# Γ = 5/3 Ideal Gas

## Introduction

 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 ΔXmin/M = .031 while the outermost is ΔXmax/M = 4.0.

The gaseous disk obeys a polytropic equation of state, P = Kρ0 Γ at t = 0. It is evolved according to the adiabatic evolution law, P = (Γ-1)ρ0ε where P is the pressure, K is the polytropic gas constant, ε is the internal specific energy, Γ is the adiabatic index, and ρ0 the rest-mass density. Changes in entropy, S - S0, are expressed via the relation S - S0 ∝ ln (K/K0), where S0 and K0 are the entropy and polytropic constant, respectively, at t = 0. The videos below visualize the case where Γ = 5/3 to simulate a disk composed of, e.g, a non-relativistic ideal gas or a radiation-dominated gas. The disk is taken to have an initial inner radius of Rin = 15M and extends to Rout = 65M. The mass of the disk is assumed to be negligible in comparison to the total black hole mass.

## Evolution of Density Profile

### Three-Dimensional Visualization of Density

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 3D density profile Fig. 2-2: 3D Density profile at t/M = 2600 Fig. 2-3: 3D Density profile at t/M = 8000 Fig. 2-4: 3D Density profile at t/M = 10000

### Two-Dimensional Visualization of Density

The following images show the densities on the equatorial plane.
 Fig. 3-1: Color map for 2D density profile Fig. 3-2: 2D Density profile at t/M = 2600 Fig. 3-3: 2D Density profile at t/M = 8000 Fig. 3-4: 2D Density profile at t/M = 10000

## Evolution of Entropy Profile

### Three-Dimensional Visualization of Entropy

In this clip, the polytropic gas constant K = P/ρ0 Γ of the disk is plotted on a logarithmic scale, normalized to its initial value. K/K0 > 1 implies entropy generation (S > S0), 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 = 2600 Fig. 4-3: 3D Entropy profile at t/M = 8000 Fig. 4-4: 3D Entropy profile at t/M = 10000

### Two-Dimensional Visualization of Entropy

The following images show the entropy profile at points in the equatorial plane.
 Fig. 5-1: Color map for 2D entropy profile Fig. 5-2: 2D Entropy profile at t/M = 2600 Fig. 5-3: 2D Entropy profile at t/M = 8000 Fig. 5-4: 2D Entropy profile at t/M = 10000

## Gravitational Waveforms

### Three-Dimensional Visualization of Gravitational Waveforms

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 hx) are shown.

 Fig. 6-1 h+ with color code Fig. 6-2 hx both hemispheres Fig. 6-3 h+ both hemispheres

### Two-Dimensional Visualization of Gravitational Waveforms

Here we show a three plane view of the h+ and hx modes.
 Fig. 7-1 Color map for h+ and hx Fig. 7-2 Planar hx profile Fig. 7-2 Planar h+ profile

## Final Parameters

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.

 JBH/MBH2 0.69 ΔMGW/M 0.038