In this expression, is the assumed pulsar distribution in terms of galactocentric radius R and height above the Galactic plane z . Note that is primarily a function of period P and luminosity L such that short period/low-luminosity pulsars have smaller detectable volumes and therefore higher values than their long period/high-luminosity counterparts.
This scaling function can be used to estimate the total number of active pulsars in the Galaxy. In practice, this is achieved by calculating 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 with luminosities above this value is
Monte Carlo simulations of the pulsar population incorporating the aforementioned selection effects have shown this approximation to be reliable . The factor f in this expression, known as the ``beaming factor'', is the fraction of steradians swept out by the radio beam during one rotation. Thus f gives the probability that the beam cuts the line-of-sight of an arbitrarily positioned observer. A naïve estimate of f is 20%; this assumes a beam width of 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 [118, 106, 33, 147]. It must be said, however, that a consensus on the beaming fraction-period relation has yet to be reached. This is shown below where we compare the period dependence of f as given by a number of models.
Adopting the Lyne & Manchester model, pulsars with periods ms beam to about 30% of the sky compared to the Narayan & Vivekanand model in which pulsars with periods below 100 ms beam to the entire sky. Note that, when many of these models were proposed, the sample of millisecond pulsars was < 5; hence their predictions about the beaming fractions of short-period pulsars relied largely on extrapolations from the normal pulsars. A recent analysis of a large sample of millisecond pulsar profiles  suggests that the beaming fraction of millisecond pulsars lies between 50% and unity.
|Binary and Millisecond Pulsars
D. R. Lorimer (email@example.com)
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