Quasiequilibrium black hole-neutron star binaries in general relativity

We construct quasiequilibrium sequences of black hole-neutron star binaries in general relativity. We solve Einstein’s constraint equations in the conformal thin-sandwich formalism, subject to black hole boundary conditions imposed on the surface of an excised sphere, together with the relativistic equations of hydrostatic equilibrium. In contrast to our previous calculations we adopt a flat spatial background geometry and do not assume extreme mass ratios. We adopt a $\Gamma =2$ polytropic equation of state and focus on irrotational neutron star configurations as well as approximately nonspinning black holes. We present numerical results for ratios of the black hole’s irreducible mass to the neutron star’s ADM mass in isolation of $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=1$, 2, 3, 5, and 10. We consider neutron stars of baryon rest mass $M_{\mathrm{B}}^{\mathrm{NS}}/M_{\mathrm{B}}^{\max }=83\%$ and 56%, where $M_{\mathrm{B}}^{\max }$ is the maximum allowed rest mass of a spherical star in isolation for our equation of state. For these sequences, we locate the onset of tidal disruption and, in cases with sufficiently large mass ratios and neutron star compactions, the innermost stable circular orbit. We compare with previous results for black hole-neutron star binaries and find excellent agreement with third-order post-Newtonian results, especially for large binary separations. We also use our results to estimate the energy spectrum of the outgoing gravitational radiation emitted during the inspiral phase for these binaries.

Figure 1

Contours of the lapse function $\alpha$ in the equatorial plane for our innermost configuration of a binary of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=3$ and neutron star mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$. The minimum value of the lapse, $\alpha \simeq 0.453$, is located on the excised surface and the maximum value, $\alpha \simeq 0.909$, at the right edge of the figure. The value of the lapse at the point of maximum baryon rest-mass density is about 0.558. The contour curves are spaced by $\delta \alpha \simeq 0.03$. The cross “$\times$” indicates the position of the rotation axis.

Figure 2

Fractional binding energy $E_{\mathrm{b}}/M_0$ as a function of $\Omega M_0$ for binaries of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=1$. The solid line with filled circles shows results for neutron stars of mass ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$, and the dashed line with squares for neutron stars of mass of 0.10. The dashed-dotted line denotes the results of the third post-Newtonian approximation [50]. These sequences end due to cusp formation—and hence the onset of tidal disruption—before the binary reaches the ISCO at the turning point of the binding energy.

Figure 3

Same as Fig. 2 but for sequences of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=2$.

Figure 4

Same as Fig. 2 but for sequences of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=3$.

Figure 5

Same as Fig. 2 but for sequences of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=5$. For the more compact neutron star the binary now encounters an ISCO before the neutron star is tidally disrupted.

Figure 6

Same as Fig. 2 but for sequences of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=10$.

Figure 7

Virial error $\delta M$ versus $\Omega M_0$ for our binary sequences.

Figure 8

Minimum of the mass-shedding indicator ${\chi }_{\min }$ as a function of $\Omega M_0$.

Figure 9

Decrease in the maximum density parameter $\delta q_{\max }$ as a function of $\Omega M_0$.

Figure 10

The energy spectrum $d{E}_{\mathrm{b}}/df$ for the sequences with $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=1$ as a function of the dimensionless gravitational wave frequency $f{M}_0$. The solid, dashed, dashed-dotted, and dotted curves show the fits to the ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$ and ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.1$ sequences and the 3PN and Newtonian expressions, respectively. Asterisks denote the approximate point where a cusp forms and tidal disruption begins, terminating the sequence, at frequencies given by Table .

Figure 11

Same as Fig. 10, but for the sequences with $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=3$.

Figure 12

Same as Fig. 10, but for the sequences with $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=5$. Note that the sequence with ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.15$ reaches a minimum binding energy, where $d{E}_{\mathrm{b}}/df=0$, prior to tidal disruption.

Figure 13

Comparison of our new results to those calculated assuming an extreme mass ratio for a spatially flat background geometry [28] for the minimum of the mass-shedding indicator ${\chi }_{\min }$. Here $M_{\mathrm{BH}}$ denotes either the black hole irreducible mass (for our new results) or the background mass (for the results of Ref. 28); see text.

Figure 14

Comparison of our new results to those calculated using a Kerr-Schild background [29] for the total angular momentum. The dashed-dotted line denotes the results of the third post-Newtonian approximation [50].

Figure 15

Comparison between our results and those of G06 for the binding energy as a function of $\Omega M_0$, for a binary of mass ratio $M_{\mathrm{irr}}^{\mathrm{BH}}/M_{\mathrm{ADM},0}^{\mathrm{NS}}=5$. The solid line with filled circles marks our results for a neutron star compaction of $M_{\mathrm{ADM},0}^{\mathrm{NS}}/R_0=0.150$, which corresponds to the baryon rest mass of ${\overline{M}}_{\mathrm{B}}^{\mathrm{NS}}=0.153$. The dashed line with triangles represents those of G06 for a neutron star compaction of 0.150. The dashed-dotted line denotes the results of the third post-Newtonian approximation [50].