5.1. Distance distribution
The inclusion of a spread in the DM in the population synthesis has two effects on the results of the simulation. The first is that the DM distribution of the simulated population changes (see Fig. 2). The second effect is a change of the sample of simulated pulsars that is retained for comparison with the real pulsars, because these pulsars are selected on the basis of the derived distance instead of the actual distance. In the new simulation both real and simulated pulsars with are placed at a derived kpc. Pulsars with an actual distance projected on the Galactic Plane kpc thus can have a projected derived distance kpc (see Fig. 3). In fact, at kpc, almost half of the pulsars in the simulated comparison sample has kpc. More importantly, the derived luminosity is based on the derived distance, and is lower than the real luminosity for pulsars above the electron layer. Thus, the luminosity distribution derived from the fluxes in the simulation shifts towards lower values; to compensate for this, a higher intrinsic luminosity distribution of the pulsars is required. (In terms of Eq. 3 of Hartman et al. (1997) for the luminosity distribution, the best value of a changes from 1.5 in their model B to 0.9 in our model.)
5.2. Cloud size
Because some of the parameters we use can be derived independently, we can determine the actual cloud size given by our model. From EM and DM measurements Reynolds (1991) derived a filling factor (see also Anantharamaiah & Bhattacharya 1986). For an average electron density (e.g. Weisberg et al. 1979) together with the obtained value of and Eq. 2we find
Remarkably, this is similar to the sizes of clouds containing both neutral and ionized hydrogen that have been found by Reynolds et al. (1995).
From we can check the assumption made in the appendix, that we can replace with infinity in the summation over the Poisson probabilities. This is strictly only possible if , i.e. . From the filling factor and the equation for L (see Appendix) we find . However since the Poisson distribution drops off rapidly, the error in replacing with in the summations in the Appendix is smaller than 1 %.
For small distances, and therefore small DM, we make an error in applying this model since the individual inhomogeneities become important. However, the sizes of the clouds are relatively small compared to the scales involved in the simulation ( kpc), and the number of pulsars in our simulations at distances less than the free path length L is negligible.
5.3. DM variations in other pulsar simulations
Lorimer et al (1993) model the spread in the dispersion measures expected at a given distance, by assuming that the logarithm of the ratio has a Gaussian distribution with width . (In a model of constant electron density this is identical to the assumption by Gunn & Ostriker (1970) that the logarithm of the ratio of real to derived distance of radio pulsars has a Gaussian distribution.) In the description by Lorimer et al. (1993) the spread in the DM is roughly proportional to DM itself. In principle, the relation between DM and can be derived from the deviation of the directly measured distances (i.e. by HI absorption, association with an object of known distance, or parallax) from the distances derived from the dispersion measure, but in practice the number of accurate distance measurements is too small.
Because of the wide applicability of the central limit theorem, the simple model discussed in Sect. 2 suggests that for a wide variety of models for small scale structure in the electron density distribution. Our simulations show that such a variance adequately describes the currently available observations.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998