Pulsations of rotating stars

4.2 Pulsations of rotating stars

Pulsations of rotating relativistic stars are traditionally studied (when possible) as a time independent, linear eigenvalue problem, but recent advances in numerical relativity also allow the study of such pulsations via numerical time evolutions. The first quasi-radial mode frequencies of rapidly rotating stars in full general relativity have been recently obtained in [105

], something that has not been achieved yet with linear perturbation theory. The fundamental quasi-radial mode in full general relativity has a similar rotational dependence as in the relativistic Cowling approximation, and an empirical relation between the full GR computation and the Cowling approximation can be constructed (Figure 18

). For higher order modes, apparent intersections of mode sequences near the mass-shedding limit do not allow for such empirical relations to be constructed.

In the relativistic Cowling approximation, 2D time evolutions have yielded frequencies for the $l = 0$ to $l = 3$ axisymmetric modes of rapidly rotating relativistic polytropes with $N = 1.0$ [104 ]. The higher order overtones of these modes show characteristic apparent crossings near mass-shedding (as was observed for the quasi-radial modes in [330 ]).

Figure 18: The first fully relativistic, quasi-radial pulsation frequencies for a sequence of rapidly rotating stars (solid lines). The frequencies of the fundamental mode $F$ (filled squares) and of the first overtone $H1$ (filled circles) are obtained through coupled hydrodynamical and spacetime evolutions. The corresponding frequencies obtained from computations in the relativistic Cowling approximation [104 ] are shown as dashed lines. (Figure 16 of Font, Goodale, Iyer, Miller, Rezzolla, Seidel, Stergioulas, Suen, and Tobias [105 ].)

Numerical relativity has also enabled the first study of nonlinear r-modes in rapidly rotating relativistic stars (in the Cowling approximation) by Stergioulas and Font [293 ]. For several dozen dynamical timescales, the study shows that nonlinear r-modes with amplitudes of order unity can exist in a star rotating near mass-shedding. However, on longer timescales, nonlinear effects may limit the r-mode amplitude to smaller values (see Section 3.5.3).