Fig. 1-1: Initial Configuration of Binary and Gaseous Disk |

Evolution is performed on an AMR (Adaptive Mesh Refinement) grid. The outermost grid is 256 M x 256 M while the two innermost grids are 1 M x 1 M. Here M is the total (ADM) mass of the intial system. The innermost resolution is ΔX_{min}/M = .031 while the outermost is ΔX_{max}/M = 4.0.

The gaseous disk obeys a polytropic equation of state, P = Kρ_{0}^{ Γ} at t = 0. It is evolved according to the adiabatic evolution law, P = (Γ-1)ρ_{0}ε where P is the pressure, K is the polytropic gas constant, ε is the internal specific energy, Γ is the adiabatic index, and ρ_{0} the rest-mass density. Changes in entropy, S - S_{0}, are expressed via the relation S - S_{0} ∝ ln (K/K_{0}), where S_{0} and K_{0} are the entropy and polytropic constant, respectively, at t = 0. The videos below visualize the case where Γ = 4/3 to simulate a disk composed of, e.g., a relativistic ideal gas or a radiation-dominated gas. The disk is taken to have an initial inner radius of R_{in} = 15M and extends to R_{out} = 65M. The mass of the disk is assumed to be negligible in comparison to the total black hole mass.

In the clip, the rest-mass density of the disk is plotted on a logarithmic scale normalized to the initial central density. For our disk initial data, we use the equilibrium solution for a stationary disk around a single Kerr BH. Keeping the binary separation fixed and the metric fixed in the rotating frame of the binary ("conformal thin-sandwich" initial data), we allow this initial disk to evolve so as to settle into a quasistationary equilibrium around the binary black hole system ("Early Inspiral Epoch"). After this, we allow the matter and the metric to evolve, where the gravitational field is evolved via the BSSN scheme using "moving puncture" gauge conditions and the relativistic hydrodynamic equations are solved using a high-resolution shock-capturing (HRSC) method ("Late Inspiral, Merger and Ringdown Epochs").

In Fig. 2-2 and Fig. 2-3 we see spiral arms form around the black holes during the early inspiral epoch and the start of the late inspiral epoch, respectively. The main effect of the tidal torque of the binary on the disk is to create a hollow region in the disk about the black hole. Once binary-disk "decoupling" occurs (the Late Inspiral Epoch), the tidal field of the binary falls off. As a result, spiral arms disappear and the mass accretion rate onto the black holes plummets. Not until viscosity or magnetic fields (omitted in these simulations) have time to act is the hollow filled about the merger remnant.

Fig. 2-1: Color map for density profile |
Fig. 2-2: 3D Density profile at t/M = 2100 |

Fig. 2-3: 3D Density profile at t/M = 6500 |
Fig. 2-4: 3D Density profile at t/M = 8000 |

Fig. 3-1: Color map for 2D density profile |
Fig. 3-2: 2D Density profile at t/M = 2100 |

Fig. 3-3: 2D Density profile at t/M = 6500 |
Fig. 3-4 Density profile 2D at t/M = 8000 |

In this clip, the polytropic gas constant K = P/ρ_{0}^{ Γ} of the disk is plotted on a logarithmic scale, normalized to its initial value. K/K_{0} > 1 implies entropy generation (S > S_{0}), or heat generation. Such entropy generation comes about due to shock heating that is a result of the tidal forces exerted by the binary system on the disk. Regions of higher entropy correspond to higher gass temperature and radiate more strongly. The spiral arms constitute such regions.

Fig. 4-1: Color map for 3D entropy profile |
Fig. 4-2: 3D Entropy profile at t/M = 2100 |

Fig. 4-3: 3D Entropy profile at t/M = 6500 |
Fig. 4-4: 3D Entropy profile at t/M = 8000 |

Fig. 5-1: Color map for 2D entropy profile |
Fig. 5-2: 2D Entropy profile at t/M = 2100 |

Fig. 5-3: 2D Entropy profile at t/M = 6500 |
Fig. 5-4: 2D Entropy profile at t/M = 8000 |

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, gravitational 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 remnant settles down to Kerr equilibrium. Both polarization modes (h_{+} and h_{x}) are shown.

Fig. 6-1 h_{+} with color map |
Fig. 6-2 h_{x} both hemispheres |

Fig. 6-3 h_{+} both hemispheres |

Fig. 7-1 Color map for h_{+} and h_{x} |
Fig. 7-2 Planar h_{x} profile |

Fig. 7-2 Planar h_{+} profile |

Listed in the table below is the dimensionless spin of the black hole remnant at the end of our simulation. Also shown is the radiated mass-energy in the form of gravitational wave emission.

J_{BH}/M_{BH}^{2} |
0.69 |

ΔM_{GW}/M |
0.038 |