Correcting the observed pulsar sample

3.2 Correcting the observed pulsar sample

In the following, we review common techniques to account for many of the aforementioned selection effects and form a less biased picture of the true pulsar population.

3.2.1 Scale factor determination

A very useful approach [288 , 390 ], is to define a scaling factor $ξ$ as the ratio of the total Galactic volume weighted by pulsar density to the volume in which a pulsar is detectable:

Here, $Σ (R,z)$ is the assumed pulsar space density distribution in terms of galactocentric radius $R$ and height above the Galactic plane $z$ . Note that $ξ$ is primarily a function of period $P$ and luminosity $L$ such that short-period/low-luminosity pulsars have smaller detectable volumes and therefore higher $ξ$ values than their long-period/high-luminosity counterparts.

In practice, $ξ$ is calculated for each pulsar separately using a Monte Carlo simulation to model the volume of the Galaxy probed by the major surveys [267 ]. For a sample of $Nobs$ observed pulsars above a minimum luminosity $Lmin$ , the total number of pulsars in the Galaxy

where $f$ is the model-dependent “beaming fraction” discussed below in Section 3.2.3. Note that this estimate applies to those pulsars with luminosities $≳ Lmin$ . Monte Carlo simulations have shown this method to be reliable, as long as $N obs$ is reasonably large [225 ].

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 [177 , 181 ] for the sample of double neutron star binaries where we know of only five systems relevant for calculations of the merging rate (see Section 3.4.1). Only when $Nobs ≳ 10$ does the sum of the scale factors become a good indicator of the true population size.

Figure 15: Small-number bias of the scale factor estimates derived from a synthetic population of sources where the true number of sources is known. Left panel: An edge-on view of a model Galactic source population. Right panel: The thick line shows $NG$ , the true number of objects in the model Galaxy, plotted against $Nobserved$ , the number detected by a flux-limited survey. The thin solid line shows $Nest$ , the median sum of the scale factors, as a function of $Nobs$ from a large number of Monte Carlo trials. Dashed lines show 25 and 75% percentiles of the $N est$ distribution.

Despite a limited sample size, it has been demonstrated [192 ] that rigorous confidence intervals of $NG$ can be derived using Bayesian techniques. Monte Carlo simulations verify that the simulated number of detected objects $Ndetected$ closely follows a Poisson distribution and that $Ndetected = αNG$ , where $α$ is a constant. By varying the value of $NG$ in the simulations, the mean of this Poisson distribution can be measured. The Bayesian analysis [192 ] finds, 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” $f$ in Equation (5 ) is the fraction of $4π$ steradians swept out by a pulsar’s radio beam during one rotation. Thus $f$ is the probability that the beam cuts the line-of-sight of an arbitrarily positioned observer. A naïve estimate for $f$ of roughly 20% assumes a circular beam of width $∼ 10∘$ and a randomly distributed inclination angle between the spin and magnetic axes [358 ]. Observational evidence summarised in Figure 16 suggests that shorter period pulsars have wider beams and therefore larger beaming fractions than their long-period counterparts [268 , 246 , 39 , 353 ].

Figure 16: Beaming fraction plotted against pulse period for four different beaming models: Narayan & Vivekanand 1983 (NV83) [268 ], Lyne & Manchester 1988 (LM88) [246 ], Biggs 1990 (JDB90) [39

] and Tauris & Manchester 1998 (TM98) [353 ].

When most of these beaming models were originally proposed, the sample of millisecond pulsars was $≲ 5$ 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 [205 ] suggests that their beaming fraction lies between 50 and 100%. Independent constraints on $f$ for millisecond pulsars come from deep Chandra observations of the globular cluster 47 Tucanae [131 ] and radio pulsar surveys [57 ] which suggest that $f > 0.4$ and likely close to unity [139 ]. The large beaming fraction and narrow pulses often observed strongly suggests a fan beam model for millisecond pulsars [262 ].