Exploring this issue quantitatively requires dynamic simulations, and the results of such simulations are reviewed in Section 3. However, the study of quasi-equilibrium sequences can also provide a guide to where separation mass shedding or dynamic merger occurs. In the following, we summarize the quantitative insights obtained from the study of quasi-equilibrium sequences. Specifically, we will review qualitative expressions that may be used to predict whether a BH-NS binary of arbitrary mass ratio and NS compactness encounters an ISCO before shedding mass or not.
First, we summarize the results for the binary separation and the orbital angular velocity at which mass
shedding from the NS surface occurs. In Newtonian gravity and semi-relativistic approaches, simple
equations may be introduced to fit the effective radius of a Roche lobe [225, 94, 151, 60]. In [202, 203
] a
fitting equation is introduced for binaries composed of a non-spinning BH and a corotating NS in general
relativity. In this section, we review how to derive a fitting equation from data in [210
] for a non-spinning
BH and an irrotational NS.
To derive a fitting formula, we need to determine the orbital angular velocity at the mass-shedding limit.
However, it is not possible to construct cusp-like configurations by the numerical code used in [210], which
is based on a spectral method and accompanied by the Gibbs phenomena. This is also the case for a
configuration with smaller values of
, even though a cusp-like configuration does not appear (here
is a mass-shedding indicator defined by Equation (99
)). Thus, the data points for such close orbits have
to be determined by extrapolation. For this purpose, Taniguchi et al. [210
] tabulated
as a
function of the orbital angular velocity and extrapolated the sequence to
by using
fitting polynomial functions to find the orbits at the onset of mass shedding. Figure 5
shows
an example of such extrapolations for sequences of NS compactness
with mass
ratios
, 2, 3, and 5 [210
]. From the extrapolated results toward
, the orbital
angular velocity at the mass-shedding limit is approximately determined for each set of
and
.
To derive a fitting formula for the orbital angular velocities at the mass-shedding limit for all the values
of and
, the qualitative Newtonian expression of Equation (7
) is useful. By fitting sequence data to
this expression, Taniguchi et al. [210
] determined the value of
for
polytropic EOS as 0.270,
i.e.,
It may be interesting to note that the value of is the same as that found for
quasi-equilibrium sequences of NS-NS binaries in general relativity in [214] and of BH-NS binaries in
general relativity where the NS is corotating in [202
, 203
]. The value of
could be widely used
for an estimation of the orbital angular velocity at the mass-shedding limit of NS in a relativistic binary
system with
.
One of the most important pieces of information for relativistic close binaries is the binary separation (or the orbital angular velocity) at which the minimum of the binding energy appears, corresponding to the ISCO. The minimum point is located by fitting three nearby points of the sequence to a second-order polynomial, because the numerical data is discrete and does not necessarily give the exact minimum.
A simple empirical fitting that predicts the angular velocity at the ISCO for an arbitrary
companion orbiting a BH may be derived in the manner of [210
]. In their approach, one searches for an
expression with three free parameters that express
as a function of the mass ratio
and the
compactness
of the companion. Then the three parameters are determined by matching to
three known values of
, namely, (1) that of a test particle orbiting a Schwarzschild BH,
(for
), (2) that of an equal-mass BH-BH system as computed in [40],
(for
and
), and finally (3) that of a BH-NS configuration as
computed in [210
],
(for
and
). A further requirement arises from
the fact that for a test particle (with
), the expression should be independent of the
companion’s compactness. A good fit to the numerical data is found in [210
] with the expression
Combining Equations (102) and (103
), we can identify the critical binary parameters that separate the final
fates. The binary encounters an ISCO before reaching mass shedding or the NS reaches the mass-shedding
limit. Figure 8
illustrates the final fate of BH-NS binaries for
. The solid curve denotes the
orbital angular velocity at the mass-shedding limit, and the long-dashed one denotes it at the ISCO. As
seen from Equations (102
) and (103
), both of these curves depend on the mass ratio
,
but in different ways, which leads to the intersection of the two curves. An inspiraling binary
evolves along horizontal lines towards increasing
, starting at the left and moving toward the
right, until reaching either the ISCO or the mass-shedding limit. After the binary reaches the
ISCO for sufficiently large mass ratio, we cannot predict the fate of the NS, because it is in a
dynamic plunge phase (see Section 3), but nevertheless the mass-shedding limit for unstable
quasi-equilibrium sequences is included as the dotted curve in Figure 8
. As shown in Figure 8
,
the model with mass ratio
(dotted-dashed line) encounters the ISCO, while that of
(dot-dot-dashed line) ends up at the mass-shedding limit. The intersection between the
mass-shedding and ISCO curves marks a critical point that separates the two distinct fates of the binary
inspiral.
When we eliminate from Equations (102
) and (103
), we can draw a curve of the critical mass
ratio, which separates BH-NS binaries that encounter an ISCO before reaching mass shedding, and vice
versa, as a function of the compactness of the NS. The equation, which gives the critical curve is expressed
as
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