Solar-like Oscillations:
Past, Present, &
Future
Colloquium @ Tsinghua University | Slides at http://hyad.es/talks
February 23, 2023
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
\[\Huge P \sim L^\alpha\]
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\small N^2 = {- g}\left.{\partial \log \rho \over \partial s}\right|_P{\mathrm d s \over \mathrm d r}\] 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
\[{c_s^2 k_r^2 \sim \omega^2 \left(1 - {{\color{blue} S_l}^2 \over \omega^2}\right)\left(1 - {{\color{orange}N}^2 \over \omega^2}\right)}\]
\[\Large \omega_+^2 \sim c_s^2 |\mathbf{k}|^2\]
\[{\color{gray} \omega_p > S_l, N}\]
\[\Large \omega_-^2 \sim N^2 {k_h^2 \over |\mathbf{k}|^2}\]
\[{\color{red} \omega_g < N, S_l}\]
\(\ell = 0\) MDI Doppler velocities
Power spectra of MDI dopplergrams
Mode frequencies satisfy \(\nu_{nl} \sim \Delta\nu\left(n + {l \over 2} + \epsilon_l(\nu)\right) + \mathcal{O}(1/\nu)\)
Resolved imaging of the Sun gives
mode frequencies at very high \(\ell\)!
Ground-based asteroseismology is expensive!
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.
Stellar Rotation
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.):
\[\scriptsize\begin{aligned}\sum_{i} c_i \left( \delta\omega_{\mathrm{rot}, i} \over m_i \beta_{i} \right) &\sim \int \Omega(r) \left(\sum_i c_i K_{i}(r)\right) \mathrm d r \\ & \to \boxed{\Omega(r_0)}\end{aligned}\]
Solar Rotation
In the Sun, a transition from solid-body to differential rotation at the base of the convective envelope
slow
fast
Inversions for Solar Structure
two standard
solar models
Inversions for Stellar Structure
Bellinger+ 2019:
first inference of structure from
a star with a convective core.
Bellinger+ 2017: comparison of interior structure in a binary
Evolved stars dominate our asteroseismic sample!
(facultative with Kepler, obligate with TESS)
(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
(Ong & Basu 2020)
\[\small\vec{\xi}_\text{mixed} \sim {\color{grey} \sum_i c_{\pi, i} \vec{\xi}_{\pi,i}} + {\color{red} \sum_j c_{\gamma, j} \vec{\xi}_{\gamma,j}}\]
\[\iff\]
\[\small\psi_\text{mol} = {\color{blue}\sum_i c_{1,i} \psi_{1,i}} + {\color{orange}\sum_j c_{2,j} \psi_{2,j}}\]
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
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
Earth 2.0
2026 (Planned)
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…)
