where
and
are ``beam-pattern'' factors, that depend on the direction of
the source
and on a polarization angle
, and
and
are gravitational waveforms corresponding to the two
polarizations of the gravitational wave (for a review, see [126]). In a source coordinate system in which the
x
-
y
plane is the plane of the sky and the
z
-direction points toward the detector, these two modes are given
by
where
represent transverse-traceless (TT) projections of the
calculated waveform of Eq. (51
), given by
where
is a unit vector pointing toward the detector. The beam pattern
factors depend on the orientation and nature of the detector. For
a wave approaching along the laboratory
z
-direction, and for a mass whose location on the
x
-
y
plane makes an angle
with the
x
axis, the beam pattern factors are given by
and
. For a resonant cylinder oriented along the laboratory
z
axis, and for source direction
, they are given by
,
(the angle
measures the relative orientation of the laboratory and source
x
-axes). For a laser interferometer with one arm along the
laboratory
x
-axis, the other along the
y
-axis, and with
defined as the
differential
displacement along the two arms, the beam pattern functions are
and
.
The waveforms
and
depend on the nature and evolution of the source. For example,
for a binary system in a circular orbit, with an inclination
i
relative to the plane of the sky, and the
x
-axis oriented along the major axis of the projected orbit, the
quadrupole approximation of Eq. (53
) gives
where
is the orbital phase.
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |