Figure 23 displays the typical gravitational waveforms for
, which clearly reflect the features of the
orbital evolution and subsequent merger processes (tidal disruption or not) as described in the following. In
the early inspiral phase in which
and
, two objects behave like point masses. In
addition, general relativistic effects to the orbital motion are not extremely strong. For such a phase, the
signal of gravitational waves is the chirp signal that can be well reproduced by the PN approximation for
the two-body problem [25].
For a close orbit in which the finite-size effect is still negligible but general relativistic gravity between
two objects plays a role, it is known that the simple PN analysis fails to provide a precise waveform.
Comparisons of the waveforms derived through PN analysis and through numerical computation for BH-BH
binaries [36, 30, 31, 5
, 179] propose that a better waveform is phenomenologically derived using the
Taylor-T4 formula. This method requires a special summation method of PN high-order terms in
the equations of motion, which include gravitational radiation reaction effects in an adiabatic
approximation. First, one needs to calculate the evolution of the orbital angular velocity
through
up to 3.5PN order by solving the following ordinary differential equations [36
]
Figure 23 shows that gravitational waveforms in the late inspiral phase before the onset of the merger
(or tidal disruption ) indeed agree with the result derived by the Taylor-T4 formula for the BH-NS binaries,
as in the case of BH-BH binaries. This is natural because of the equivalence principle for general
relativity [227]. (Note that in the first few wave cycles, the agreement is not very good. This
is because the initial condition given for their simulation was not in an exact quasi-circular
orbit.)
The waveforms may deviate from the prediction by the Taylor-T4 formula before the onset of the merger
for a small value of or for a large NS radius. The reason for this is that the NS is tidally deformed by
the BH, and, as a result, the pure Taylor-T4 formula, in which the tidal-deformation effect is not taken into
account, is not a good formula for such a phase. The features of gravitational waveforms in the final inspiral
phase are summarized later.
By contrast, for a sufficiently large value of or for a sufficiently compact NS, the tidal-deformation
effect is negligible, and hence, the waveforms are quite similar to those for a BH-BH binary as mentioned
above. For a small degree of tidal deformation and mass shedding, most of the NS material falls
into the BH simultaneously (this case corresponds to type-II according to the definition of
Figure 18
). In such a case, a fundamental QNM of a BH is excited (see the waveform for the
soft EOS in Figure 23
), and the highest frequency of gravitational waves is determined by the
QNM.
The degree of the QNM excitation depends strongly on the degree of tidal deformation and mass
shedding. The primary reason for this is that a phase cancellation is concerned in the excitation; here, the
phase cancellation is the amount that the gravitational waves emitted in a non-coherent manner (with
different phases) interfere with each other to suppress the amplitude of gravitational waves [141, 187, 139].
With increasing degree of mass shedding, the phase cancellation effect plays an increasingly
important role and the amplitude of the QNM-ringdown gravitational waves decreases. For the case
in which a NS is tidally disrupted far outside the ISCO, this effect is significantly enhanced
because the NS material does not simultaneously fall into the BH. Rather, a widely spread
material, for which the density is much smaller than the typical NS’s density, falls into the
BH from a wide region of the BH surface spending a relatively long time duration (this case
corresponds to type-I according to the definition of Figure 18). Here, it is appropriate to point out
why infall occurs from a wide region of the BH surface; the BH mass is small for the case in
which mass shedding occurs for
, and thus, the areal radius of the BH is smaller than or
as small as the NS radius. All these effects are discouraging for efficiently exciting a QNM,
and therefore, the amplitude of the QNM-ringdown gravitational waves is strongly suppressed
for the case in which tidal disruption occurs (see the waveform for EOS 2H in Figure 23
).
For the case in which tidal disruption occurs, the highest frequency of gravitational waves is
approximately determined by the orbital frequency at tidal disruption, not by the frequency of a
QNM.
One important remark here is that this highest, characteristic frequency is not in general determined by the frequency at the onset of mass shedding. Even after the onset of mass shedding, the NS continues to be a self-gravitating star for a while and gravitational waves associated with an approximately-inspiral motion are emitted. After a substantial fraction of gravitational waves is emitted and thus, the orbital separation becomes sufficiently small, tidal disruption occurs. At such a moment, the amplitude of the gravitational waves damps steeply, and hence, the highest frequency of gravitational waves should be determined by the tidal-disruption event.
The qualitative features summarized above depend on the BH spin; for binaries composed of a BH of high spin, tidal disruption may occur for a high mass ratio, and hence, the infall process of the tidally-disrupted material into the BH may be qualitatively modified. This is well reflected in gravitational waveforms, as described in the next Section 3.6.2.
Gravitational waveforms are significantly modified in the presence of BH spin. Figure 24 plots gravitational
waveforms for
with the same stiff EOS (HB EOS) and with the same initial angular velocity
(
) but with different values of the BH spin. This obviously shows that with
increasing the BH spin, the lifetime of the binary system increases and hence the number of
wave cycles increases. This is explained primarily by the spin-orbit coupling effect (see also
Section 3.3), which brings a repulsive force into the BH-NS binary for the prograde spin of the BH.
Due to the presence of this repulsive force, the orbital separation of the ISCO (the absolute
value of the binding energy there) can be smaller (larger) than that for the non-spinning BH.
This effect increases the lifetime of the binary, and furthermore, enhances the chance for tidal
disruption of the NS because a circular orbit with a closer orbital separation is allowed. Second,
the repulsive force reduces the orbital velocity for a given separation, because the centrifugal
force may be weaker for a given separation to maintain a quasi-circular orbit. The decrease
of the orbital velocity results in the decrease of the gravitational-wave luminosity, and this
decelerates the orbital evolution as a result of gravitational radiation reaction, making the lifetime of
the binary longer and increasing the number of cycles of gravitational waves. We note that
all these effects are also clearly reflected in the gravitational-wave spectrum, as is shown in
Section 3.7.
Figure 24 shows that for
, a ringdown waveform associated with a QNM of the BH is clearly
seen, whereas for
, such a feature is absent. This reflects the fact that tidal disruption
of the NS occurs for
far outside the ISCO, whereas it does not for
. For
, tidal disruption occurs but a ringdown waveform associated with a QNM is seen.
This is a new type of gravitational waveform. In this case, tidal disruption occurs near the
ISCO and a large fraction of the NS material falls into the BH. The infall occurs approximately
simultaneously and proceeds from a narrow region of the BH surface. This new type appears for the
case in which the BH mass (or mass ratio
) is large enough that the surface of the event
horizon is wider than the extent of the infalling material, as explained in Section 3.3.2 (see
Figure 18
).
The inspiral waveform matches well to that of the Taylor-T4 formula for binaries composed of a spinning
BH as well as for a non-spinning BH. Figure 24 also shows that matching is achieved as well as for
,
irrespective of the spin, except for the final phase just before the onset of the merger. As in the case of
, the deviation of gravitational waveforms from the prediction by the Taylor-T4 formula is
enhanced with increasing degree of tidal deformation, and with subsequent mass shedding and tidal
disruption.
http://www.livingreviews.org/lrr-2011-6 |
Living Rev. Relativity 14, (2011), 6
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