BHNS

1. Introduction
2. Evolution of Density Profile
3. Gravitational Waveforms
4. Final Black Hole Parameters

Introduction

Fig. 1-1 Initial Configuration of Binary

Evolution is performed on an AMR (Adaptive Mesh Refinement) grid. The outermost grid is 394 M x 394 M, the innermost refinement level for the BH is 2.8 M x 2.8 M and the innermost refinement level for the NS is 3.3 M x 3.3 M. Here M is the total (ADM) mass of the intial system. The innermost resolution is ΔX_min/M = 1/32.5 while the outermost is ΔX_max/M = 3.94. In this simulation, the initial binary coordinate separation is D₀/M = 8.81, the initial angular momentum of the system is J/M² = 0.702, the mass ratio is M_BH:M_NS = 3:1, and the black hole spin parameter is J_BH/M_BH² = 0.00. (Note: The open circle in the lower right-hand corner of the above figure is a clock.)

The neutron star obeys a polytropic equation of state, P = Kρ₀  ^Γ, at t = 0. It is evolved according to the adiabatic evolution law, P = (Γ-1)ρ₀&epsilon, where P is the pressure, K the polytropic constant, &epsilon is the internal specific energy, Γ is the adiabatic index, and ρ₀ the rest-mass density. We chose Γ = 2 to mimick stiff nuclear matter. The star is irrotational (nonspinning) and in a quasiequilibrium circular orbit.

The initial compaction of the neutron star is M_NS/R_NS = 0.145. The rest mass is 83% of the maximum rest mass of an isolated, nonrotating NS with the same polytropic EOS.

The initial binary is in a quasiequilibrium circular orbit. The matter and gravitational field variables are determined by using the conformal thin-sandwich formalism. BH equilibrium boundary conditions are imposed on the BH horizon.

Evolution of Density Profile

In the clip, the rest-mass density of the neutron star is plotted on a logarithmic scale normalized to the initial central density. The gravitational field is evolved via the BSSN scheme using "moving puncture" gauge conditions. The relativistic hydrodynamic equations are solved using a high-resolution shock-capturing (HRSC) method.

After two orbits the density contours are for the most part undisturbed. After 3.5 orbits the first few density contours have crossed into the apparent horizon. The NS tail deforms into a quasistationary disk, as the bulk of the matter is all accreted onto the hole in a few periods.

Fig. 2-1 Color code for density profile	Fig. 2-2 Density Profile at t/M = 450
Fig. 2-3 Density Profile at t/M = 500	Fig. 2-4 Density Profile at t/M = 900

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Gravitational Waveforms

The gravitational wavetrain from a compact binary system may be separated into three qualitatively different phases: the inspiral, merger, and ringdown. During the inspiral phase, which takes up most of the binary's lifetime, gravity wave emission gradually reduces the binary separation. The merger phase of the gravitational wavetrain is characterized by tidal disruption of the neutron star. Finally, ringdown radiation is emitted as the distorted black hole settles down to Kerr-like equilibrium (Note: Only in the case of a vacuum spacetime does the spinning BH obey the Kerr solution. The BHs formed here are surrounded by gaseous disks with small, but nonnegligible, rest mass). Both polarization modes (h₊ and h_x) are shown. Total energy conservation is obeyed to within 0.02%. Total angular momentum conservation is obeyed to within 2.2%

Fig. 3-1 h+ in both hemispheres	Fig. 3-2 h+ in one hemisphere
Fig. 3-1 h× both hemispheres

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Final Black Hole Parameters

Listed in the table below is the dimensionless spin of the Kerr-like black hole at the end of our simulation. Also shown are the radiated energy, angular momentum, and linear recoil velocity resulting from gravitational wave emission. Additional parameters for the disk are given in the film clip.

JBH/MBH2	0.559
ΔJGW/J	17.4%
ΔEGW/M	0.93%
Recoil velocity	33 km/s