The Quasi-Equilibrium Approximation

The fundamental assumption in the quasi-equilibrium approximation is that the orbital timescale is much shorter than the gravitational radiation reaction timescale. The binary inspiral then proceeds along a sequence of quasi-equilibrium configurations in nearly circular orbits at a constant rest mass. In a quasi-equilibrium (QE) dynamical simulation, each QE binary configuration provides a periodic matter source for Einstein's field equations of general relativity. The dynamical field equations are integrated in time using the QE source to determine the outgoing gravitational wave emission at each binary separation.

## Quasi-Equilibrium Binary Sequence

The following movie shows the orbit and gravitational wavetrain of the QE configuration with total mass-energy M and gravitational wave luminosity dM/dt. The waveform is measured by distant observers along the binary rotation axis. The locations of the binary in the binding energy and luminosity diagrams above are indicated by green dots.

## Construction of Quasi-Equilibrium Sequence

A finite number of QE configurations with different values of separation ZA (and hence r) are constructed and then numerically evolved. The binding energy curve for the sequence of binaries is shown in the top panel. For each configuration, indicated by a green dot, the gravitational wave luminosity dM/dt and gravitational wave amplitude are determined by solving Einstein's equations. Results for the luminosity dM/dt are indicated in the bottom panel.

A smooth fit to the data gives M/M0 and dM/dt as continuous functions of ZA (and hence r).

The innermost stable circular orbit (ISCO), where the quasi-equilibrium approximation breaks down, can be located by finding the turning point dM/dZA = 0 in the binding energy curve.

The gravity wave luminosity, dM/dt, and the slope of the binding energy curve can be used to find the inspiral rate dr/dt, as indicated in the left-hand panel. By smoothly joining the wave data for each discrete member of the binary sequence, the complete gravitational wavetrain can be assembled for the late inspiral, as shown in the right-hand panel.

last updated 6 Nov 14 aakhan3