BH-NS binaries have not been observed yet even in our galaxy in contrast to NS-NS binaries [205, 131].
However, many of statistical studies based on the stellar evolution synthesis suggest that the coalescence
will occur by 1 – 10% as frequently as that of NS-NS binaries in our galaxy and hence in the normal
spiral galaxies [144, 160, 223, 98, 97, 20, 21, 148] (every 106 – 107 years). In addition,
coalescence in elliptic galaxies could contribute to the total coalescence rate of the universe by a
significant fraction [147]. This implies that coalescence is likely to occur frequently in the Hubble
volume, and therefore, the evolution process and the final fate of BH-NS binaries deserve a
detailed theoretical study. In particular, the following two facts have recently enhanced the
motivation for the study of BH-NS binaries: First, BH-NS binaries in close orbits are among
the most promising sources for the large laser-interferometric gravitational-wave detectors,
such as LIGO [126
, 2
, 1], VIRGO [222, 4, 3], LCGT [120], and Einstein Telescope [91
, 92
]:
The frequency and amplitude of gravitational waves near the last orbit are estimated to give
The final fate of BH-NS binaries is classified into two categories; a NS is tidally disrupted by its companion BH before it is swallowed by the BH or a NS is simply swallowed by its companion BH in the final phase. There is a third possibility, in which stable mass transfer occurs after the onset of mass shedding of the NS by the BH tidal field. Although this may be possible, numerical simulations performed so far have not shown this to be the case, as will be mentioned in Section 1.6.
The final fate of a NS depends primarily on the mass of its companion BH and the compactness of the NS. When the BH mass is small enough or the NS radius is large enough, the NS will be tidally disrupted before it is swallowed by the BH. A necessary (not sufficient) condition for this is semi-quantitatively derived from the following analysis. Mass shedding of a non-spinning NS occurs when the tidal force of its companion BH at the surface of the NS is stronger than the self-gravity of the NS. This condition is approximately (assuming Newtonian gravity) written as
where We emphasize here that Equation (5) is the necessary condition for the onset of mass shedding, strictly
speaking. Tidal disruption occurs after substantial mass is stripped from the surface of the NS, during the
decrease of the orbital separation due to gravitational-wave emission. Thus, tidal disruption should
occur for a smaller orbital separation (larger orbital angular velocity) than that derived from
Equation (5
). We also note that the NS radius is assumed to depend weakly on the NS mass. If the
radius quickly increases with mass loss, tidal disruption may occur soon after the onset of mass
shedding.
Assuming that the binary is in a circular orbit with the Keplerian angular velocity, Equation (5) may be
written in terms of the angular velocity
as
Up to now, we implicitly assume that orbits with arbitrary orbital separations may be possible for the
binary system. However, BH-NS binaries always have the innermost stable circular orbit (ISCO) determined
by the general relativistic effect, and hence, we should impose the condition that mass shedding (and tidal
disruption) has to occur before the binary orbit reaches the ISCO in this analysis. According to PN
analysis [25], a non-dimensional orbital compactness parameter,
, at the ISCO, is
for
for a system composed of a non-spinning BH and a NS. (The tidal-deformation effect reduces
this value by
10 – 20% [209
, 210
]). In the presence of the ISCO, the condition for the onset of mass
shedding is written by
In the above simple estimate, the tidal-deformation effect of the NS to the orbital motion is not taken
into account. As a result of the tidal deformation, the gravitational force between two stars is modified, so
are the orbital evolution and the criterion for tidal disruption. Lai, Rasio, and Shapiro [111, 113, 112]
thoroughly investigated the effects associated with tidal deformation in the Newtonian framework with the
ellipsoidal approximation. They found that the two-body attractive force is strengthen by the
effect of the tidal deformation and, by this, the orbital separation of the ISCO is increased (see
[165, 166, 114
, 190] for related issues) and that gravitational waveforms in the late inspiral phase
are modified (see [66, 68, 67, 72] for related issues). Uryū and Eriguchi confirmed the fact
that the tidal force modifies the orbital motion of BH-NS binaries by numerically computing
equilibrium states of a binary composed of a point mass and a fluid star. The essence is that in
addition to the Newtonian potential, which has the form
, the correction term of the form
appears when a NS is tidally deformed. The magnitude of this correction increases
steeply with the decrease of the orbital separation, and modifies the location of the ISCO.
Also, the tidal effect accelerates the orbital evolution, because the orbital velocity (and the
centrifugal force) has to be increased to maintain circular orbits in an enhanced attractive
force, and then, the emissivity of gravitational waves is enhanced, leading to a shorter inspiral
time.
http://www.livingreviews.org/lrr-2011-6 |
Living Rev. Relativity 14, (2011), 6
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