Solar-like Oscillations:
Past, Present, & Future

Colloquium @ Tsinghua University | Slides at

Joel Ong

February 23, 2023

Long, Long Ago…

At first sight it would seem that the deep interior of the sun and stars is less accessible to scientific investigation than any other region of the universe. Our telescopes may probe farther and farther into the depths of space; but how can we ever obtain certain knowledge of that which is hidden behind substantial barriers? What appliance can pierce through the outer layers of a star and test the conditions within? The problem does not appear so hopeless when misleading metaphor is discarded… And the interior of a star is not wholly cut off from such communication.

The Internal Constitution of Stars, 1926

Arthur Eddington,

Arthur Eddington
Henrietta Swan Leavitt
from Jeffery & Saio (2016)
δ Cep: from Elver+ 2014

\[\Huge P \sim L^\alpha\]

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 \(k^2(r, \omega) < 0\).

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

\[\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\] entropy gradient (\(=0\) in CZ)

\[\small S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\] wave angular momentum

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

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

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

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

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

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

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

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

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

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

Eckart-Scuflaire-Osaki Classification Scheme


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

\[\small\Delta\nu \sim \left(2 \int {\mathrm d r \over c_s}\right)^{-1}\]


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

\[\small\Delta\Pi_l \sim {2\pi^2 \over \sqrt{l(l+1)}}\left(\int {N \over r} \mathrm d r\right)^{-1}\]

Much easier to see
in the Sun…

Others: f-modes (surface), r-modes (inertial), …

The Sun as a Star

SOHO EIT Image (2016)

HMI Dopplergram (2017)

\(\ell = 0\) MDI Doppler velocities

Power spectra of MDI dopplergrams

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

(HMI Power Spectra at different \ell)

Resolved imaging of the Sun gives
mode frequencies at very high \(\ell\)!

Helioseismology: Greatest Hits

  • Solar Neutrino Problem (Nobel!)
  • How do stars work??
  • Rotational Structure
  • Solar Abundance Problem
  • Penetrative Convection
  • Far-side imaging (helioseismic holography)


From Helio- to Asteroseismology

Ground-based asteroseismology is expensive!

Procyon as seen through 11 PRV spectrographs: from Arentoft+ (2008)

Solar-like oscillators from 1995 onwards: n = 15; from Arentoft+ (2008)

Yesterday and Today

Kepler launching in 2009
2013: Transition to K2

Key Application: Stellar Properties

from Chaplin & Miglio (2013)

Power spectra of MDI dopplergrams

\[ \begin{aligned} \Delta\nu &\sim \sqrt{M / R^3} \\ \nu_\text{max} &\sim M / R^2 \sqrt{T_\text{eff}} \end{aligned} \]

\[ \begin{aligned} {M \over M_\odot} &\sim \left(\nu \over \nu_{\text{max},\odot}\right)^{3}\left(\Delta\nu \over \Delta\nu_\odot\right)^{-4} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{3/2} \\ {R \over R_\odot} &\sim \left(\nu \over \nu_{\text{max},\odot}\right)\left(\Delta\nu \over \Delta\nu_\odot\right)^{-2} \left(T_\text{eff} \over T_{\text{eff},\odot}\right)^{1/2} \end{aligned} \]

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

\[ \color{red} \text{and } \Delta\nu, \nu_{\text{max}}, \left\{\nu_{n,l}\right\} \]

The Forward Problem: Global Parameter Estimation

Hare “Zebedee”, Cunha+ (incl. Ong, 2021)

Precise measurements of field stars: \[ {\sigma_R \over R} \lesssim 2 \%; {\sigma_M \over M} \lesssim 5 \% \]

Key Application: Rotation

  • Gyrochronology: rotation gives ages (e.g. Skumanich 1972, Barnes 2007).
  • 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.

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

Zonal (\(m = 0\))

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

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

Stellar Rotation

Hall+ 2021

Hall+ 2021: Verification of weakened-magnetic-braking hypothesis from
ensemble asteroseismology

Rotational Inversion

For slow rotation, \[\boxed{\delta\omega_{nlm} \sim m \beta_{nl} \int \Omega(r) K_{nl}(r) \mathrm d r}\]

OLA (Backus & Gilbert 1968; Gough 1985;
Pijpers & Thompson 1992; Schunker 2016; etc.):

\[\scriptsize\begin{aligned}\sum_{i} c_i \left( \delta\omega_{\mathrm{rot}, i} \over m_i \beta_{i} \right) &\sim \int \Omega(r) \left(\sum_i c_i K_{i}(r)\right) \mathrm d r \\ & \to \boxed{\Omega(r_0)}\end{aligned}\]

(HMI Power Spectra at different \ell)

Solar Rotation

GONG measurements of solar differential rotation
“shellular” rotation: from Glatzmeier (1985)

In the Sun, a transition from solid-body to differential rotation at the base of the convective envelope



Inversions for Solar Structure

from Basu (2020)

two standard
solar models

Inversions for Stellar Structure

Bellinger+ 2017

Bellinger+ 2019:
first inference of structure from
a star with a convective core.

Bellinger+ 2017: comparison of interior structure in a binary

TESS Mission: 2018—Present

Evolved stars dominate our asteroseismic sample!

(facultative with Kepler, obligate with TESS)

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

(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: \[{1 \over \nu_{n,l}} \sim \Delta\Pi_l \left(n_g + {l \over 2} + \epsilon_{g, n,l}\right)\]

\[\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\] entropy gradient (\(=0\) in CZ)

\[\small S_l^2 = c_s^2 k_h^2 = {l(l+1) c_s^2 \over r^2}\] wave angular momentum

Over the course of post-main-sequence evolution, \(N^2\) increases dramatically —
an interior g-mode cavity develops.

Evolved Stars Exhibit Mixed Modes

Main Sequence
Red Giant

(Ong & Basu 2020)

\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]


\[\small\psi_\text{mol} = {\color{blue}\sum_i c_{1,i} \psi_{1,i}} + {\color{orange}\sum_j c_{2,j} \psi_{2,j}}\]

Applications for determination of (sub)giant structure and
properties (Ong+ 2021a, b, c), and internal rotation
(Ong+ 2022, and 2023, in review)

We know more about giant cores
than about the core of our own Sun!

Subgiant Fundamental Properties

Joint posterior distributions for TOI 197
(reference values: Huber+ incl. Ong, 2019)
with one sector (27 days)
of TESS data!

(Ong+ 2021c)

Naive approach

Ong+ 2021b

Radiative core contracts dramatically off main sequence
\(\implies\) core spins up (if conserving angular momentum)

Mode mixing yields avoided crossings
between multiplet components of identical \(m\)

(cf. Mosser+ 2012, Ouazzani+ 2013, Deheuvels+ 2017, Ong+ 2022)

Red Giant Core Diagnostics: Rotation

Period-echelle power diagram showing rotational splitting (from Ong & Gehan 2023, in review)
Core rotation measurements: Gehan+ 2018

Core rotation rates appear not to increase significantly
as cores contract \(\implies\) angular momentum transport?

Red Giant Core Diagnostics: Evolution

from Mosser+ (2014)

Clustering on \(\Delta\nu\)-\(\Delta\Pi_1\) diagram:
diagnostic of core helium burning.
(RC stars = standard candles!)

Single-star electron degeneracy sequence:
deviations → merger remnants?
(Rui+ 2021, Deheuvels+ 2021)

Red Giant Core Diagnostics: Magnetism

Symmetric g-mode splitting: pure rotation
Asymmetric g-mode splitting: core magnetic fields (Li+ 2022)


The Party Isn’t Over.

ESA PLATO Mission: 2026 (Planned)

Earth 2.0
2026 (Planned)

Hard Problems remain…

Evolutionary models (e.g. MESA, ASTEC) yield poor descriptions of stars with convective cores — improvements?

(e.g. Pedersen+ 2018;
Lindsay, Ong, Basu 2022;
Vanlaer+ 2023, in review)

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

(e.g. Ball+ 2014, 2017; Sonoi+ 2015; Roxburgh 2018;
Ong+ 2021a,c; Li+ 2022…)

Mode amplitudes are usually ignored,
but are entirely determined by turbulent convective driving.

How do we predict amplitudes of convectively excited modes?

(e.g. Philidet+ 2021, 2022; Zhou+ 2021)


We can understand stars very well by going to space,
staring at them for very long,
and blinking very fast.

Solar-like oscillations probe a very large range of
stellar structure, dynamics, and evolution.
In many respects we are left with more questions than answers.

\[\mathrm{j}\mathrm{o}\mathrm{e}\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\ \text{@}\ \text{hawaii}.\text{edu}\]