Rethinking
the Surface Term

Aarhus University

Joel Ong

February 11 2021 | Slides at http://hyad.es/talks

Background

Animation: NASA

Data: \(y_\text{obs} \in Y\)

Models: \(x_i \in X\);\[F: X \to Y\]

Best-fitting model: \[x = \mathop{\mathrm{argmax}}_{x_j \in X}\mathcal{L}\left(x_j\right)\]

\[F: \underbrace{\left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right)}_{x \in X} \mapsto \underbrace{\left(L, T_\text{eff}, [\text{M/H}], \log g, \ldots\right)}_{y \in Y}\]

Power spectra of MDI dopplergrams

\[F: \left(M, t, Y_0, Z_0, \alpha_\text{mlt}, \ldots\right) \mapsto \left(L, T_\text{eff}, [\text{M/H}], \log g, \color{blue}{\Delta\nu, \nu_\text{max}, \left\{\nu_{nl}\right\}}\right)\]

From Basu (2020)

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

Surface Corrections (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 \(\delta\nu_{nl,\text{surf}} \sim f(\nu_{nl}; x; \theta)\).

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

Alternative Approach 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)

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

Alternative Approach II:
\(\epsilon\)-matching

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

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

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 \(f^\text{model}(\nu_\text{model})\)).

Comparing Surface Corrections

from Basu & Kinnane (2018)

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, ceteris paribus

*Not actually very large

I.

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.

\(z\)-score for initial helium abundance

II.

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?

Disagreements on the mass distribution?

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?

Understanding
Mixed Modes

Mixed Modes

Main Sequence
Subgiant
Red Giant

from Deheuvels and Michel (2011)

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

\[ \mathbf{H} \begin{bmatrix} {\color{blue}\mathbf{c}_1} \\ {\color{orange}\mathbf{c}_2} \end{bmatrix} = E \cdot \mathbf{D} \begin{bmatrix} {\color{blue}\mathbf{c}_1} \\ {\color{orange}\mathbf{c}_2} \end{bmatrix} \] \[ \small H_{{\color{blue}i}{\color{orange}j}} = \langle {\color{blue}\psi_i}|\hat{H}|{\color{orange}\psi_j}\rangle;~~~~D_{{\color{blue}i}{\color{orange}j}} = \langle {\color{blue}\psi_i}|{\color{orange}\psi_j}\rangle \]

\[\Large \hat{\mathcal{L}} \xi_n = -\omega_n^2 \xi_n\]

\[\Large \hat{\mathcal{L}} = {\color{red}\hat{\mathcal{L}}_\gamma} + \hat{\mathcal{R}}_\gamma\]

\[\Large {\color{red}\hat{\mathcal{L}}_\gamma \xi_{\gamma,n} = -\omega^2_{\gamma,n}\xi_{\gamma,n}}\]

\[\Large \hat{\mathcal{L}} = \hat{\mathcal{R}}_\pi + {\color{grey}\hat{\mathcal{L}}_\pi}\]

\[\Large {\color{grey}\hat{\mathcal{L}}_\pi \xi_{\pi,n} = -\omega^2_{\pi,n}\xi_{\pi,n}}\]

\[\Large \hat{\mathcal{L}} = {\color{grey}\hat{\mathcal{L}}_\pi} + \hat{\mathcal{R}}_\pi = \hat{\mathcal{R}}_\gamma + {\color{red}\hat{\mathcal{L}}_\gamma}\]

\[\Large \xi_\text{mixed} \sim {\color{grey}c_\pi \xi_\pi} + {\color{red}c_\gamma \xi_\gamma}\]

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)

The Surface Term and Mixed Modes

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

Order of Operations matters!

  • The surface term acts on the bare \({\color{grey}\pi\text{-mode}}\) system, but is measured by the effect this has on mixed modes.
  • All surface term corrections in the literature work backwards from mixed-mode frequencies, to infer perturbations on \({\color{grey}\pi\text{-modes}}\): \[ \small\delta\nu_{\pi, \text{surf}} \sim f(\nu_{nl}) / I_\pi\left(\nu_{nl}\right) \iff \delta\nu_{\text{surf}} \sim f(\nu_{nl}) / I_{nl} \equiv \delta\nu_{\pi,\text{surf}} {I_{\pi,nl}\over I_{nl}} \]

These two operations do not necessarily commute.

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)
Full matrix coupling

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

Subgiant: \(\Delta\nu = 67.5\ \mu\text{Hz}\)

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

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

traditional approx. OK

perturbative series diverges

???

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

The Future

Subgiants Dominate the TESS ATL

from Schofield+ 2019

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

Summary

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:

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 (in prep.)

Ong, Basu, Lund, Viani, Bieryla, Latham (in prep.)

Summary

The surface term behaves differently on red giants vs. main-sequence stars. To study its behaviour in the intermediate regime, in subgiants, we must generalise conventional surface corrections to apply to mixed modes. We do this by analytically decoupling mixed modes into \(\pi\)- and \(\gamma\)-like components.

Backup Slides I:
Outer Turning Points

Backup Slides II:
Generalising Classical Parameterisations

The Surface Term as a
Matrix Perturbation

From operator perturbation problem, \(\hat{\mathcal{L}} = \hat{\mathcal{L}}_0 + \lambda {\hat{\mathcal{V}}},\) construct corresponding matrix perturbation problem by evaluating \[V_{ij} = \int \rho\ \vec{\xi}_{\pi,i} \cdot \hat{\mathcal{V}}\vec{\xi}_{\pi,j}\ \mathrm d^3 x.\]

\[\text{``Variational" expression: }\delta\omega_i^2 \sim {\int \rho\ \vec \xi_i\ \delta\hat{\mathcal{L}} (\vec \xi_i)\ \mathrm d^3 x \over \int \rho\ |\vec \xi_i|^2\ \mathrm d^3 x}\]

Matrix Parametrisations
for the Surface Term

So long as the off-diagonal entries of the matrix \(\mathbf{V}\) can be specified, existing parametrisations can adapted as (e.g. for BG14): \[ \large \begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} + \mathbf{V}_{(a_{-1}, a_{3})} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix} = -\omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \begin{bmatrix} {\color{grey}\mathbf{c}_\pi} \\ {\color{red}\mathbf{c}_\gamma} \end{bmatrix}. \]

// reveal.js plugins