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:
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 . For a sample of observed pulsars above a minimum luminosity , the total number of pulsars in the Galaxy is.
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.
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 . 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  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 .
© Max Planck Society and the author(s)