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 [71, 137, 135, 191, 225, 94, 69, 70, 154]. 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

where is the affine parameter of a geodesic around the BH, is a coordinate orthogonal to the geodesic, denotes the internal velocity of the fluid star, is the rest-mass density, is the pressure, is the Newtonian potential, which obeys , and 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 [94].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, . The reason for this is that the angular velocity at the ISCO depends sensitively on the BH spin; for the non-spinning BH whereas it is 1/2 for the BH of maximum possible spin, [16, 186]. Here, the spin is non-dimensional and defined by where is the angular momentum of the BH. As Equation (10) indicates, the increase of results in the increase of the critical value of for mass shedding (tidal disruption). Indeed, analyses [191, 225] indicate that the critical value for (or BH mass) increases by a factor of 15 if the spin is changed from zero to the maximum value (); e.g., for an incompressible fluid star, the maximum possible mass of a BH, which can cause mass shedding, was derived as

where for for for for , and for .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 [225, 94]: 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 (12) is modified by the EOS. In the calculation with compressible models [225, 94], 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 22).

Living Rev. Relativity 14, (2011), 6
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