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## The waveform

The Teukolsky formalism gives the two components and of the gravitational waves produced by a test mass in a fixed, circular orbit of radius as

 (7.1.159)

where is the standard retarded time. Here is Newton's constant and the speed of light. Here the functions are the spherical harmonics of spin weight , is the angular velocity and is the total mass of the system. The angles and are the usual spherical coordinates, as defined in Section , Figure ( and ). Since the motion is symmetric around the -axis does not have an intrinsic meaning. Throughout this section angles are measured in radians. The amplitudes are constants and have to be calculated numerically by solving the Teukolsky equation. We provide tabulated in a datafile [45] as a function of the orbital velocity (measured in units of the speed of light )
 (7.1.160)

Here is the orbital frequency measured in units on Hertz.

To account for the decay of the orbit due to the emission of gravitational waves we use an adiabatic approximation: The energy radiated away as calculated from expression () is used to calculate the change in orbital frequency. By doing so, , and thus also , become functions of time and we have to replace the product by an integral .

Following [46] we find that the time evolution of the velocity is governed by

 (7.1.161)

where is the reduced mass of the system. The function , which is determined by the 's, is defined by the equation

where is the quadrupole-formula expression for the gravitational-wave luminosity. We also provide in tabulated form [45]. The function relates the energy and frequency of the orbiting particle:

This can be calculated by solving the geodesic equation for circular orbits in the Schwarzschild spacetime.

Note that can be calculated from equation () once and for all: First (numerically) calculate the solution of equation () with the factor set to one; then the solution for general is simply

 (7.1.162)

As mentioned before, we have to replace the product in equation () by an appropriate integral. We first note that we want to look at waves at a fixed radius . We can thus simply ignore the dependence on because it will only contribute a fixed phase . Since our data are tabulated as functions of the orbital velocity , it is convenient to use instead of the time as an independent variable. This is permissible because depends monotonically on . Thus the phase becomes:
 (7.1.163)

It is important to note that is not unique, but depends on an arbitrary parameter . As with , a change in will only affect the phase . Since can become rather large, the freedom in choosing can be used to keep small in the region of interest. A good choice of will improve the numerical accuracy tremendously. (For example, the standard trigonometric functions in C become hopelessly inaccurate for arguments for floats and for doubles).

The signal is now given by

 (7.1.164)

where is given by equation ().

To extract and independently we use the fact that and, since and is real, we have [26]. We can now split the sum () into real and imaginary part. This gives the two components as

 (7.1.165)

Here we have introduced the notation , and and .

The GRASP routine which calculates and uses expression () truncated to a finite number of terms determined by the user. Finally we note that a change in has the same effect on the waveforms as a change in and .

Next: A note on the Up: GRASP Routines: Waveforms from Previous: GRASP Routines: Waveforms from   Contents
Bruce Allen 2000-11-19