5.3 Detection of the QNM ringing

It is well known in astrophysics that many stars will end their lives with a violent supernova explosion. This will leave behind a compact object which will oscillate violently in the first few seconds. Huge amounts of gravitational radiation will be emitted and the initial oscillations will consequently damp out. The gravitational waves will carry away information about the compact object. If the supernova remnant is a black hole we will observe a short monochromatic burst lasting a few tenths of a millisecond (see Figures 1View Image and 4View Image). If it is a neutron star we will observe a more complicated signal which will be the overlapping of several frequencies (see Figure 6View Image). The stellar signal will consist of a short burst (similar to that from a black hole) followed by a long lasting sinusoidal wave. Supernovae are quite frequent, the expected event rate is 5.8 ± 2.4 events per century per galaxy [205]. The amount of energy emitted in such an event depends on the details of the collapse. A spherical collapse will not produce any gravitational waves at all, while as much as 10− 3M c2 ⊙ [179Jump To The Next Citation Point] can be radiated away in a highly non-spherical one. The collapse releases an enormous amount of energy, at least equal to the binding energy of a neutron star, about 2 0.15 M ⊙c. Most of this energy should be carried away by neutrinos, and this is supported by the neutrino observations at the time of supernova SN1987A. But even if only 1% of the energy released in neutrinos is radiated in gravitational waves then the above number makes sense. The present numerical codes used to simulate collapse predict that the energy emitted as gravitational waves will be of the order of −4 −7 2 10 – 10 M ⊙c [47]. However, most of these codes are based on Newtonian dynamics and the few fully general relativistic ones are not 3-dimensional. Modern computers are still not able to perform realistic simulations of gravitational collapse in 3D, including all the important nuclear reactions and neutrino and photon transport. For example, most of the codes fail to explain the high average pulsar velocity which is believed to be a result of a boost that the neutron star gets during the collapse due to anisotropy in the neutrino distribution [51].

Although collapse may be the most frequent source for excitation of black hole and stellar oscillations there are other situations in which significant pulsations take place. For example, after the merger of two coalescing black holes or neutron stars it is natural to expect that the final object will oscillate. Thus the well known waveform for inspiralling binaries [45] will be followed by a short, but not yet properly known, period (the merger phase) and will end with the characteristic quasi-normal signal (ringing) of the newly created neutron star or black hole. During the inspiralling phase the stellar oscillations can be excited by the tidal fields of the two stars [127]. A detailed description of the gravitational wave emission and detection from binary black hole coalescences can be found in two recent articles by Flanagan and Hughes [90Jump To The Next Citation Point91Jump To The Next Citation Point]. In the same way smaller bodies falling on a neutron star or black hole will excite oscillations. Stellar or black hole oscillations can also be excited by a close encounter with another compact object [2038428].

Another potential excitation mechanism for stellar pulsation is a starquake, e.g., associated with a pulsar glitch. The typical energy released in this process may be of the order of 10 −10M c2 ⊙. This is an interesting possibility considering the recent discovery of so-called magnetars: Neutron stars with extreme magnetic fields [81]. These objects are sometimes seen as soft gamma-ray repeaters, and it has been suggested that the observed gamma rays are associated with starquakes. If this is the case, a fraction of the total energy could be released through nonradial oscillations in the star. As a consequence, a burst from a soft gamma-ray repeater may be associated with a gravitational wave signal.

Finally, a phase-transition could lead to a sudden contraction during which a considerable part of the stars gravitational binding energy would be released, and it seems inevitable that part of this energy would be channeled into pulsations of the remnant. Transformation of a neutron star into a strange star is likely to induce pulsations in a similar fashion.

One way of calibrating the sensitivity of detectors is to calculate the amplitude of the gravitational wave that would be produced if a certain fraction of the released energy were converted into gravitational waves. To obtain rough estimates for the typical gravitational wave amplitudes from a pulsating star we use the standard relation for the gravitational wave flux which is valid far away from the star [178Jump To The Next Citation Point]

--c3-- ˙ -1---dE- F = 16 πG |h| = 4πr2 dt , (62 )
where h is the gravitational wave amplitude and r the distance of the detector from the source. Combining this with i) dE ∕dt = E ∕2τ where τ is the damping time of the pulsation and E is the available energy, ii) the assumption that the signal is monochromatic (with frequency f), and iii) the knowledge that the effective amplitude achievable after matched filtering scales as the square root of the number of observed cycles, t = h √n--= h√f--τ eff, we get the estimates [18Jump To The Next Citation Point19Jump To The Next Citation Point]
( )1 ∕2( )1∕2( ) heff ∼ 2.2 × 10−21 ----E------ 2 kHz- 50-kpc- (63 ) 10− 6M ⊙c2 fgw r
for the f-mode, and
( ) ( ) ( ) − 22 E 1∕2 10 kHz 1∕2 50 kpc heff ∼ 9.7 × 10 10-−6M---c2 --f---- ---r--- (64 ) ⊙ gw
for the fundamental w-mode. Here we have used typical parameters for the pulsation modes, and the distance scale used is that to SN1987A. In this volume of space one would not expect to see more than one event per ten years or so. However, the assumption that the energy release in gravitational waves in a supernova is of the order of −6 2 10 M ⊙c is very conservative [179].

Similar relations can be found for black holes [178]:

( )1∕2( )1∕2( ) heff ∼ 5 × 10−22 ----E------ 1-kHz- 15-Mpc-- , (65 ) 10 −3M ⊙c2 fgw r
for stellar black holes, and
( )1 ∕2( )1 ∕2 ( ) heff ∼ 3 × 10−18 ----E---- 1 mHz-- 3-Gpc- , (66 ) 103 M ⊙c2 fgw r
for galactic black holes.

An important factor for the detection of gravitational waves are the pulsation mode frequencies. Existing resonant gravitational wave detectors, as well as laser interferometric ones which are under construction, are only sensitive in a certain bandwidth. The spherical and bar detectors are typically tuned to 0.6 – 3 kHz, while the interferometers are sensitive within 10 – 2000 Hz. The initial part of the QNM waveform, which carries away whatever deformation a collapse left in the spacetime, is expected to be for a neutron star in the frequency range of 5 – 12 kHz (w-mode). The subsequent part of the waveform is constructed from combination of the f- and p-modes. Still the present gravitational wave detectors are sensitive only in the frequencies of the f-mode. For a black hole the frequency will depend on the mass and rotation rate3, thus for a 10 solar mass black hole the frequency of the signal will be around 1 kHz, around 100 Hz for a 100 M ⊙ black hole and around 1 mHz for galactic black holes.

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