Nonaxisymmetric oscillations of rapidly rotating relativistic stars by conformal flatness approximation

We present a new numerical code to compute non-axisymmetric eigenmodes of rapidly rotating relativistic stars by adopting spatially conformally flat approximation of general relativity. The approximation suppresses the radiative degree of freedom of relativistic gravity and the field equations are cast into a set of elliptic equations. The code is tested against the low-order $f$ and $p$ modes of slowly rotating stars for which a good agreement is observed in frequencies computed by our new code and those computed by the full theory. Entire sequences of the low order counter-rotating $f$-modes are computed, which are susceptible to an instability driven by gravitational radiation.

Figure 1

Dimensionless pattern speed ${\omega }_p$ (defined as $\sigma /m$ where $\sigma$ is the eigenfrequency and $m$ is the azimuthal quantum number of the eigenmode) of counterrotating $f$ modes as a function of a stellar rotational parameter $T/W$. These modes become $l=m$ $f$ modes in the nonrotating limit of the star. For a slowly rotating star their pattern rotates against the stellar spin. These modes becomes unstable to the CFS instability at their zeroes when viscosity is neglected. The rightmost points of the mode sequence correspond to the mass-shedding limit of the star. The black dots are the neutral points of the instability computed by Ref. 1. From right to left they correspond to $m=2$, 3, 4 $f$ modes.

Figure 2

Perturbed fluid variables of the counterrotating bar mode near its neutral point of CFS instability ($T/W=0.0843$, the ratio of the polar to the equatorial radius is 0.650). The value of functions along each “$\theta =\mathrm{const}$” direction is plotted as a function of $r$. Each panel corresponds to the following functions: top-left, $\delta h$; top-right, $\delta {\pi }_r$; bottom-left, $\delta {\pi }_{\theta }/r$; bottom-right, $\delta {\pi }_{\phi }/r\sin \theta$.

Figure 3

Metric perturbations for the same model as Fig. 2 are plotted inside the star. Each panel corresponds to the following functions: top-left, $\delta \alpha$; top-right, $\delta \psi$; center-left, $\delta {\beta }_r$; center-right, $\delta {\beta }_{\theta }/r$; and bottom, $\delta {\beta }_{\phi }/r\sin \theta$.