# 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

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

## 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

where the rotation splitting kernel is

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 [1] 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. (7) are replaced by

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 \(\ell\)[2] and azimuthal order \(m\), and the spin parameter

### 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,

and \(\lambda \rightarrow \ell(\ell+1)\). With these expressions, the solution forms (12) reduce to those given in eqn. (7).

Conversely, for the Rossby family

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

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

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