2.4 Endpoint of sequences

One of the most important questions in the context of BH-NS binaries is whether the coalescence leads to mass shedding of the NS before reaching an ISCO, or whether the NS is swallowed by the BH before the onset of mass shedding. This question arises from the perspective of gravitational-wave observations and from the relation with SGRB. Gravitational waveforms in the final inspiral phase depend strongly on this issue, and hence, a precise understanding of this is necessary to predict gravitational waveforms theoretically. For launching a SGRB, the formation of an accretion disk surrounding the BH is one of the most promising models, and can occur only if the NS is disrupted prior to reaching an ISCO.

Exploring this issue quantitatively requires dynamic simulations, and the results of such simulations are reviewed in Section 3. However, the study of quasi-equilibrium sequences can also provide a guide to where separation mass shedding or dynamic merger occurs. In the following, we summarize the quantitative insights obtained from the study of quasi-equilibrium sequences. Specifically, we will review qualitative expressions that may be used to predict whether a BH-NS binary of arbitrary mass ratio and NS compactness encounters an ISCO before shedding mass or not.

2.4.1 Mass-shedding limit

First, we summarize the results for the binary separation and the orbital angular velocity at which mass shedding from the NS surface occurs. In Newtonian gravity and semi-relativistic approaches, simple equations may be introduced to fit the effective radius of a Roche lobe [225, 94, 151, 60]. In [202Jump To The Next Citation Point, 203Jump To The Next Citation Point] a fitting equation is introduced for binaries composed of a non-spinning BH and a corotating NS in general relativity. In this section, we review how to derive a fitting equation from data in [210Jump To The Next Citation Point] for a non-spinning BH and an irrotational NS.

To derive a fitting formula, we need to determine the orbital angular velocity at the mass-shedding limit. However, it is not possible to construct cusp-like configurations by the numerical code used in [210Jump To The Next Citation Point], which is based on a spectral method and accompanied by the Gibbs phenomena. This is also the case for a configuration with smaller values of χ ≤ 0.5, even though a cusp-like configuration does not appear (here χ is a mass-shedding indicator defined by Equation (99View Equation)). Thus, the data points for such close orbits have to be determined by extrapolation. For this purpose, Taniguchi et al. [210Jump To The Next Citation Point] tabulated χ as a function of the orbital angular velocity and extrapolated the sequence to χ = 0 by using fitting polynomial functions to find the orbits at the onset of mass shedding. Figure 5View Image shows an example of such extrapolations for sequences of NS compactness 𝒞 = 0.145 with mass ratios Q = 1, 2, 3, and 5 [210Jump To The Next Citation Point]. From the extrapolated results toward χ = 0, the orbital angular velocity at the mass-shedding limit is approximately determined for each set of 𝒞 and Q.

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Figure 5: Extrapolation of sequences for NS compactness 𝒞 = 0.145 to the mass-shedding limit (χ = 0). The thick curves are sequences constructed using numerical data, and the thin curves are extrapolated sequences [210Jump To The Next Citation Point]. Note that the horizontal axis is the orbital angular velocity in polytropic units, ¯Ω = ΩRpoly.
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Figure 6: Fits of the mass-shedding limit by the analytic expression (101View Equation[210Jump To The Next Citation Point]. The mass-shedding limit for each NS compactness and mass ratio is computed by the extrapolation of the numerical data.

To derive a fitting formula for the orbital angular velocities at the mass-shedding limit for all the values of 𝒞 and Q, the qualitative Newtonian expression of Equation (7View Equation) is useful. By fitting sequence data to this expression, Taniguchi et al. [210Jump To The Next Citation Point] determined the value of C Ω for Γ = 2 polytropic EOS as 0.270, i.e.,

𝒞3∕2 ( )1∕2 ¯Ωms = 0.270 ----- 1 + Q −1 , (101 ) M¯NS
or equivalently
Ωmsm0 = 0.270 𝒞3∕2(1 + Q )(1 + Q −1)1∕2 . (102 )
Figure 6View Image shows the results of the fitting for the mass-shedding limit. The solid curve denotes Equation (101View Equation) and the points are the numerical results. The agreement is not perfect, but fairly good for Q ≥ 2.

It may be interesting to note that the value of C Ω = 0.270 is the same as that found for quasi-equilibrium sequences of NS-NS binaries in general relativity in [214] and of BH-NS binaries in general relativity where the NS is corotating in [202Jump To The Next Citation Point, 203Jump To The Next Citation Point]. The value of C Ω = 0.270 could be widely used for an estimation of the orbital angular velocity at the mass-shedding limit of NS in a relativistic binary system with Γ = 2.

2.4.2 Innermost stable circular orbit

One of the most important pieces of information for relativistic close binaries is the binary separation (or the orbital angular velocity) at which the minimum of the binding energy appears, corresponding to the ISCO. The minimum point is located by fitting three nearby points of the sequence to a second-order polynomial, because the numerical data is discrete and does not necessarily give the exact minimum.

A simple empirical fitting that predicts the angular velocity ΩISCO at the ISCO for an arbitrary companion orbiting a BH may be derived in the manner of [210Jump To The Next Citation Point]. In their approach, one searches for an expression with three free parameters that express Ω ISCO as a function of the mass ratio Q and the compactness 𝒞 of the companion. Then the three parameters are determined by matching to three known values of ΩISCO, namely, (1) that of a test particle orbiting a Schwarzschild BH, ΩISCOm0 = 6−3∕2 (for Q = ∞), (2) that of an equal-mass BH-BH system as computed in [40], ΩISCOm0 = 0.1227 (for Q = 1 and 𝒞 = 0.5), and finally (3) that of a BH-NS configuration as computed in [210Jump To The Next Citation Point], Ω m = 0.0854 ISCO 0 (for Q = 5 and 𝒞 = 0.1452). A further requirement arises from the fact that for a test particle (with Q = ∞), the expression should be independent of the companion’s compactness. A good fit to the numerical data is found in [210Jump To The Next Citation Point] with the expression

[ ] Ω m = 0.0680 1 − 0.444-(1 − 3.54 𝒞1∕3) , (103 ) ISCO 0 Q0.25
as demonstrated in Figure 7View Image. The exponents of Q and 𝒞 in Equation (103View Equation) are empirically determined by requiring that the fitted curves lie near the data points for all the models. The agreement is not perfect, but sufficient for finding the orbital angular velocity at the ISCO within ∼ 10% error.
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Figure 7: Fits of the minimum point of the binding energy curve by expression (103View Equation[210Jump To The Next Citation Point].

2.4.3 Critical mass ratio

Combining Equations (102View Equation) and (103View Equation), we can identify the critical binary parameters that separate the final fates. The binary encounters an ISCO before reaching mass shedding or the NS reaches the mass-shedding limit. Figure 8View Image illustrates the final fate of BH-NS binaries for 𝒞 = 0.145. The solid curve denotes the orbital angular velocity at the mass-shedding limit, and the long-dashed one denotes it at the ISCO. As seen from Equations (102View Equation) and (103View Equation), both of these curves depend on the mass ratio Q, but in different ways, which leads to the intersection of the two curves. An inspiraling binary evolves along horizontal lines towards increasing Ω¯, starting at the left and moving toward the right, until reaching either the ISCO or the mass-shedding limit. After the binary reaches the ISCO for sufficiently large mass ratio, we cannot predict the fate of the NS, because it is in a dynamic plunge phase (see Section 3), but nevertheless the mass-shedding limit for unstable quasi-equilibrium sequences is included as the dotted curve in Figure 8View Image. As shown in Figure 8View Image, the model with mass ratio Q = 6 (dotted-dashed line) encounters the ISCO, while that of Q = 3 (dot-dot-dashed line) ends up at the mass-shedding limit. The intersection between the mass-shedding and ISCO curves marks a critical point that separates the two distinct fates of the binary inspiral.

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Figure 8: An example of the boundary between the mass-shedding limit and the ISCO [210Jump To The Next Citation Point]. The selected model is the case of 𝒞 = 0.145. The solid curve denotes the mass-shedding limit, and the long-dashed one the ISCO for each mass ratio as a function of the orbital angular velocity in polytropic units. The dotted curve denotes the mass-shedding limit for unstable quasi-equilibrium sequences.
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Figure 9: Critical mass ratio, which separates BH-NS binaries that encounter an ISCO before reaching mass shedding and undergoing tidal disruption, as a function of the compactness of NS. This figure is drawn for the model of Γ = 2 polytropic EOS [210Jump To The Next Citation Point].

When we eliminate Ωm0 from Equations (102View Equation) and (103View Equation), we can draw 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 equation, which gives the critical curve is expressed as

[ ] 3∕2 ( − 1)1∕2 0.444-( 1∕3) 0.270 𝒞 (1 + Q) 1 + Q = 0.0680 1 − Q0.25 1 − 3.54 𝒞 , (104 )
and the curve that separates those two regions is shown in Figure 9View Image. The solid curve denotes the critical mass ratio for each compactness. If the mass ratio of a BH-NS binary is larger than the critical one, the quasi-equilibrium sequence terminates by encountering the ISCO, while if smaller, it ends at the mass-shedding limit of the NS.
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