Mode analysis

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 [202 , 198 , 171 , 199 , 200 , 141 , 146 , 79 , 147 ]. 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 [128 ], Kokkotas and Schutz showed the existence of a new family of modes: the $w$ -modes [130 ]. 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 [138 , 21 ]; a part of the spectrum was found earlier by Kojima [119 ] and it has been shown that they exist also for odd parity (axial) oscillations [125 ]. 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. [138 ] (see also [17 ]), and the trapped modes found by Chandrasekhar and Ferrari [60 ] (see also [125 , 122 , 17 ]). 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) [22 ]. 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 [98 , 99 ] and [146 , 147 ]. Tables of frequencies and damping times of neutron star oscillations for twelve equations of state can be found in a recent work [19 ] which verifies and extends earlier work [141 ]. 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 [87

], $w1$ 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

The $f$ -mode (fundamental) is a stable mode which exists only for non-radial oscillations. The frequency is proportional to the mean density of the star and it is nearly independent of the details of the stellar structure. An exact formula for the frequency can be derived for Newtonian uniform density stars

This relation is approximately correct also for the relativistic case [17 ] (see also the discussion in Section 5.4). The $f$ -mode eigenfunctions have no nodes inside the star, and they grow towards the surface. A typical neutron star has an $f$ -mode with a frequency of 1.5 – 3 kHz and the damping time of this oscillation is less than a second (0.1 – 0.5 sec). Detailed data for the frequencies and damping times (due to gravitational radiation) of the $f$ -mode for various equations of state can be found in [141 , 19 ]. Estimates for the damping times due to viscosity can be found in [69 , 71 ].
The $p$ -modes (pressure or acoustic) exist for both radial and non-radial oscillations. There are infinitely many of them. The pressure is the restoring force and it experiences substantial fluctuations when these modes are excited. Usually, the radial component of the fluid displacement vector is significantly larger than the tangential component. The oscillations are thus nearly radial. The frequencies depend on the travel time of an acoustic wave across the star. For a neutron star the frequencies are typically higher than 4 – 7 kHz ( $p 1$ -mode) and the damping times for the first few $p$ -modes are of the order of a few seconds. Their frequencies and damping times increase with the order of the mode. Detailed data for the frequencies and damping times (due to gravitational radiation) of the $p1$ -mode for various equations of state can be found in [19 ].
The $g$ -modes (gravity) arise because gravity tends to smooth out material inhomogeneities along equipotential level-surfaces and buoyancy is the restoring force. The changes in the pressure are very small along the star. Usually, the tangential components of the fluid displacement vector are dominant in the fluid motion. The $g$ -modes require a non-zero Schwarzschild discriminant in order to have non-zero frequency, and if they exist there are infinitely many of them. If the perturbation is stable to convection, the $g$ -modes will be stable ( $ω2 > 0$ ); if unstable to convection the $g$ -modes are unstable ( $ω2 < 0$ ); and if marginally stable to convection, the $g$ -mode frequency vanishes. For typical neutron stars they have frequencies smaller than a hundred Hz (the frequency decreases with the order of the mode), and they usually damp out in time much longer than a few days or even years. For an extensive discussion about $g$ -modes in relativistic stars refer to [87 , 88 ]; and for a study of the instability of the $g$ -modes of rotating stars to gravitational radiation reaction refer to [134 ].
The $r$ -modes (rotational) in a non-rotating star are purely toroidal (axial) modes with vanishing frequency. In a rotating star, the displacement vector acquires spheroidal components and the frequency in the rotating frame, to first order in the rotational frequency $Ω$ of the star, becomes

An inertial observer measures a frequency of

From (58 ) and (59 ) it can be deduced that a counter-rotating (with respect to the star, as defined in the co-rotating frame) $r$ -mode appears as co-rotating with the star to a distant inertial observer. Thus, all $r$ -modes with $ℓ ≥ 2$ are generically unstable to the emission of gravitational radiation, due to the Chandrasekhar–Friedman–Schutz (CFS) mechanism [53 , 95 ]. The instability is active as long as its growth-time is shorter than the damping-time due to the viscosity of neutron star matter. Its effect is to slow down, within a year, a rapidly rotating neutron star to slow rotation rates and this explains why only slowly rotating pulsars are associated with supernova remnants [23 , 142 , 131 ]. This suggests that the $r$ -mode instability might not allow millisecond pulsars to be formed after an accretion induced collapse of a white dwarf [23 ]. It seems that millisecond pulsars can only be formed by the accretion induced spin-up of old, cold neutron stars. It is also possible that the gravitational radiation emitted due to this instability by a newly formed neutron star could be detectable by the advanced versions of the gravitational wave detectors presently under construction [161 ]. Recently, Andersson, Kokkotas and Stergioulas [24 ] have suggested that the $r$ -instability might be responsible for stalling the neutron star spin-up in strongly accreting Low Mass X-ray Binaries (LMXBs). Additionally, they suggested that the gravitational waves from the neutron stars, in such LMXBs, rotating at the instability limit may well be detectable. This idea was also suggested by Bildsten [44 ] and studied in detail by Levin [139 ].

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.

The curvature modes are the standard $w$ -modes [130 ]. They are the most important for astrophysical applications. They are clearly related to the spacetime curvature and exist for all relativistic stars. Their main characteristic is the rapid damping of the oscillations. The damping rate increases as the compactness of the star decreases: For nearly Newtonian stars (e.g. white dwarfs) these modes have not been calculated due to numerical instabilities in the various codes, but this case is of marginal importance due to the very fast damping that these modes will undergo. One of their main characteristics is the absence of significant fluid motion (this is a common feature for all families of $w$ -modes). Numerical studies have indicated the existence of an infinite number of modes; model problems suggest this too [128 , 40 , 13 ]. For a typical neutron star the frequency of the first $w$ -mode is around 5 – 12 kHz and increases with the order of the mode. Meanwhile, the typical damping time is of the order of a few tenths of a millisecond and decreases slowly with the order of the mode.
The trapped modes exist only for supercompact stars ( $R ≤ 3M$ ) i.e. when the surface of the star is inside the peak of the gravitational field’s potential barrier [60 , 125 ]. Practically, the first few curvature modes become trapped as the star becomes more and more compact, and even the $f$ -mode shows similar behavior [122 , 17 ]. The trapped modes, as with all the spacetime modes, do not induce any significant fluid motions and there are only a finite number of them (usually less than seven or so). The number of trapped modes increases as the potential well becomes deeper, i.e. with increasing compactness of the star. Their damping is quite slow since the gravitational waves have to penetrate the potential barrier. Their frequencies can be of the order of a few hundred Hz to a few kHz, while their damping times can be of the order of a few tenths of a second. In general no realistic equations of state are known that would allow the formation of a sufficiently compact star for the trapped modes to be relevant.
The interface modes [138 ] are extremely rapidly damped modes. It seems that there is only a finite number of such modes (2 – 3 modes only) [17 ], and they are in some ways similar to the modes for acoustic waves scattered off a hard sphere. They do not induce any significant fluid motion and their frequencies can be from 2 to 15 kHz for typical neutron stars while their damping times are of the order of less than a tenth of a millisecond.