2.5 Summary: quasi-equilibrium states

In this section, we have given an overview of the current status of the study of quasi-equilibrium BH-NS binary sequences in general relativity. Broadly speaking, two formulations have been proposed to construct quasi-equilibrium BH-NS binaries. The difference in these formulations comes primarily from the difference in the method for handling the BH singularity, the excision approach or the puncture approach. In these formulations, only five of ten components of Einstein’s equation are solved: constraint equations and the slicing condition. Deriving an improved formulation, in which full components of Einstein’s equation are solved, is left for the future (but see [204, 46] for proposed formulations and [218, 219] for a study of NS-NS binaries).

For the quasi-equilibrium sequences numerically derived so far, we mainly review those in [210Jump To The Next Citation Point], because it is the only systematic study. In particular, we highlight 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 result is shown in Equation (104View Equation) and in Figure 9View Image. Such a critical curve clearly classifies the possible final fate of BH-NS binaries, which depends on the ratio and the compactness of the NS for a given EOS of NS and BH spin.

As seen in Table 1, the parameter space surveyed so far is quite narrow. A systematic study of quasi-equilibrium sequences has been done only for binaries composed of a non-spinning BH and an irrotational NS with Γ = 2 polytropic EOS. It is desirable that the remaining parameter space is systematically surveyed near the future. Systematic numerical results for such a study will be quite helpful for predicting the final fate of BH-NS binaries and for checking the results derived in numerical simulations. Specifically, it is necessary to study in detail quasi-equilibrium sequences of binaries composed of a spinning BH and a NS with an EOS other than single polytrope.

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