The population of normal and millisecond pulsars

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 1.5 kpc of the Sun [247 ]. 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.

Figure 17: Left panel: The corrected luminosity distribution (solid histogram with error bars) for normal pulsars. The corrected distribution before the beaming model has been applied is shown by the dot-dashed line. Right panel: The corresponding distribution for millisecond pulsars. In both cases, the observed distribution is shown by the dashed line and the thick solid line is a power law with a slope of –1. The difference between the observed and corrected distributions highlights the severe under-sampling of low-luminosity pulsars.

The 430-MHz luminosity distributions obtained from this analysis are shown in Figure 17 . For the normal pulsars, integrating the corrected distribution above 1 mJy kpc² and dividing by $π×$ (1.5)² kpc² yields a local surface density, assuming a beaming model [39 ], of 156±31 pulsars kpc^–2 for 430-MHz luminosities above 1 mJy kpc². The same analysis for the millisecond pulsars, assuming a mean beaming fraction of 75% [205 ], leads to a local surface density of 38±16 pulsars kpc^–2 also for 430-MHz luminosities above 1 mJy kpc².

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 [166 , 223 ]. 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 1000±250 kpc² [310 ]. This implies a population of $∼$ 160,000 active normal pulsars and $∼$ 40,000 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 $˙ P –P$ diagram in Figure 3 , we infer a typical lifetime for normal pulsars of $∼$ 10⁷ yr, corresponding to a Galactic birth rate of $∼$ 1 per 60 yr – consistent with the rate of supernovae [382 ]. As noted in Section 2.2, the millisecond pulsars are much older, with ages close to that of the Universe $τu$ (we assume here $τ u$ = 13.8 Gyr [405 ]). Taking the maximum age of the millisecond pulsars to be $τ u$ , we infer a mean birth rate of at least 1 per 345,000 yr. This is consistent, within the uncertainties, of other studies of the millisecond pulsar population [110 , 350 ] and with the birth-rate of low-mass X-ray binaries [234 , 190 ].