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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 quadrupoleformula
expression for the gravitationalwave 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
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
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Bruce Allen
20001119