Introduction
Fig. 1-1 Initial Configuration of Binary |
Evolution is performed with the BSSN scheme on an AMR grid with 9 levels
of refinement. The outer boundary of our grid is 320M and the mesh
spacing ranges from
ΔXmax = 8.0 M in the
outermost level to
ΔXmin = M/32 in the
innermost level. In this simulation, the initial coordinate radius of
the binary orbit is
D0/M = 9.89. Here M is the
total initial binary ADM mass. The black hole interiors, bounded by their
apparent horizons, are denoted by black spheres. Their motion is shown
in the orbital plane. The evolution is performed with "moving puncture" gauge conditions. (Note: The circle in the
lower right-hand corner of the above figure is a clock.)
Binary Inspiral and Merger
The binary makes approximately
six orbits prior to merging at
t ≈ 870 M. As the
inital binary merges, we see the development of a common horizon which
oscillates until settling down at t ≈ 930 M. The
simulation continues until
t = 1246 M to demonstrate the
stability of the resulting Kerr black hole. The early growth of the
apparent horizons is a gauge (coordinate) effect.
Fig. 2-1 Evolution at t/M = 0 |
Fig. 2-2 Evolution at t/M = 650 |
Fig. 2-3 Evolution at t/M = 800 |
Fig. 2-4 Evolution at t/M = 1250 |
Gravitational Radiation Waveform
The gravitational wavetrain from
a compact binary system may be separated into three qualitatively
different phases: 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 as the BHs
maintain a quasicircular orbit. Here, we see the
gravitational radiation waveform during the late inspiral and merger
stages of our binary black hole coalescence simulation. Finally, we see
a ringdown as the distorted black hole settles down to Kerr equilibrium.
Both polarization modes (h
+ and h
x) are shown.
Fig. 3-1 h+ in both hemispheres |
Fig. 3-2 h+ in one hemisphere |
Fig. 3-3 h× both hemispheres |
Final Black Hole Parameters
Listed in the table below is the dimensionless spin of the black hole at
the end of our simulation. Also shown are the radiated energy and
angular momentum from gravitational wave emission. Here, M is the
initial binary ADM mass whereas M
BH is the final ADM mass of the
black hole remnant.
| MBH/M |
0.962 |
| JBH/MBH2 |
0.685 |
| ΔE GW/M |
0.038 |
| ΔJGW/M2 |
0.331 |
| δE ≡ (M-MBH-ΔEGW)/M |
4 x 10-4 |
| δJ ≡ (J-JBH-ΔJGW)/M |
4 x 10-3 |
Our simulation maintains excellent conservation of energy and momentum,
since δE and δJ are on the order of 10
-4 and
10
-3, respectively.
