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 [90,
91
]. 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 [203,
84,
27].
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
. 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 [178]
where
h
is the gravitational wave amplitude and
r
the distance of the detector from the source. Combining this
with i)
where
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,
, we get the estimates [18
,
19
]
for the f -mode, and
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
is very conservative [179].
Similar relations can be found for black holes [178]:
for stellar black holes, and
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 rate
, thus for a 10 solar mass black hole the frequency of the signal
will be around 1 kHz, around 100 Hz for a 100
black hole and around 1 mHz for galactic black holes.
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Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |