Relativistic black hole-neutron star binaries in quasiequilibrium: Effects of the black hole excision boundary condition

We construct new models of black hole-neutron star binaries in quasiequilibrium circular orbits by solving Einstein’s constraint equations in the conformal thin-sandwich decomposition together with the relativistic equations of hydrostationary equilibrium. We adopt maximal slicing, assume spatial conformal flatness, and impose equilibrium boundary conditions on an excision surface (i.e., the apparent horizon) to model the black hole. In our previous treatment we adopted a “leading-order” approximation for a parameter related to the black hole spin in these boundary conditions to construct approximately nonspinning black holes. Here we improve on the models by computing the black hole’s quasilocal spin angular momentum and setting it to zero. As before, we adopt a polytropic equation of state with adiabatic index $\Gamma =2$ and assume the neutron star to be irrotational. In addition to recomputing several sequences for comparison with our earlier results, we study a wider range of neutron star masses and binary mass ratios. To locate the innermost stable circular orbit we search for turning points along both the binding energy and total angular momentum curves for these sequences. Unlike for our previous approximate boundary condition, these two minima now coincide. We also identify the formation of cusps on the neutron star surface, indicating the onset of tidal disruption. Comparing these two critical binary separations for different mass ratios and neutron star compactions we distinguish those regions that will lead to a tidal disruption of the neutron star from those that will result in the plunge into the black hole of a neutron star more or less intact, albeit distorted by tidal forces.

Figure 1

Mass-radius relation for a spherical, polytropic star with adiabatic index $\Gamma =2$. The horizontal axis denotes the isotropic radius and the vertical axis the baryon rest mass (solid curve) and gravitational mass (dashed curve), where the radius and masses are in polytropic units. The gravitational mass is equivalent to the ADM mass for an isolated, spherical neutron star. Filled circles denote the models we chose in this paper. The compaction parameter $\mathcal{C}$ for each model is also shown.

Figure 2

Contours of the conformal factor $\psi$ in the equatorial plane for our innermost configuration with binary mass ratio $\widehat{q}=3$ and neutron star baryon rest mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$. The cross “$\times$” indicates the position of the rotation axis.

Figure 3

Binding energy $E_{\mathrm{b}}/M_0$ (upper panel) and total angular momentum $J/M_0^2$ (lower panel) as a function of $\Omega M_0$ for binaries of mass ratio $\widehat{q}=1$ and neutron star mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$. The solid line with filled circles shows new results, and the dashed line the old results in Paper I. The dotted-dashed line denotes the results of the 3PN approximation [57]. The numerical sequences end due to cusp formation—and hence the onset of tidal disruption—before the binary reaches the innermost stable circular orbit at the turning point of the binding energy. 3PN sequences all exhibit turning points and cannot reveal cusps.

Figure 4

Same as Fig. 3 but for sequences of mass ratio $\widehat{q}=3$.

Figure 5

Same as Fig. 3 but for sequences of mass ratio $\widehat{q}=5$. The binary encounters an ISCO before the neutron star is tidally disrupted.

Figure 6

The relative difference between the 3PN and numerical total angular momentum as a function of $\Omega M_0$ for binaries of mass ratio $\widehat{q}=5$ and neutron star mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$. The solid curve shows new results in this paper, the dashed curve the results of the old formulation in Paper I but using the new method to compute ${\Omega }_r$, and the dotted-dashed curve the old results in Paper I. This figure demonstrates that the new spin condition improves the results predominantly at small binary separation, while the new formulation of the equations has a larger effect at large binary separations.

Figure 7

The binding energy as a function of total angular momentum for binaries of mass ratio $\widehat{q}=5$, and different neutron star compactions.

Figure 8

Quasilocal spin angular momentum of the black hole as a function of the orbital angular velocity (along a quasiequilibrium constant-mass sequence). Solid and dotted-dashed lines correspond to the condition $S=0$, while dashed and dotted curves result from ${\Omega }_r=\Omega$. (Compare the curves with the inset in Fig. 7 in 43.)

Figure 9

Same as Fig. 8, except for the black hole spin parameter ${\Omega }_r$.

Figure 10

Extrapolation of sequences for the neutron star baryon rest mass ${\overline{M}}_{\mathrm{B}}=0.15$ to the mass-shedding point ($\chi =0$). Thick lines are sequences constructed using numerical data, and thin lines are extrapolated sequences.

Figure 11

Fits of the mass-shedding limit by the analytic expression (29). The mass-shedding limit for each neutron star mass and mass ratio is computed by the extrapolation of the numerical data.

Figure 12

Fits of the minimum point of the binding energy curve by expression (33).

Figure 13

An example of the boundary between the mass-shedding limit and the ISCO. We select the model of neutron star mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$. 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 quasiequilibrium sequences.

Figure 14

Endpoint of sequences for different neutron star masses. Left panel is the mass ratio as a function of the orbital angular velocity in polytropic units, while the right panel is normalized by the ADM mass at infinite orbital separation.

Figure 15

Critical mass ratio which separates BHNS binaries that encounter an ISCO before reaching mass-shedding and undergoing tidal disruption, as a function of the compaction of neutron star.