Rotation Effects

The differential equations and boundary conditions laid out in the Dimensionless Equations section are formulated for a non-rotating star. Solving the oscillation equations for a rotating star is a challenging task, and a complete treatment lies beyond the scope of GYRE. However, GYRE does include two important modifications arising from rotation.

Doppler Shift

The lowest-order effect of rotation appears in the Doppler shift that arises when transforming between the inertial reference frame and the local co-rotating reference frame. To incorporate this effect in the oscillation equations, all instances of the inertial-frame frequency \(\sigma\) are replaced by the co-rotating frequency

\[\sigmac \equiv \sigma - m \Omega,\]

where \(m\) is the azimuthal order of the mode and \(\Omega\) is the rotation angular frequency. Because GYRE assumes so-called shellular rotation (see, e.g., Meynet & Maeder, 1997), both \(\Omega\) and \(\sigmac\) are functions of radial coordinate \(r\).

Coriolis Force

The Traditional Approximation of Rotation

Higher-order effects of rotation arise through the Coriolis force, which appears in the linearized momentum equation to correct for the non-inertial nature of the co-rotating reference frame. GYRE incorporates an approximate treatment of the Coriolis force based on the traditional approximation of rotation (TAR), which was first introduced by Eckart (1960; Hydrodynamics of Oceans and Atmospheres) and has since then been used extensively within the pulsation community (see, e.g., Bildsten et al., 1996; Lee & Saio, 1997; Townsend, 2003; Bouabid et al., 2013; Townsend, 2020).

Within the TAR, the solution forms given in equation (5) are replaced by

(6)\[\begin{split}\begin{aligned} \xir(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txir(r) \, \houghr(\theta) \, \exp(\ii m \phi -\ii \sigma t) \right], \\ \xit(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \frac{\hought(\theta)}{\sin\theta} \, \exp(\ii m \phi -\ii \sigma t) \right], \\ \xip(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \txih(r) \, \frac{\houghp(\theta)}{\ii \sin\theta} \, \exp(\ii m \phi -\ii \sigma t) \right], \\ f'(r,\theta,\phi;t) &= \operatorname{Re} \left[ \sqrt{4\pi} \, \tf'(r) \, \houghr(\theta) \, \exp(\ii m \phi -\ii \sigma t) \right] \end{aligned}\end{split}\]

(cf. equations 1-3 of Townsend, 2020). Here, the Hough functions \(\houghr\), \(\hought\) and \(\houghp\) are the eigenfunctions obtained by solving Laplace’s tidal equations (TEs), a second-order system of differential equations and boundary conditions in the polar (\(\theta\)) coordinate. Together with the associated eigenvalue \(\lambda\), depend on the harmonic degree \(\ell\)[1] and azimuthal order \(m\), and the spin parameter

\[q \equiv \frac{2 \Omega}{\sigmac}.\]

Solution Families

Solutions to the TEs can be grouped into two families based on the behavior of the eigenfunctions and eigenvalue in the limit \(\Omega \rightarrow 0\). For the gravito-acoustic family,

(7)\[\begin{split}\left. \begin{aligned} \houghr(\theta) \ \rightarrow & \ Y^{m}_{\ell}(\theta,0) \\ \hought(\theta) \ \rightarrow & \ \sin\theta \pderiv{}{\theta} Y^{m}_{\ell}(\theta,0) \\ \houghp(\theta) \ \rightarrow & \ - m Y^{m}_{\ell}(\theta,0) \end{aligned} \right\} \quad \text{as } \Omega \rightarrow 0.\end{split}\]

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms in equation (6) reduce to those given in equation (5).

Conversely, for the Rossby family

(8)\[\begin{split}\left. \begin{aligned} \houghr(\theta) \ \rightarrow & \ 0 \\ \hought(\theta) \ \rightarrow & \ m Y^{m}_{\ell}(\theta,0) \\ \houghp(\theta) \ \rightarrow & \ - \sin\theta \pderiv{}{\theta} Y^{m}_{\ell}(\theta,0) \end{aligned} \right\} \quad \text{as } \Omega \rightarrow 0.\end{split}\]

and \(\lambda \rightarrow 0\). Moreover, Rossby-mode eigenfrequencies also show the limiting behavior

(9)\[\sigmac = \frac{2 m \Omega}{\ell(\ell+1)} \quad \text{as } \Omega \rightarrow 0,\]

which is independent of the stellar structure.

Incorporating the TAR

To incorporate the TAR in the oscillation equations, all instances of the term \(\ell(\ell+1)\) are replaced by the TE eigenvalue \(\lambda\). Then, all instances of the harmonic degree \(\ell\) are replaced by \(\elli\), an effective harmonic degree found by solving

\[\elli(\elli+1) = \lambda\]

at the inner boundary (remember, because \(\sigmac\) is a function of radial coordinate, so too are \(q\) and \(\lambda\)).

Footnotes

[1]The harmonic degree isn’t formally a ‘good’ quantum number in the TAR; however, it can still be used to identify Hough functions by considering their behavior in the limit \(\Omega \rightarrow 0\), as given in eqns (7) and (8).