# Rotation Effects

The oscillation equations presented in the preceding sections are formulated for a non-rotating star. The corresponding equations for a rotating star are significantly more complicated, and a complete treatment of rotation lies beyond the scope of GYRE. However, GYRE can include two important effects arising from rotation.

## Doppler Shift

A lowest-order effect of rotation arises in the Doppler shift from transforming between the inertial reference frame and the local co-rotating reference frame. To incorporate this effect in the separated equations, all instances of the inertial-frame frequency $$\sigma$$ are replaced by the co-rotating frequency

(14)$\sigmac \equiv \sigma - m \Orot,$

where $$m$$ is the azimuthal order of the mode and $$\Orot$$ is the rotation angular frequency. GYRE assumes shellular rotation (see, e.g., Meynet & Maeder, 1997), and so the latter can in principle be a function of radial coordinate $$r$$. The corresponding modifications to the dimensionless formulation involve replacing the dimensionless inertial-frame frequency $$\omega$$ with the dimensionless co-rotating frequency

$\omegac \equiv \omega - m \Orot \sqrt{\frac{R^{3}}{GM}}.$

## Perturbative Coriolis Force Treatment

Another lowest-order effect of rotation arises from the Coriolis force. For slow rotation, this effect can be determined through a perturbation expansion technique (see, e.g., section 19.2 of Unno et al., 1989). To first order in $$\Orot$$, the frequency of a mode is shifted by the amount

$\Delta \sigma = m \int_{0}^{R} \Orot \, \deriv{\beta}{r} \diff{r},$

where the rotation splitting kernel is

$\deriv{\beta}{r} = \frac{\left\{ \txir^{2} + [\ell(\ell+1) - 1] \txih^{2} - 2 \txir \txih \right\} \rho r^{2}} {\int_{0}^{R} \left\{ \txir^{2} + \ell(\ell+1) \txih^{2} \right\} \rho r^{2} \diff{r}}$

In this latter expression, the eigenfunctions $$\txir$$ and $$\txih$$ are evaluated from solutions to the oscillation equations without rotation. Therefore, the expression above for $$\Delta \sigma$$ can be applied as a post-calculation correction to non-rotating eigenfrequencies.

## Non-Perturbative Coriolis Force Treatment

The perturbation expansion technique above breaks down when $$\Orot/\sigmac \gtrsim 1$$. To deal with such cases, the gyre frontend  can incorporate a non-perturbative treatment of the Coriolis force based on the ‘traditional approximation of rotation’ (TAR). The TAR was first introduced by Eckart (1960; Hydrodynamics of Oceans and Atmospheres) and has since 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 eqn. (8) are replaced by

(15)\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}

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 (see Townsend 2020). Together with the associated eigenvalue $$\lambda$$, they depend on the harmonic degree $$\ell$$ and azimuthal order $$m$$, and the spin parameter

$q \equiv \frac{2 \Orot}{\sigmac}.$

### Solution Families

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

(16)\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 } \Orot \rightarrow 0.\end{split}

and $$\lambda \rightarrow \ell(\ell+1)$$. With these expressions, the solution forms (15) reduce to those given in eqn. (8).

Conversely, for the Rossby family

(17)\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 } \Orot \rightarrow 0.\end{split}

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

(18)$\sigmac = \frac{2 m \Orot}{\ell(\ell+1)} \quad \text{as } \Orot \rightarrow 0,$

which is independent of the stellar structure.

### Implementing the TAR

To implement the TAR in the separated equations and boundary conditions, 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 $$\elle$$, an effective harmonic degree found by solving

$\elle(\elle+1) = \lambda.$

Similar steps are taken in the dimensionless formulation, but in the definitions of the dependent variables $$\{y_{1},y_{2},\ldots,y_{6}\}$$, $$\ell$$ is replaced by $$\elli$$, the effective harmonic degree evaluated at the inner boundary.

Footnotes