Figure 1 displays contours of the conformal factor for a BH-NS binary with mass ratio and NS compactness . This contour plot shows the configuration at the smallest orbital separation, for which Taniguchi’s code can achieve a converged solution. The thick solid circle on the left-hand side denotes the position of the excised surface (the apparent horizon), while that on the right-hand side denotes the position of the NS surface. A saddle point presents between the BH and NS, and for this close orbit, it is located in the vicinity of the NS surface, suggesting that the orbit of the binary is close to the mass-shedding limit. The value of on the excised surface is not constant because a Neumann boundary condition (47) is imposed.

Figure 2 shows the binding energy () and the total angular momentum () as functions of the orbital angular velocity () for a NS with baryon rest mass () and for mass ratio . All the quantities shown are normalized to be dimensionless. The solid curves with the filled circles denote the numerical results, and the dashed curves, the results in the 3PN approximation [25]. The numerical sequences terminate at an orbit of cusp formation (i.e., at an orbit of the mass-shedding limit) before the ISCO is encountered, i.e., before a turning point (minimum) of the binding energy appears.

From the qualitative argument described in Section 1.1, the binary separation at the onset of mass shedding of a NS can be approximately derived as (see Equation (5))

If is greater by a sufficient amount than the ISCO separation , we may expect the NS to start shedding mass and to be tidally disrupted before being swallowed by the BH. Relation (100) suggests that the mass-shedding separation decreases with increasing mass ratio, , and compactness of the NS, . It is natural to expect to encounter minima in the binding energy and total angular momentum for binaries with sufficiently large mass ratio and NS compactness. Figure 3 shows the binding energy () and the total angular momentum () as functions of the orbital angular velocity () for a NS with baryon rest mass () and for . The NS compactness is the same as, but the mass ratio is larger than, that shown in Figure 2. In this model, the binary encounters an ISCO before the onset of mass shedding, i.e., we see minima in the binding energy and angular momentum just before the end of the sequence.Figure 3 shows that the turning points in the binding energy and the total angular momentum appear simultaneously to within numerical accuracy. This fact is more clearly seen in Figure 4 in which the binding energy versus total angular momentum for sequences of mass ratio but with different NS compactness is plotted. A simultaneous turning point in the binding energy and total angular momentum leads to a cusp in these curves. As suggested by Equation (100), turning points are not found for small compactness (e.g., the case of ), since the sequences terminate at mass shedding before encountering an ISCO. However, for larger compactness, these curves indeed form a cusp. Note that 3PN sequences cannot identify mass shedding and therefore always exhibit turning points.

Living Rev. Relativity 14, (2011), 6
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