Modeling The Christodoulou Memory Of A Coalescing Extreme Mass Ratio Binary

Gary M. Stange, University of FloridaDr. Daniel Kennefick, University of Arkansas

IntroductionIntroduction

GravitationalGravitational WavesWaves

ChristodoulouChristodoulou MemoryMemoryModelingModeling

AcknowledgementsAcknowledgements

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The final aim of this project is to determine the ability of the space-based gravitational wave detector LISA to detect the Christodoulou memory waves emitted by a binary system with an extreme mass ratio. The first step in this process is to model the signal of the memory waves by producing a function that accurately outputs the amplitude of the memory wave as a function of the time until the coalescence of the binary system.

In Einstein’s theory of general relativity it is stated that a mass undergoing acceleration will emit gravitational radiation. These gravitational waves are often compared to the electromagnetic waves produce by accelerating charges; however the form of the wave is vastly different. Gravitational waves are characterized by oscillations in the very fabric of space-time, causing space and everything in it to grow and shrink as the waves pass by.

Two masses spiraling around each other will generate gravitational waves.

An emitted gravitational wave will have energy of its own. According to relativity theory energy is equivalent to mass; therefore a gravitational wave qualifies as a mass in motion that will emit its own gravitational wave. This so called “wave of the wave” is known as the Christodoulou Memory. It is given this name because a memory wave will leave a permanent relative displacement in a system of free masses. Christodoulou showed that these memory waves are emitted by compact binary systems, such as black hole-black hole binaries.

It is theorized that there is a black hole at the center of every galaxy with a mass from 100,000 solar masses to over one billion solar masses. These super massive black holes, or SMBHs, are thought to capture many of the nearby stars or stellar mass black holes of the galactic core. This means that there should be many occurrences of extreme mass ratio binary systems. Because gravitational wave detectors such as LISA will have a better chance of detecting such systems, the project will focus on modeling the Christodoulou Memory amplitude for a binary system with an extreme mass ratio.

LISA will consist of 3 spacecraft put in an orbit trailing Earth. Gravitational waves and memory waves will be detected by interferometry.

To model the memory, the amplitude function

(ι is equal to 90 degrees in this approximation)

was coded into FORTRAN. A cutoff time was established for the memory curve by finding the time at which the two bodies would first touch. To find when the two bodies would first touch, the orbital radius of the lesser object was set equal to the radius of the larger object. Because the larger object is a black hole, its radius is given by the Schwarzschild equation.

Solving this expression for the time gives:

After this cutoff time, a flat line is coded in at the last memory amplitude to show that the two masses have coalesced and stopped emitting gravitational waves. At the point where the two curves join there is an unphysical cusp which will cause problems in later stages of the project. Although it is unclear exactly what shape the curve would take at the time of coalescence, a smooth curve was interpolated to remove the discontinuity.

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Dr. Daniel Kennefick and Olga Petrova

3. A graph of the memory amplitude (vertical) versus the time until coalescence (horizontal).

1. Binary Black Hole System

2. LISA