3.2 Correcting the observed pulsar sample
In the following, we review common techniques to account for the various selection effects and form a
less biased picture of the true pulsar population.
3.2.1 Scale factor determination
A very useful technique [250, 334], is to define a scaling factor
as the ratio of the total Galactic volume
weighted by pulsar density to the volume in which a pulsar is detectable:
Here,
is the assumed pulsar space density distribution in terms of galactocentric radius
and
height above the Galactic plane
. Note that
is primarily a function of period
and luminosity
such that short-period/low-luminosity pulsars have smaller detectable volumes and therefore higher
values than their long-period/high-luminosity counterparts.
In practice,
is calculated for each pulsar separately using a Monte Carlo simulation to
model the volume of the Galaxy probed by the major surveys [222]. For a sample of
observed pulsars above a minimum luminosity
, the total number of pulsars in the Galaxy is
where
is the model-dependent “beaming fraction” discussed below in Section 3.2.3. Note that this
estimate applies to those pulsars with luminosities
. Monte Carlo simulations have shown this
method to be reliable, as long as
is reasonably large [182].
3.2.2 The small-number bias
For small samples of observationally-selected objects, the detected sources are likely to be those with
larger-than-average luminosities. The sum of the scale factors (5), therefore, will tend to underestimate the
true size of the population. This “small-number bias” was first pointed out [145, 151
] for the sample of
double neutron star binaries where we know of only four systems relevant for calculations of the merging
rate (see Section 3.4.1). Only when
does the sum of the scale factors become a good indicator
of the true population size.
Despite a limited sample size, recent work [156
] has demonstrated that rigorous confidence intervals of
can be derived. Monte Carlo simulations verify that the simulated number of detected objects
closely follows a Poisson distribution and that
, where
is a constant. By
varying the value of
in the simulations, the mean of this Poisson distribution can be measured. Using
a Bayesian analysis it was shown [156] that, for a single object, the probability density function of the total
population is
Adopting the necessary assumptions required in the Monte Carlo population about the underlying pulsar
distribution, this technique can be used to place interesting constraints on the size and, as we shall see later,
birth rate of the underlying population.
3.2.3 The beaming correction
The “beaming fraction”
in Equation (5) is the fraction of
steradians swept out by a pulsar’s radio
beam during one rotation. Thus
is the probability that the beam cuts the line-of-sight of an arbitrarily
positioned observer. A naïve estimate for
of roughly 20% assumes a circular beam of width
and a randomly distributed inclination angle between the spin and magnetic axes [311].
Observational evidence suggests that shorter period pulsars have wider beams and therefore
larger beaming fractions than their long-period counterparts [223
, 202
, 34
, 306
]. As can be
seen in Figure 16, however, a consensus on the beaming fraction-period relation has yet to be
reached.
When most of these beaming models were originally proposed, the sample of millisecond pulsars was
and hence their predictions about the beaming fractions of short-period pulsars relied largely on
extrapolations from the normal pulsars. An analysis of a large sample of millisecond pulsar
profiles [165
] suggests that their beaming fraction lies between 50 and 100%. The large beaming
fraction and narrow pulses often observed strongly suggests a fan beam model for millisecond
pulsars [218
].