > 0} {N \over r} \mathrm d r \\&= 2 \pi \left(n + {l \over 2} + \epsilon_{g, l}\right) \end{aligned} \]

\[\large{1 \over \nu} = \Delta\Pi_l \left(n + {l \over 2} + \epsilon_{g, l}\right)\]

Problem Statement Asteroseismology as an Instrument, or, “How do we use this?”

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

From Basu (2020)

Stellar Surfaces
are Complicated.

For the Sun, even standard solar models don’t give the right frequencies!

The asteroseismic “surface term”: a (mostly) smooth function of frequency.

Differential Systematics from Surface Corrections (Ong, Basu & McKeever 2021; ApJ, 906, 54)

Comparing surface corrections?

from Basu & Kinnane (2018)

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

Differences between surface corrections

For each parameter \(P\), we consider normalised differences \[z_{P, \text{BG14 vs. other}} = {\mu_{P, \text{other}} - \mu_{P, \text{BG14}} \over \sqrt{\sigma^2_{P, \text{BG14}} + \sigma^2_{P, \text{other}}}}\]

We examine the distribution of these differences
over a large* sample of stars, all else being equal

*Not actually very large


Main-sequence stars

\[\tiny{(\text{Kepler LEGACY sample}: N = 66)}\]

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


First-ascent red giants

\[\tiny{(\text{NGC 6791}: N = 29; l=0,2\text{ only})}\]

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

What’s going on?

NGC 6791 is an open cluster — what about the
actual distribution of stellar properties?

Stars should be coeval, so red giants should be of similar mass and 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

from Jørgensen et al. (2020)

Other parametric methods yield comparable internal scatter!

Different treatments of the surface term exhibit qualitative systematic differences for main-sequence stars vs. red giants.

What happens in between?

Why have we excluded dipole (\(l = 1\)) modes in this analysis?

Mixed Modes

Evolved Stars Exhibit Mixed Modes

Main Sequence
Red Giant

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)

adapted from Deheuvels and Michel (2011)

Relationship between \((\omega_i, \alpha_i)\) and stellar structure unclear…

A New Analytic Treatment of Mixed Modes (Ong & Basu 2020; ApJ, 898, 127)

LCAOs give Molecular Orbitals

I stole this from a chemistry textbook

\[\Large \hat H \psi_n = \left(\hat T + \hat V\right)\psi_n = E_n \psi_n\]

\[\large {\color{blue}{\hat H_1}} \psi_n = \left(\hat T + \hat V_1\right)\psi_n = {\color{blue}E_n \psi_n}\]

\[\large {\color{orange}{\hat H_2}} \psi_n = \left(\hat T + \hat V_2\right)\psi_n = {\color{orange}E_n \psi_n}\]

\[\huge \hat H = \hat T + \hat V_1 + \hat V_2 = {\color{blue}{\hat H_1}} + \hat V_2 = \hat V_1 + {\color{orange}{\hat H_2}}\]

\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} + {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 - \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]

\[\Large\psi \sim {1 \over \sqrt{2}} \left({\color{blue}\psi_1} - {\color{orange}\psi_2}\right)\] \[\Large E \sim E_1 + \left|\int {\color{blue}\psi_1^*} \hat H {\color{orange}\psi_2}\ \mathrm dV\right|\]

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)

Mixed Modes and the Surface Term (Ong, Basu & Roxburgh 2021, ApJ 920, 8; Ong, Basu, Lund et al. 2021, ApJ 922, 18)

The Surface Term as a
Structural Perturbation

Consider two stars with identical \(M\) and \(R\),
differing only in the near surface layers: \[\hat{\mathcal{L}}_0\xi_{0,n} = -\omega_{0,n}^2 \xi_{0,n};~~~~\hat{\mathcal{L}}\xi_{n} = -\omega_{n}^2 \xi_{n}.\] We write \[\hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}},\] so that \(\lambda\) interpolates linearly between the two structures.

\[\Large \hat{\mathcal{V}}{\color{red}\xi_\gamma} \to 0\]

\({\color{red}\gamma\text{-modes}}\) 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.

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Full matrix coupling
\(\lambda\) from \(0 \to 1\)

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Traditional approximation to coupling

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Full vs. Traditional mode coupling

Young red giant: \(\Delta\nu = 17.2\ \mu\text{Hz}\)

Evolved red giant: \(\Delta\nu = 3.9\ \mu\text{Hz}\)

Surface-term corrections
should not reorder mixed modes!
(cf. Ball+ 2018)

traditional approx. OK

perturbative series diverges


Ignore mixing altogether? (a la Ball et al. 2018)

Correcting the Surface Term
for Mixed Modes

  1. Determine an appropriate correction for p-modes
  2. Construct modified coupling matrices
  3. Solve the perturbed GHEP:

\[ \left( \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} + \lambda \begin{bmatrix} {\color{blue}\mathbf{V}} & 0 \\ 0 & 0 \end{bmatrix} \right) \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \]

We may now continue our methodological inventory.



\[\tiny{(\text{Various Kepler/K2/TESS}: N = 47;\ l=0, 1, 2)}\]

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


Parametric vs Nonparametric

Parametric vs Nonparametric

Benchmark: differences between grids

Looking Forward

Evolved Stars Dominate our Asteroseismic Sample!

Kepler Sample (from Yu+ 2020)
TESS ATL (from Schofield+ 2019)

e.g. Measuring \(Y_0\) from the TESS CVZs (TASOC WG2 2.12)

Asymmetric Rotational Splitting

Dipole-mode asymmetry from mode mixing (generalising Deheuvels et al. 2017)


Treatments of the Surface Term:

Nonparametric treatments of the surface term may yield
qualitatively different results from parametric ones,
particularly for measuring \(Y_0\), and for evolved red giants.

Ong, Basu, McKeever (2021)

Decoupling of Mixed Modes:

Decomposition of wave operator into purely acoustic/buoyant propagation terms and their remainder operators
permits closed-form evaluation of coupling matrix elements.

Ong and Basu (2020)

Mixed Modes and the Surface Term I:

Traditional surface term corrections handle mode coupling
only to first order, if at all.

More sophisticated techniques are required
as stars get more evolved.

Ong, Basu, Roxburgh (2021)

Mixed Modes and the Surface Term II:

Inference of stellar masses and compositions depend on
how mode mixing is treated
when correcting for the surface term.

Ong, Basu, Lund, et al. (2021)


We generalise conventional surface corrections to apply to mixed modes, by analytically decoupling mixed modes into p- and g-like components. While the surface term behaves differently on red giants vs. main-sequence stars, it exhibits intermediate behaviour in subgiants. Measurements of stellar masses and compositions depend significantly on how the surface term is treated.


Supplementary Slides

Corrections for the Surface Term (see Basu+ 2018, Jørgensen+ 2020 for review)

(Fully) calibrated corrections

\[\nu_\text{model} \mapsto \nu_\text{corr} = \nu_\text{model} + \delta\nu_{nl,\text{surf}},\] with corrections depending only on the model as \(\delta\nu_{nl,\text{surf}} \sim f(\nu_{nl}; x)\).

Parametric corrections

Corrections with free parameters \(\theta \in \Theta\) as \(\delta\nu_{nl,\text{surf}} \sim f(\nu_{nl}; x; \theta)\).

\[\scriptsize {\text{e.g. Ball \& Gizon (2014) correction: }\delta\nu_{nl,\text{surf}} \sim \left(a_{-1} \left(\nu_{nl}\right)^{-1} + a_{3} \left(\nu_{nl}\right)^{3}\right) / I_{nl}}\]

Nonparametric Treatments

Use of transformed variables such that \[O_{nl,\text{surf}} \sim f(\nu_{nl})\]

(where the structure of \(f\) is left underspecified)

Interpolation required to compare
\(f^\text{obs}(\nu_\text{obs})\) vs. \(f^\text{model}(\nu_\text{obs})\) (instead of \(f^\text{model}(\nu_\text{model})\)).

Method I:
Separation Ratios

\[\begin{aligned} \text{e.g. } r_{02, n} = {\nu_{n,0} - \nu_{n-1,2} \over \nu_{n,1} - \nu_{n-1,1}} \end{aligned}\]

Interpretation as surface-independent functions of frequency:
use these to constrain model directly (Otí Floranes et al. 2005)

\[\begin{aligned} r_{02, n} & \sim \delta_2(\nu_{n,0})\\ r_{01, n} \sim \delta_1(\nu_{n,0})&; ~~~ r_{10, n} \sim \delta_1(\nu_{n,1}) \end{aligned}\]

Method II:

\[\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_{nl}\right), \implies \epsilon_{nl} = {\nu_{nl} \over \Delta\nu} - n - {l \over 2} \equiv{\epsilon_l(\nu_{nl})}\] \[ \text{Compute } \mathcal{E}_l(\nu^\text{obs}_{nl}) = \epsilon^\text{obs}_{l}(\nu^\text{obs}_{nl}) - \epsilon^\text{model}_{l}(\nu^\text{obs}_{nl}). \]

All of the \(\mathcal{E}_l\) should collapse to a single function if the structural differences are localised to the surface (Roxburgh 2016).

Bad: scatter around best-fit function

Better: data points collapse to single function

Analytic Construction

1. Self-gravitating fluid:

\[ \begin{aligned} {\partial \rho \over \partial t} + \nabla \cdot \left(\rho \mathbf{v}\right) &= 0 \\ \rho \left({\partial \mathbf{v} \over \partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) &= -\nabla P - \rho \nabla \Phi\\ \nabla^2 \Phi &= 4 \pi G \rho \end{aligned} \]

2. Linearised Adiabatic Lagrangian Perturbations give Normal Modes:

\[\rho(t, \mathbf{x}) = \rho_0(r) + \underbrace{\left(\rho'(r) + \vec{\xi}(r) \cdot \nabla\rho_0(r)\right)}_{\mathclap{\delta \rho\text{, the Lagrangian perturbation in $\rho$}}}e^{\pm i \omega t} Y_l^m(\theta,\phi)\]

\[\delta s = 0 \implies \delta P = c_s^2 \delta \rho\]

\[\mathbf{u}(t, r, \theta, \phi) = e^{\pm i \omega t} \left[u_r(r) \mathbf{Y}_l^m(\theta,\phi) + u_h(r) \mathbf{\Psi}_l^m (\theta, \phi) + u_t(r) \mathbf{\Phi}_l^m (\theta, \phi)\right].\]

3. Boundary Value Problem:

\[ \Large \begin{aligned} \mathcal{L}\vec{\xi} &= \nabla \left(\rho \vec\xi \cdot \vec{g} + c_s^2 \rho \nabla \cdot \vec\xi\right)\\ &- \vec{g} \nabla \cdot (\rho \vec\xi) - \rho G \nabla\left(\int \mathrm d^3 x' {\nabla' \cdot(\rho \vec\xi) \over |x - x'|}\right)\\ &= -\omega^2 \rho \vec{\xi}. \end{aligned} \]

4. Outer Turning Points

for \(r/R \sim 1\),

\[ \Large \left(c_s {\mathrm d \over \mathrm d r}\right)^2 \xi_r \sim (\omega_\text{cutoff}^2(r) - \omega^2)\xi_r \]

Additional Work


  • Various boutique-modelling papers:
    • TASOC WG1: 1.1, 1.7, 1.8, 1.10, 1.14
    • TASOC WG2: 2.1, 2.15, 2.19, 2.25, 2.26, 2.27, 2.28
    • TASOC WG7: 7.12
  • pbjam (Nielsen et al. 2021)
  • Scientific planning for PLATO (Cunha et al. 2021, ongoing)
  • Red Giant modelling (Lillo-Box et al. 2021, Campante et al. in review, Lindsay et al. in review, Saunders et al. in prep.)