It is not easy to strictly determine the condition of tidal disruption, because its concept is not as clear as that of mass shedding. One way to determine the condition is to use the property of gravitational waveforms in the merger phase; if the “cutoff” frequency is smaller than the frequency of a quasi-normal mode (QNM) of the remnant BH, we may conclude that tidal disruption occurs (see Section 3.6 for details of the cutoff frequency). Another way is to use the property of the merger remnant; we may recognize that tidal disruption occurs if the mass of a remnant disk surrounding a BH is substantial, say larger than 1% of the total rest mass at after the onset of mass shedding. We employ both criteria here to determine whether tidal disruption occurs or not.

First, we summarize the criterion for the case in which the BH spin is zero (). In this case, the condition of tidal disruption depends strongly on the mass ratio and compactness of the NS. According to the criterion associated with a gravitational-wave spectrum (see Figure 28, as an example), the condition of tidal disruption is for and for .

Figure 19, which displays the result of the disk mass for with several piecewise polytropic EOS, also illustrates that the condition of tidal disruption depends strongly on the values of and : the approximate condition for is for and for . We note that for , this conclusion is consistent with those of the CCCW [57] and UIUC [63] groups. These criteria agree approximately with those derived from the gravitational-wave spectrum. The AEI group studied the case of with -law EOS () and showed that the condition is for [154].

For a spinning BH, in particular for the case in which the spin vector aligns with the orbital angular momentum vector, the condition for tidal disruption is highly relaxed. Here, we focus only on the case in which the BH spin vector is aligned with the orbital angular momentum vector. From the disk mass as a function of compactness shown in Figure 21 derived by the KT group, we can read the criteria as follows. For , tidal disruption occurs irrespective of the value of for . However, for a counter-rotating spin (), the criterion for tidal disruption is significantly restricted (). For , tidal disruption occurs irrespective of the value of as far as .

Finally, we compare the conditions for tidal disruption and mass shedding for . Figure 22 plots threshold curves of tidal disruption, mass shedding in general relativity, and mass shedding in a tidal approximation in the plane of . If the value of or is smaller than that of the threshold curves, tidal disruption or mass shedding occurs. The points with error bar approximately denote the numerical results for the criterion of tidal disruption, based on the results by the KT group for , by the CCCW, KT, and UIUC groups for , and by the AEI group for . The solid and dashed curves denote the critical curves for the onset of mass shedding for EOS in general relativity [209, 210], and for the incompressible fluid in a tidal approximation (see Equation (12)), respectively. This shows that, for a realistically compact NS, , the condition for tidal disruption is more restricted than that for mass shedding. The primary reason for this is that for such large compactness with , tidal disruption occurs only for a small value of . For such a system, the time scale for gravitational radiation reaction is as short as the orbital period in close orbits. This indicates that at the onset of mass shedding, the radial approaching velocity induced by gravitational radiation reaction is high enough to significantly decrease (increase) the orbital radius (angular frequency) during the subsequent mass-shedding phase up to final tidal disruption. If the fraction of this decrease in the orbital radius is large enough to enforce the orbit inside the ISCO, tidal disruption is prohibited. This mechanism makes the condition of tidal disruption for BH-NS binaries more restricted than that of mass shedding.

On the other hand, if the value of is small, tidal disruption may occur even for a large value of . With increase of , the ratio of the time scale of gravitational radiation reaction to the orbital period at the ISCO increases (e.g., Equation (2)). For such a high- case, the effect of orbital decrease by gravitational radiation reaction after the onset of mass shedding becomes relatively minor, and therefore, the critical curves of tidal disruption and mass shedding approach each other.

For a BH-NS binary with high BH spin, mass shedding may occur even for high mass ratio (say 0 for , e.g., Equation (12)). For such a case, the conditions for tidal disruption and mass shedding may approximately agree with each other. This point should be clarified through future study. If this is the case, the quasi-equilibrium study plays an important role in determining the condition of mass shedding, because this also gives the (approximate) condition of tidal disruption.

Living Rev. Relativity 14, (2011), 6
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