2.3 Numerical results

Several works, summarized in Table 1, have been done to calculate quasi-equilibrium sequences. In this section, we focus on the results reported in [210Jump To The Next Citation Point] because a systematic survey for a wider parameter space was done only there.

Figure 1View Image displays contours of the conformal factor ψ for a BH-NS binary with mass ratio Q = 3 and NS compactness 𝒞 = 0.145. This contour plot shows the configuration at the smallest orbital separation, for which Taniguchi’s code can achieve a converged solution. The thick solid circle on the left-hand side denotes the position of the excised surface (the apparent horizon), while that on the right-hand side denotes the position of the NS surface. A saddle point presents between the BH and NS, and for this close orbit, it is located in the vicinity of the NS surface, suggesting that the orbit of the binary is close to the mass-shedding limit. The value of ψ on the excised surface is not constant because a Neumann boundary condition (47View Equation) is imposed.

View Image

Figure 1: Contours of the conformal factor ψ in the equatorial plane for the innermost configuration with Q = 3 and 𝒞 = 0.145 shown in [210Jump To The Next Citation Point]. The cross “×” indicates the position of the rotation axis.

2.3.1 Binding energy and total angular momentum

Figure 2View Image shows the binding energy (Eb∕m0) and the total angular momentum (2 J ∕m 0) as functions of the orbital angular velocity (Ωm0) for a NS with baryon rest mass ¯ MB = 0.15 (𝒞 = 0.145) and for mass ratio Q = 3. All the quantities shown are normalized to be dimensionless. The solid curves with the filled circles denote the numerical results, and the dashed curves, the results in the 3PN approximation [25Jump To The Next Citation Point]. The numerical sequences terminate at an orbit of cusp formation (i.e., at an orbit of the mass-shedding limit) before the ISCO is encountered, i.e., before a turning point (minimum) of the binding energy appears.

From the qualitative argument described in Section 1.1, the binary separation dms at the onset of mass shedding of a NS can be approximately derived as (see Equation (5View Equation))

-dms-≃ --1---. (100 ) MBH Q2 ∕3𝒞
If dms is greater by a sufficient amount than the ISCO separation dISCO, we may expect the NS to start shedding mass and to be tidally disrupted before being swallowed by the BH. Relation (100View Equation) suggests that the mass-shedding separation decreases with increasing mass ratio, Q, and compactness of the NS, 𝒞. It is natural to expect to encounter minima in the binding energy and total angular momentum for binaries with sufficiently large mass ratio and NS compactness. Figure 3View Image shows the binding energy (Eb ∕m0) and the total angular momentum (2 J ∕m 0) as functions of the orbital angular velocity (Ωm0) for a NS with baryon rest mass ¯MB = 0.15 (𝒞 = 0.145) and for Q = 5. The NS compactness is the same as, but the mass ratio is larger than, that shown in Figure 2View Image. In this model, the binary encounters an ISCO before the onset of mass shedding, i.e., we see minima in the binding energy and angular momentum just before the end of the sequence.
View Image

Figure 2: Binding energy Eb∕m0 (left panel) and total angular momentum 2 J∕m 0 (right panel) as functions of Ωm0 for binaries of mass ratio Q = 3 and NS mass ¯ MB = 0.15 (𝒞 = 0.145[210Jump To The Next Citation Point]. The solid curve with filled circles show numerical results, and the dashed curve denotes the results in the 3PN approximation [25Jump To The Next Citation Point].
View Image

Figure 3: The same as Figure 2View Image but for the sequence of mass ratio Q = 5 [210Jump To The Next Citation Point].

Figure 3View Image shows that the turning points in the binding energy and the total angular momentum appear simultaneously to within numerical accuracy. This fact is more clearly seen in Figure 4View Image in which the binding energy versus total angular momentum for sequences of mass ratio Q = 5 but with different NS compactness is plotted. A simultaneous turning point in the binding energy and total angular momentum leads to a cusp in these curves. As suggested by Equation (100View Equation), turning points are not found for small compactness (e.g., the case of 𝒞 = 0.120), since the sequences terminate at mass shedding before encountering an ISCO. However, for larger compactness, these curves indeed form a cusp. Note that 3PN sequences cannot identify mass shedding and therefore always exhibit turning points.

View Image

Figure 4: The binding energy as a function of total angular momentum for binaries of mass ratio Q = 5, and different NS compactness [210Jump To The Next Citation Point]. The solid curve denotes the results in the 3PN approximation [25Jump To The Next Citation Point].

  Go to previous page Go up Go to next page