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 [116]. 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 [95]. 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 [150]. 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 [90] suggests that the beaming fraction of millisecond pulsars lies between 50% and unity.

Binary and Millisecond Pulsars
D. R. Lorimer (dunc@mpifr-bonn.mpg.de)
http://www.livingreviews.org/lrr-1998-10
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |