Rotation Effects

The differential equations and boundary conditions laid out in the Dimensionless Formulation 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, 2003a; Bouabid et al., 2013; Townsend, 2020).

Within the TAR, the solution forms given in equation (6) are replaced by

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

(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,

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

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms in equation (7) reduce to those given in equation (6).

Conversely, for the Rossby family

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

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

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

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 \(\lambda\)).



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 (8) and (9).