### 3.3 The population of normal and millisecond pulsars

#### 3.3.1 Luminosity distributions and local number estimates

Based on a number of all-sky surveys carried out in the 1990s, the scale factor approach has been used to
derive the characteristics of the true normal and millisecond pulsar populations and is based on the sample
of pulsars within of the Sun [205]. Within this region, the selection effects are well understood
and easier to quantify than in the rest of the Galaxy. These calculations should therefore give a reliable local
pulsar population estimate.

The luminosity distributions obtained from this analysis are shown in Figure 17. For the normal pulsars,
integrating the corrected distribution above and dividing by yields a local
surface density, assuming a beaming model [34], of for luminosities above
. The same analysis for the millisecond pulsars, assuming a mean beaming fraction of
75% [165], leads to a local surface density of also for luminosities above
.
#### 3.3.2 Galactic population and birth-rates

Integrating the local surface densities of pulsars over the whole Galaxy requires a knowledge of the
presently rather uncertain Galactocentric radial distribution [137, 181]. One approach is to
assume that pulsars have a radial distribution similar to that of other stellar populations and to
scale the local number density with this distribution in order to estimate the total Galactic
population. The corresponding local-to-Galactic scaling is [268]. This implies a
population of active normal pulsars and millisecond pulsars in the
Galaxy.

Based on these estimates, we are in a position to deduce the corresponding rate of formation or
birth-rate required to sustain the observed population. From the diagram in Figure 3, we infer a
typical lifetime for normal pulsars of , corresponding to a Galactic birth rate of per
- consistent with the rate of supernovae [328]. As noted in Section 2.2, the millisecond pulsars are
much older, with ages close to that of the Universe (we assume here [347]). Taking
the maximum age of the millisecond pulsars to be , we infer a mean birth rate of at least 1 per
. This is consistent, within the uncertainties, with the birth-rate of low-mass X-ray
binaries [189].