Solar-Like Oscillations:
Life After Kepler

TASC7/KASC14 | Slides at http://hyad.es/talks

Joel Ong

July 17, 2023

Solar-like Oscillations

\(\ell = 0\) MDI Doppler velocities

Power spectra of MDI dopplergrams

\[ \begin{aligned} {\Delta\nu_\odot} &\sim 135\ \mathrm{\mu Hz} \\ {\nu_{\text{max},\odot}} &\sim 3090\ \mathrm{\mu Hz} \end{aligned} \]

(roughly 5-minute oscillations)

p-mode frequencies satisfy \(\nu_{n\ell} \sim \Delta\nu\left(n + {\ell \over 2} + \epsilon_\ell(\nu)\right) + \mathcal{O}(1/\nu)\)

Stochastic,
broad-band
excitation

Helioseismology: Greatest Hits

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

\[\vdots\]

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

Telescopes can only point at one star at time…

…and space is big.

Required photometric stability not achievable from ground :\

Kepler launching in 2009 — *Kepler* launching in 2009

Seismology as a Tool

Key Application: Stellar Properties

\[ \begin{aligned} \Delta\nu &\sim \int \left({\mathrm d r \over c_s }\right)^{-1}\sim t_\mathrm{ff}^{-1} \sim \sqrt{G \rho_\text{avg}} \\ \nu_\text{max} &\sim \omega_\text{cutoff} \sim g_\text{atm}/\sqrt{T_\text{atm}} \end{aligned} \]

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

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

The Forward Problem: Global Parameter Estimation

Hare “Zebedee”, Cunha+ (2021)

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

Key Application: Measuring Rotation

Further applications:
calibrating gyrochronology with
ensemble asteroseismology
(e.g. Hall+ 2021).

cf. Zachary Claytor’s talk tomorrow,
+ Rotation & Activity session

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.):

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

Directly probing the stellar interior!

Inversions for Stellar Structure

Direct probe into structure of
stellar interiors —
tests of evolutionary modelling (e.g. Bellinger+ 2019)

cf. Alexander Kosovichev, poster #22

\(\scriptsize u'(r) = P(r) R / \rho(r) M\)

From Kepler to TESS

TESS Mission: 2018—Present — *TESS* Mission: 2018—Present

Big Sample!

20 Second Cadence!

Evolved stars dominate our asteroseismic sample.

(facultative with Kepler, obligate with TESS)

Kepler Sample (from Yu+ 2020) — *Kepler* Sample (from Yu+ 2020)

TESS ATL (from Schofield+ 2019) — *TESS* ATL (from Schofield+ 2019)

Pure p-modes: \[\boxed{\nu_{n,\ell} \sim \Delta\nu \left(n_p + {\ell \over 2} + \epsilon_{n,\ell}\right)}\]

Pure g-modes: \[\boxed{{1 \over \nu_{n,\ell}} \sim \Delta\Pi_\ell \left(n_g + {\ell \over 2} + \epsilon_{g, n,\ell}\right)}\]

g-modes: Core Rotation

e.g. Mosser et al. 2012, 2015, 2017…; Gehan et al. 2018, 2021

\[\delta P_{\text{rot}, g, \ell=1} \sim - {m \Omega_\text{core} \over 2 \nu^2}\]

g-modes: Core Magnetism

Li et al. 2022:
Asymmetric splittings probe
core magnetic fields

(cf. Jérôme Ballot’s talk after this)

Population studies
of rotation vs. magnetism

(cf. Emily Hatt’s talk after this)

\[\scriptsize \delta \nu_{\text{mag}, g, \ell=1} \sim {m^2 \over \nu^3}\]

Mixed Modes: Core Sizes/Evolutionary Diagnostics

from Mosser+ (2014)

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

Mixed Modes: Structural Diagnostics

Glitches in CHeB stars (cf. Vrard+ 2022) — **Glitches** in CHeB stars
(cf. Vrard+ 2022)

Main-sequence progenitors (cf. Christopher Lindsay’s talk on Thursday) — Main-sequence **progenitors**
(cf. Christopher Lindsay’s talk on Thursday)

Mixed Mode Diagnostics in TESS?

Hon et al. (in prep.)

Back to Basics

\(\nu_\text{max}\) and Amplitudes

(RHD simulations courtesy of Joel D. Tanner)

Pulsations emerging from
3D hydrodynamic simulations
(e.g. Zhou+ 2020, 2021)

cf. Zhou Yixiao’s talk on Thursday

Stochastic Driving & Damping

Describing pulsations
in the Fokker-Planck picture
with a stochastic wave equation:
Philidet et al. 2021, 2022

\[\tiny\begin{aligned} & \mathrm{d} A_\mu = \left(\mathcal{G}_\mu - \dfrac{1}{2}\sum_{\lambda\nu} \dfrac{\partial \mathcal{D}^{1/2}_{\mu\lambda}}{\partial A_\nu} \mathcal{D}^{1/2}_{\lambda\nu} \right) ~ \mathrm{d} t + \sum_{\nu} \mathcal{D}^{1/2}_{\mu\nu} \circ \mathrm{d} W_{A\nu}~,\\ & \mathrm{d} \Phi_\mu = \left(\mathcal{H}_\mu - \dfrac{1}{2}\sum_{\lambda\nu} \dfrac{\partial \mathcal{F}^{1/2}_{\mu\lambda}}{\partial \Phi_\nu} \mathcal{F}^{1/2}_{\lambda\nu} \right) ~ \mathrm{d} t + \sum_{\nu} \mathcal{F}^{1/2}_{\mu\nu} \circ \mathrm{d} W_{\Phi\nu}~, \end{aligned}\] \[ \tiny \dfrac{\partial w}{\partial t} = -\dfrac{\partial w\mathcal{G}_\mu}{\partial A_\mu} - \dfrac{\partial w\mathcal{H}_\mu}{\partial \Phi_\mu} + \dfrac{1}{2}\dfrac{\partial^2 w\mathcal{D}_{\mu\nu}}{\partial A_\mu\partial A_\nu} + \dfrac{1}{2}\dfrac{\partial^2 w\mathcal{E}_{\mu\nu}}{\partial A_\mu\partial \Phi_\nu} + \dfrac{1}{2}\dfrac{\partial^2 w\mathcal{F}_{\mu\nu}}{\partial \Phi_\mu \partial\Phi_\nu}~. \]