### 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 [210], 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 (104) and
in Figure 9. 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 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.