4.2 Mode analysis

The study of stellar oscillations in a general relativistic context already has a history of 30 years. Nevertheless, recent results have shown remarkable features which had previously been overlooked.

Until recently most studies treated the stellar oscillations in a nearly Newtonian manner, thus practically ignoring the dynamical properties of the spacetime [202198171199Jump To The Next Citation Point200141Jump To The Next Citation Point146Jump To The Next Citation Point79Jump To The Next Citation Point147Jump To The Next Citation Point]. The spacetime was used as the medium upon which the gravitational waves, produced by the oscillating star, propagate. In this way all the families of modes known from Newtonian theory were found for relativistic stars while in addition the damping times due to gravitational radiation were calculated.

Inspired by a simple but instructive model  [128Jump To The Next Citation Point], Kokkotas and Schutz showed the existence of a new family of modes: the w-modes [130Jump To The Next Citation Point]. These are spacetime modes and their properties, although different, are closer to the black hole QNMs than to the standard fluid stellar modes. The main characteristics of the w-modes are high frequencies accompanied with very rapid damping. Furthermore, these modes hardly excite any fluid motion. The existence of these modes has been verified by subsequent work [138Jump To The Next Citation Point21Jump To The Next Citation Point]; a part of the spectrum was found earlier by Kojima [119Jump To The Next Citation Point] and it has been shown that they exist also for odd parity (axial) oscillations [125Jump To The Next Citation Point]. Moreover, sub-families of w-modes have been found for both the polar and axial oscillations i.e. the interface modes found by Leins et al. [138Jump To The Next Citation Point] (see also [17Jump To The Next Citation Point]), and the trapped modes found by Chandrasekhar and Ferrari [60Jump To The Next Citation Point] (see also [125Jump To The Next Citation Point122Jump To The Next Citation Point17Jump To The Next Citation Point]). Recently, it has been proven that one can reveal all the properties of the w-modes even if one “freezes” the fluid oscillations (Inverse Cowling Approximation) [22Jump To The Next Citation Point]. In the rest of this section we shall describe the features of both families of oscillation modes, fluid and spacetime, for the case ℓ = 2.

4.2.1 Families of fluid modes

For non-rotating stars the fluid modes exist only for polar oscillations. Here we will describe the properties of the most important modes for gravitational wave emission. These are the fundamental, the pressure and the gravity modes; this division has been done in a phenomenological way by Cowling [64]. For an extensive discussion of other families of fluid modes we refer the reader to [9899] and [146147]. Tables of frequencies and damping times of neutron star oscillations for twelve equations of state can be found in a recent work [19Jump To The Next Citation Point] which verifies and extends earlier work [141Jump To The Next Citation Point]. In Table 2 we show characteristic frequencies and damping times of various QNM modes for a typical neutron star.


Table 2: Typical values of the frequencies and the damping times of various families of modes for a polytropic star (N = 1) with R = 8.86 km and M = 1.27 M ⊙ are given. p 1 is the first p-mode, g1 is the first g-mode [87Jump To The Next Citation Point], w1 stands for the first curvature mode and wII for the slowest damped interface mode. For this stellar model there are no trapped modes.
mode frequency damping time
f 2.87 kHz 0.11 sec
p1 6.57 kHz 0.61 sec
g1 19.85 Hz years
w1 12.84 kHz 0.024 ms
wII 8.79 kHz 0.016 ms

View Image

Figure 3: A graph which shows all the w-modes: curvature, trapped and interface both for axial and polar perturbations for a very compact uniform density star with M ∕R = 0.44. The black hole spectrum is also drawn for comparison. As the star becomes less compact the number of trapped modes decreases and for a typical neutron star (M ∕R = 0.2) they disappear. The Im (ω ) = 1∕damping of the curvature modes increases with decreasing compactness, and for a typical neutron star the first curvature mode nearly coincides with the fundamental black hole mode. The behavior of the interface modes changes slightly with the compactness. The similarity of the axial and polar spectra is apparent.

4.2.2 Families of spacetime or w-modes

The spectra of the three known families of w-modes are different but the spectrum of each family is similar both for polar and axial stellar oscillations. As we have mentioned earlier they are clearly modes of the spacetime and from numerical calculations appear to be stable.


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