The final BH spin depends sensitively on the mass ratio and initial BH spin. This can be understood by
the following simple analysis. In Newtonian gravity, the total orbital angular momentum for two point
masses in a circular orbit with an angular velocity is
Equation (123) gives rather qualitative estimate for the spin of the remnant BH. Nevertheless, it still
gives a good approximate value of the final spin with the choice of
as large as the
remnant disk mass is small. With this choice,
, and
for
, and
and for
. These values agree with the results derived by the CCCW [74
], KT [194
, 107
, 108
], and
UIUC [63
] groups within the error of
.
For a large BH spin with , a disk of a large mass (
) is often formed even for
(see below). In such cases, Equation (123
) overestimates the final BH spin. However, this
equation still captures the qualitative tendency of the final spin; e.g., for small BH spin, the final spin is
determined by the value of
and the larger values of
results in smaller final BH spin; for larger
values of
with a large BH spin
, the final BH spin is primarily determined by the initial BH
spin.
The mass and characteristic density of the remnant disk surrounding the BH depend sensitively
on the mass ratio (), the BH spin (
), and the EOS (or the compactness) of the NS.
Figures 19
– 21
illustrate this fact. Figure 19
displays a result of the disk mass as a function of the
NS compactness for
(left) and
(right) for various piecewise polytropic EOS
and for various values of
, reported by the KT group [109
]. This shows that the disk mass
decreases steeply and systematically with the increase of the compactness irrespective of
and
.
The left panel of Figure 20 plots together the results obtained by the UIUC, CCCW, and KT groups for
EOS with
and
(the revised result by the KT group is plotted here; results in
early work by the KT group do not agree with the result shown here [228
, 194
]; see below for the
reason). This shows that the disk mass increases steeply with BH spin (
) for given values
of
and
. The results by these three groups agree approximately with each other for
. The CCCW group showed for the first time that the disk mass decreases with the
increase of the inclination angle of the BH spin, and toward the limit to 90 degree, the disk
mass approaches to that of
. The right panel of Figure 20
plots the results by the KT
group for different compactness (using EOS HB with
; see Table 4). This
shows again that the disk mass increases with the increase of the BH spin, and also that for
high BH spin (e.g.,
), the disk mass is larger than
even for
with
.
Figure 21 shows disk mass as a function of NS compactness for
and
as performed
by the KT group [109
]. A steep decrease in disk mass with increasing compactness is found irrespective of
the values of
and
. Simulations for BH-NS binaries with a spinning BH for particular values of the
compactness or mass ratio were also performed by the CCCW and UIUC groups for
and 0.75,
respectively. The results of the CCCW group for
and
with various EOS agree with those
of Figure 21
within
10 – 20%. We note that the disk mass may be different in the different EOS even
with the same compactness of the NS, because the density profile of the NS, and the resulting
susceptibility to the BH tidal force is different. Thus, this disagreement is reasonable. Also, the
results of the UIUC group for
,
, and
with the
-law EOS
(the disk mass
) agree with the relation expected from Figure 21
within
20%
error.
To clarify the dependence of the disk mass on relevant parameters, we consider three types of
comparisons. First, we consider the case in which ,
, and the EOS are fixed, but
is
varied. In this case, the disk mass monotonically decreases with the increase of
for many
cases. For example, for
and
with
EOS, the disk mass is larger
(smaller) than
for
(
) [63
, 74
, 154
]. However, we should point out the
exception to this rule, because for the case in which a large-mass disk is formed, this rule may
not hold. For example, the comparison between the left and right panels of Figure 19
shows
that for relatively small compactness (
), a large-mass disk is formed for high BH
spins and the disk mass depends only weakly on the value of
for given values of
and
.
Second, we consider the case in which and
are fixed, but
is varied. The UIUC group
compared the results for
, and
for
and
with the
-law
EOS (
), and the resulting disk mass at
10 ms after the onset of the merger is
0.008, 0.039, and
for
, and
, respectively [63
]. The CCCW group
performed a similar study for
, and
for
and
with the
-law
EOS (
), and found that the disk mass at 10 ms after the onset of the merger is 0.034,
0.126, and
for
, and
, respectively [74
]. Both groups found the
systematic steep increase of the disk mass with the increase of the BH spin (cf. the left panel of
Figure 20
). This was also reconfirmed by the KT group [109
] (see the right panel of Figure 19
and
Figure 21
).
Third, we consider the case in which and
are fixed, but the compactness,
, is varied.
Systematic work was recently performed by the KT group [107
, 108
, 109
], employing a piecewise
polytropic EOS. Figures 19
and 21
show that the disk mass decreases monotonically with the increase of
for given values of
and
. For producing a disk of mass larger than
for
,
should be smaller than
for
and
for
according to their results. For
, the condition is significantly relaxed.
Finally, the KT group [107, 108] found that the disk mass depends not only on the compactness but
weakly on the density profile. For the NS with a more centrally concentrated density profile, the disk mass
is smaller. The reason for this is that if the degree of the central mass concentration is smaller,
tidal deformation is enhanced and it encourages earlier tidal disruption after the onset of mass
shedding.
It is interesting to note that for and
, the mass of a disk surrounding a BH
can be larger than
for
with
and for
with
. For
,
the disk mass will be
for any realistic NS with
. For a BH of higher spin, the disk
mass will be even larger (cf. the left panel of Figure 20
). In addition, the disk mass does not decrease
steeply with the increase of
for such a high spin. In particular, for a small value of
, the disk mass
depends very weakly on
. All these results indicate that the disk mass is likely to be large even for a
higher value of
with a high BH spin (
). The maximum density of the disk increases
monotonically with the disk mass. For a disk mass larger than
, the maximum density is larger
than
for
and
for
[109
]. Hence, a high-mass disk with a
relatively small value of
is likely to be universally opaque against thermal neutrinos
for the typical geometrical thickness of the disk; a neutrino-dominated accretion disk is the
outcome and this is favorable for copious neutrino emission. This leads to the conclusion that the
coalescence of BH-NS binaries with a high-spin BH with
and
is a promising
progenitor for forming a BH plus a massive disk system; that is the candidate for a central-engine of
SGRB.
Before closing this subsection, we should note that different groups have reported different
quantitative results for the disk mass in their earlier work, which has been improved upon.
For example, the earlier work of the KT group presented a small disk mass [228, 194]. This
is mainly due to an unsuitable choice of computational domain and partly due to a spurious
numerical effect associated with insufficient resolution and an unsuitable choice of AMR grid.
The first work of the UIUC group also underestimated the disk mass [62]. This is due to an
unsuitable prescription for handling the atmosphere. However, these have been subsequently
fixed.
Generally speaking, the quantitative disagreement is due to the numerics. First, the fluid elements in the
disk have to acquire a sufficiently large specific angular momentum which is larger than that at the ISCO of
the remnant BH. The material that forms a disk obtains angular momentum by a hydrodynamic angular
momentum transport process from the inner part of the material. This implies that such a transport process
has to be accurately computed in a numerical simulation. However, it is well known that this
is one of the challenging tasks in computational astrophysics. Second, to avoid spurious loss
and transport of the angular momentum, a high-resolution computation is required. However,
the disk material is located in a relatively distant orbit around the central BH. In the AMR
scheme, which is employed in all the groups, the resolution in this region is usually poorer
than that in the central region. This might induce a spurious loss of angular momentum and
resulting decrease of disk mass, even by a factor of 2. However, these issues are being
resolved with the improvement of computational resources, the efficiency of the numerical code,
and the skill for computation with the AMR algorithm. The left panel of Figure 20
shows as
much.
http://www.livingreviews.org/lrr-2011-6 |
Living Rev. Relativity 14, (2011), 6
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