as the ratio of the total Galactic volume weighted by pulsar
density to the volume in which a pulsar could be detected by the
surveys:
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
approach is similar to the classic
technique first used to correct observationally-biased samples
of quasars [211].
This technique 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 [170]. For a sample of
observed pulsars above a minimum luminosity
, the total number of pulsars in the Galaxy with luminosities
above this value is simply
where
f
is the model-dependent ``beaming fraction'' discussed below in
§
3.2.3
. Monte Carlo simulations of the pulsar population incorporating
the aforementioned selection effects have shown this method to be
reliable, as long as
is reasonably large [131].
)), therefore, will tend to underestimate the true size of the
population. This ``small-number bias'' was first pointed out by
Kalogera et al. [112
,
113] for the sample of double neutron star binaries where we know of
only three clear-cut examples (§
3.4.1). Only when the number of sources in the sample gets past 10 or
so does the sum of the scale factors become a good indicator of
the true population size.
, the true number of objects in the model Galaxy, plotted against
, the number detected by a flux-limited survey. The thin solid
line shows
, the median sum of the scale factors, as a function of
from a large number of Monte-Carlo trials. Dashed lines show 25
and 75% percentiles of the
distribution.
) is simply 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 [238]. Observational evidence suggests that shorter period pulsars
have wider beams and therefore larger beaming fractions than
their long-period counterparts [171,
149,
32,
231]. It must be said, however, that a consensus on the beaming
fraction-period relation has yet to be reached. This is shown in
Fig.
16
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.
]) and Narayan & Vivekanand 1983 (NV83; [171]).
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. A recent analysis of a large sample of
millisecond pulsar profiles by Kramer et al. [122] suggests that the beaming fraction of millisecond pulsars lies
between 50 and 100%.
|
Binary and Millisecond Pulsars at the New Millennium
Duncan R. Lorimer http://www.livingreviews.org/lrr-2001-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
