New Techniques for Rotational Measurements from Mixed-Mode Asteroseismology

AAS 241

Joel Ong (IfA), Lisa Bugnet (CCA), Sarbani Basu (Yale),
Emily Hatt (Birmingham), Martin Nielsen (Birmingham)

12 January 2023


Why Rotation?

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

Measuring Rotation

(sun-like star)

Spectroscopic \(v \sin i\):
\(\sim \text{PHz}\) regime \(\to\)

Photometric \(P_\text{rot}\):
\(\ll 1\ \mu\text{Hz}\)

Mixed Modes

p-modes: \[ \nu \sim \Delta\nu \left(n + {\ell \over 2} + \epsilon_p\right) \]

g-modes: \[ {1 \over \nu} \sim \Delta\Pi_l \left(n + {\ell \over 2} + \epsilon_g\right) \]

Mixed modes:

\[ \tan \pi\left(\nu - \nu_p \over \Delta\nu\right) \tan \pi \left({1 \over \nu_g} - {1 \over \nu} \over \Delta\Pi_l\right) \sim q(\nu) \]

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

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

I. Reggae

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


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

(Ong & Basu 2020)

Ong, Bugnet & Basu (2022):
generalisation to include rotational effects


II. Period Echelle CCFs

Stretched Echelle Power Diagrams

Stretching function: Ong & Gehan (in review)

Cross-Correlation Functions

Hatt et al., in prep

\[ CCF(C_\text{mag}, \delta\nu_\text{rot}) = \int \mathrm{PS}(\nu) \cdot \sum_j \delta(\nu - \tau^{-1}(\nu_j - m \delta\nu_\text{rot})) \mathrm d \nu \]

Hatt et al., in prep.



Construction of likelihood functions directly against the power spectrum permits rotational and other astrophysical properties of evolved stars to be inferred without conventional peakbagging

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