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.,

or equivalently Figure 6 shows the results of the fitting for the mass-shedding limit. The solid curve denotes Equation (101) and the points are the numerical results. The agreement is not perfect, but fairly good for .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

as demonstrated in Figure 7. The exponents of and in Equation (103) are empirically determined by requiring that the fitted curves lie near the data points for all the models. The agreement is not perfect, but sufficient for finding the orbital angular velocity at the ISCO within 10% error.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

and the curve that separates those two regions is shown in Figure 9. The solid curve denotes the critical mass ratio for each compactness. If the mass ratio of a BH-NS binary is larger than the critical one, the quasi-equilibrium sequence terminates by encountering the ISCO, while if smaller, it ends at the mass-shedding limit of the NS.
Living Rev. Relativity 14, (2011), 6
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