1.2 Tidal problem around a BH: Effects of BH spin and the NS EOS

As shown by a simple estimate in the previous Section 1.1, the final fate of BH-NS binaries depends primarily on the mass ratio, Q, and the compactness of the NS, 𝒞. However, a detailed analysis has shown that the BH spin and the NS EOS also play an important role in determining the final fate. In particular, the effect of the BH spin can be significant.

A general relativistic study for the criterion of mass shedding (note again that this is not the criterion of tidal disruption) was first performed using an analysis of the tidal interaction between a fluid star and a Kerr BH in circular orbit [71Jump To The Next Citation Point, 137, 135Jump To The Next Citation Point, 191Jump To The Next Citation Point, 225Jump To The Next Citation Point, 94Jump To The Next Citation Point, 69Jump To The Next Citation Point, 70Jump To The Next Citation Point, 154Jump To The Next Citation Point]. In this class of analysis, one handles a fluid star orbiting a Kerr BH in a circular orbit (the center of the fluid star has a geodesic motion around the Kerr BH), and analyzes the structure of the star taking into account the tidal field of the Kerr BH located at the center. Specifically, one analyzes the Euler equation in the following form

du 1 ∂P ∂ ϕ --i-= − ----i-− ---i − Cijxj, (11 ) dτ ρ ∂x ∂x
where τ is the affine parameter of a geodesic around the BH, xi is a coordinate orthogonal to the geodesic, u i denotes the internal velocity of the fluid star, ρ is the rest-mass density, P is the pressure, ϕ is the Newtonian potential, which obeys Δ Ï• = 4πG ρ, and Cij determines the lowest-order tidal field [135]. Higher-order corrections of the tidal potential based on the Manasse–Misner formalism [134] are also given in [94Jump To The Next Citation Point].

The gravitational effect of the fluid star on the orbital motion, the gravitational radiation reaction, and the relativistic effect on the self-gravity of the star are in general neglected (see [69] for a more detailed phenomenological analysis). Thus, this analysis is valid only for the case in which the BH mass is much larger than the NS mass and t the radius of the NS is relatively large: For the case of BH-NS binaries of mass ratio ≲ 10, this analysis can provide only a qualitative or semi-quantitative nature of the tidal effect and orbital evolution. However, by this analysis, several important qualitative properties of the criterion of mass shedding have been found. One of the most important findings is that the criteria for the onset of mass shedding (and/or for tidal disruption) depend sensitively on the BH spin, a. The reason for this is that the angular velocity at the ISCO depends sensitively on the BH spin; 3 3∕2 GΩISCOMBH ∕c = 1∕6 for the non-spinning BH whereas it is 1/2 for the BH of maximum possible spin, a = 1 [16Jump To The Next Citation Point, 186Jump To The Next Citation Point]. Here, the spin is non-dimensional and defined by a ≡ cJBH∕GM B2H where JBH is the angular momentum of the BH. As Equation (10View Equation) indicates, the increase of ΩISCO results in the increase of the critical value of Q for mass shedding (tidal disruption). Indeed, analyses [191Jump To The Next Citation Point, 225Jump To The Next Citation Point] indicate that the critical value for Q (or BH mass) increases by a factor of ∼ 15 if the spin is changed from zero to the maximum value (a = 1); e.g., for an incompressible fluid star, the maximum possible mass of a BH, which can cause mass shedding, was derived as

-MNS--- −1∕2 -RNS---3∕2 MBH = CM M ⊙(1.4M ⊙ ) (10 km ) , (12 )
where CM ≈ 4.6 for a = 0,7.8 for a = 0.5,12 for a = 0.75,19 for a = 0.9, and 68 for a = 1.

Another important finding is that the criteria for mass shedding (and tidal disruption) depend on the NS EOS even if the mass and radius of the NS are identical [225Jump To The Next Citation Point, 94Jump To The Next Citation Point]: NSs with stiffer EOS, (i.e., with high adiabatic index) have a relatively uniform density profile and are susceptible to tidal deformation, mass shedding, and tidal disruption by the BH tidal field. Consequently, condition (12View Equation) is modified by the EOS. In the calculation with compressible models [225Jump To The Next Citation Point, 94Jump To The Next Citation Point], the maximum possible mass of a BH, which can cause mass shedding, may be reduced by 10 – 20% for soft EOS (with the relatively small adiabatic index). This implies that not only the compactness of the NS but also its EOS, which is still unknown, is reflected in the tidal disruption event. General relativistic studies for quasi-equilibria of BH-NS binaries in addition show that the self-gravity of the NS significantly reduces the maximum mass of the BH for the onset of mass shedding (see Section 1.4 and Figure 22View Image).


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