The most prominent selection effect is the inverse square law, i.e. for a given intrinsic
luminosity1,
the observed flux density falls off as the inverse square of the distance. This results in the observed sample
being dominated by nearby and/or high luminosity objects. Beyond distances of a few kpc from the Sun,
the apparent flux density falls below the detection thresholds of most surveys. Following [88], we
express this threshold as follows:
It follows from Equation (3) that the sensitivity decreases as
and hence
increases. Also
note that if
, the pulsed signal is smeared into the background emission and is no longer
detectable, regardless of how luminous the source may be. The detected pulse width
may be broader
than the intrinsic value largely as a result of pulse dispersion and multipath scattering by free
electrons in the interstellar medium. The dispersive smearing scales as
, where
is the
observing frequency. This can largely be removed by dividing the pass-band into a number
of channels and applying successively longer time delays to higher frequency channels before
summing over all channels to produce a sharp profile. This process is known as incoherent
dedispersion.
The smearing across the individual frequency channels, however, still remains and becomes significant at
high dispersions when searching for short-period pulsars. Multipath scattering from electron density
irregularities results in a one-sided broadening due to the delay in arrival times. A simple scattering model
is shown in Figure 12 in which the scattering electrons are assumed to lie in a thin screen between the
pulsar and the observer [278]. The timescale of this effect varies roughly as
, which can not be
removed by instrumental means.
Standard pulsar searches use Fourier techniques [185] to search for a-priori unknown periodic signals and
usually assume that the apparent pulse period remains constant throughout the observation. For searches
with integration times much greater than a few minutes, this assumption is only valid for solitary
pulsars or binary systems with orbital periods longer than about a day. For shorter-period
binary systems, the Doppler-shifting of the period results in a spreading of the signal power
over a number of frequency bins in the Fourier domain, leading to a reduction in S/N [136].
An observer will perceive the frequency of a pulsar to shift by an amount
, where
is the (assumed constant) line-of-sight acceleration during the observation of length
,
is the (constant) pulsar period in its rest frame and
is the speed of light. Given that
the width of a frequency bin in the Fourier domain is
, we see that the signal will drift
into more than one spectral bin if
. Survey sensitivities to rapidly-spinning
pulsars in tight orbits are therefore significantly compromised when the integration times are
large.
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It is clearly desirable to employ a technique to recover the loss in sensitivity due to Doppler smearing.
One such technique, the so-called “acceleration search” [219], assumes the pulsar has a constant
acceleration during the observation. Each time series can then be re-sampled to refer it to the
frame of an inertial observer using the Doppler formula to relate a time interval in the
pulsar frame to that in the observed frame at time
, as
. Searching over a
range of accelerations is desirable to find the time series for which the trial acceleration most
closely matches the true value. In the ideal case, a time series is produced with a signal of
constant period for which full sensitivity is recovered (see right panel of Figure 13
). This technique
was first used to find PSR B2127+11C [5], a double neutron star binary in M15 which has
parameters similar to B1913+16. More recently, its application to 47 Tucanae [53
] resulted in
the discovery of nine binary millisecond pulsars, including one in a
orbit around a
low-mass (
) companion. This is currently the shortest binary period for any known radio
pulsar.
For intermediate orbital periods, in the range -several hours, another promising technique is
the dynamic power spectrum search. Here the time series is split into a number of smaller contiguous
segments which are Fourier-transformed separately. The individual spectra are displayed as a
two-dimensional (frequency versus time) image. Orbitally modulated pulsar signals appear as sinusoidal
signals in this plane as shown in Figure 14
.
For the shortest orbital periods, the assumption of a constant acceleration during the observation clearly
breaks down. In this case, a particularly efficient algorithm has been developed [262, 143, 263] which is
optimised to finding binaries with periods so short that many orbits can take place during an observation.
This “phase modulation” technique exploits the fact that the Fourier components are modulated by the
orbit to create a family of periodic sidebands around the nominal spin frequency of the pulsar.
While this technique has so far not resulted in any new discoveries, the existence of short period
binaries in 47 Tucanae [53
], Terzan 5 [266
] and the
X-ray binary X1820
303 in
NGC 6624 [301], suggests that there may be more ultra-compact binary pulsars that await
discovery.
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