3.3 Axisymmetric perturbations

3.3.1 Secular and dynamical axisymmetric instability

Along a sequence of nonrotating relativistic stars with increasing central energy density, there is always a model for which the mass becomes maximum. The maximum-mass turning point marks the onset of an instability in the fundamental radial pulsation mode of the star.

Applying the turning point theorem provided by Sorkin [285], Friedman, Ipser, and Sorkin [115] show that in the case of rotating stars a secular axisymmetric instability sets in when the mass becomes maximum along a sequence of constant angular momentum. An equivalent criterion (implied in [115]) is provided by Cook et al. [69]: The secular axisymmetric instability sets in when the angular momentum becomes minimum along a sequence of constant rest mass. The instability first develops on a secular timescale that is set by the time required for viscosity to redistribute the star’s angular momentum. This timescale is long compared to the dynamical timescale and comparable to the spin-up time following a pulsar glitch. Eventually, the star encounters the onset of dynamical instability and collapses to a black hole (see [273] for recent numerical simulations). Thus, the onset of the secular instability to axisymmetric perturbations separates stable neutron stars from neutron stars that will collapse to a black hole.

Goussard et al. [134] extend the stability criterion to hot proto-neutron stars with nonzero total entropy. In this case, the loss of stability is marked by the configuration with minimum angular momentum along a sequence of both constant rest mass and total entropy. In the nonrotating limit, Gondek et al. [127] compute frequencies and eigenfunctions of radial pulsations of hot proto-neutron stars and verify that the secular instability sets in at the maximum mass turning point, as is the case for cold neutron stars.

3.3.2 Axisymmetric pulsation modes

Figure 4: Apparent intersection (due to avoided crossing) of the axisymmetric first quasi-radial overtone () and the first overtone of the p-mode (in the Cowling approximation). Frequencies are normalized by , where is the central energy density of the star. The rotational frequency at the mass-shedding limit is 0.597 (in the above units). Along continuous sequences of computed frequencies, mode eigenfunctions are exchanged at the avoided crossing. Defining quasi-normal mode sequences by the shape of their eigenfunction, the sequence (filled boxes) appears to intersect with the sequence (triangle), but each sequence shows a discontinuity, when the region of apparent intersection is well resolved. (Figure 3 of Yoshida and Eriguchi [330]; used with permission.)

Axisymmetric () pulsations in rotating relativistic stars could be excited in a number of different astrophysical scenarios, such as during core collapse, in star quakes induced by the secular spin-down of a pulsar or during a large phase transition, or in the merger of two relativistic stars in a binary system, among others. Due to rotational couplings, the eigenfunction of any axisymmetric mode will involve a sum of various spherical harmonics , so that even the quasi-radial modes (with lowest order contribution) would, in principle, radiate gravitational waves.

Quasi-radial modes in rotating relativistic stars have been studied by Hartle and Friedman [145] and by Datta et al. [83] in the slow rotation approximation. Yoshida and Eriguchi [330] study quasi-radial modes of rapidly rotating stars in the relativistic Cowling approximation and find that apparent intersections between quasi-radial and other axisymmetric modes can appear near the mass-shedding limit (see Figure 4). These apparent intersections are due to avoided crossings between mode sequences, which are also known to occur for axisymmetric modes of rotating Newtonian stars. Along a continuous sequence of computed mode frequencies an avoided crossing occurs when another sequence is encountered. In the region of the avoided crossing, the eigenfunctions of the two modes become of mixed character. Away from the avoided crossing and along the continuous sequences of computed mode frequencies, the eigenfunctions are exchanged. However, each “quasi-normal mode” is characterized by the shape of its eigenfunction and thus, the sequences of computed frequencies that belong to particular quasi-normal modes are discontinuous at avoided crossings (see Figure 4 for more details). The discontinuities can be found in numerical calculations, when quasi-normal mode sequences are well resolved in the region of avoided crossings. Otherwise, quasi-normal mode sequences will appear as intersecting.

Figure 5: Frequencies of several axisymmetric modes along a sequence of rapidly rotating relativistic polytropes of , in the Cowling approximation. On the horizontal axis, the angular velocity of each model is scaled to the angular velocity of the model at the mass-shedding limit. Lower order modes are weakly affected by rapid rotation, while higher order modes show apparent mode intersections. (Figure 10 of Font, Dimmelmeier, Gupta, and Stergioulas [104].)

Several axisymmetric modes have recently been computed for rapidly rotating relativistic stars in the Cowling approximation, using time evolutions of the nonlinear hydrodynamical equations [104] (see [106] for a description of the 2D numerical evolution scheme). As in [330], Font et al. [104] find that apparent mode intersections are common for various higher order axisymmetric modes (see Figure 5). Axisymmetric inertial modes also appear in the numerical evolutions.

The first fully relativistic frequencies of quasi-radial modes for rapidly rotating stars (without assuming the Cowling approximation) have been obtained recently, again through nonlinear time evolutions [105] (see Section 4.2).

Going further   The stabilization, by an external gravitational field, of a relativistic star that is marginally stable to axisymmetric perturbations is discussed in [307].