As seen by a detector, the gravitational wave frequency of a pulsar is
virtually constant when the detector-to-pulsar distance is not
changing. But many pulsars are found in systems of two stars, called
binary star systems, where the two stars orbit one another. In this
case the detector-to-pulsar distance is constantly changing, so the
gravitational wave frequency of the pulsar is modulated by the orbital
motion, and no longer has a fixed frequency.
Another example arises
if a pulsar is falling toward a local concentration of
mass. The resulting acceleration would complicate the gravitational
waveform somewhat, introducing a linear-with-time frequency shift.
This is actually a simplification. In fact if the pulsar is emitting gravitational waves then these carry away energy and the spin frequency of the star does change - but only very slightly, so for now let's forget about this effect.
A common example of a random walk is a
mathematical problem called ``the drunken sailor''. A drunken sailor
on a road walks back and forth, changing direction at random. After a
time much larger than the time to take a single step has elapsed,
the average (in the sense of the root mean square) distance that the
sailor has reached from the starting point is proportional to the square
root of .
The spectral line artifacts whose origins have been identified
include, for example, 60 Hz power line harmonics, violin modes of the mirror
suspensions (343-347 Hz and harmonics) and 37 Hz harmonics arising from imperfection in
the synthesized oscillator used to modulate the laser phase.
: Gravitational wave strain amplitude.
: Gravitational wave frequency in Hz.
: Inclination angle of the pulsar rotating axis to the line of
the sight in radians.
: Polarization angle of the gravitational wave in radians.
: Initial phase of the gravitational wave in radians.
: Right ascension of the pulsar in radians (Equatorial coordinate.)
: Declination of the pulsar in radians (Equatorial coordinate.)
: Expected number of coincidences.
The precise definitions of these parameters (except for )
are given in .