3 Detected pulsars

3.1 Positional correlation

The radio pulsar catalog (Taylor et al. 1993; Taylor et al. 1995) contains 84 entries in the part of the sky that was covered by the WENSS area. The typical positional uncertainty for a pulsar with a flux density greater than 10 mJy is 0.1 $^{\prime \prime }$ or less. In most cases the uncertainty in the pulsar position is negligible compared to the positional uncertainty of WENSS sources.

The pulsar proper motions are neglected, since for each pulsar the change in position between the epoch of discovery and the epoch of the WENSS is less than 0.5 $^{\prime \prime }$ in right ascension and less than 0.8 $^{\prime \prime }$ in declination for all these pulsars. This is much smaller than the uncertainties in the WENSS positions.

Seven pulsars in the WENSS area have large positional errors of about 4$^\prime$. The probability that a WENSS source is located by chance within a circle of three times this positional error is more than 90 percent, if it assumed that the WENSS sources are uniformly distributed over the sky. Therefore, I have excluded these 7 pulsars (PSRs J0417+35, B1639+36B, J1758+30, J1900+30, J1931+30, J2002+30 and J2304+60) from further analysis. PSR B2000+40 was also excluded, since it is located in the Cygnus A region, where no WENSS map could be made.

Figure 2: Histogram of differences between the position of a pulsar and its nearest WENSS source in units of the combined positional uncertainty $\sigma$. The dotted line markes the 3$\sigma$ limit. The dashed lines is the expected distribution for 25 correlations, i.e. the number of found pairs with a positional difference below 3$\sigma$

I have taken the J2000 positions of the remaining 76 pulsars and compared them with the positions of the sources in the WENSS catalog. Twenty-five pulsars have a WENSS source located within three times their combined positional uncertainty $\sigma$, with

$$% \sigma = \sqrt{ \left( \frac{\Delta\alpha}{\sigma_\alpha} \right)^2 + \left( \frac{\Delta\delta}{\sigma_\delta} \right)^2 },$$ (1)

where $\Delta\alpha$ and $\Delta\delta$ are the positional differences in right ascension and declination, respectively, and

$$% \sigma_\alpha = \sqrt{\sigma_{\alpha,{\rm WENSS}}^2 + \sigma_{\alpha,{\rm PSR}}^2}$$ (2)

and similarly for $\sigma_\delta$. Table 1 lists the J2000 positions of the correlated pulsars and WENSS sources and their offset, both in arcseconds and in $\sigma$.

In Fig. 1 it can be seen that the pulsars are located in areas with different values for the WENSS source density. The probability of a change coincidence can be approximated by a probability calculation that assumes a uniform distribution of the WENSS sources. In that case, the probability that an individual correlation is just by chance is 0.0012. The binomial probability, that one out of 76 trials gives a chance correlation is 0.083. The probability that two correlations occur by chance is 0.004.

Figure 2 displays the distribution of positional differences between a pulsar and its nearest WENSS source in units of their combined positional uncertainty $\sigma$. There is a clear gap between the correlated pairs (difference less than 3$\sigma$) and the non-correlated ones. The distribution of the positional difference ($\Delta$) for the related pairs is

$$% P(\Delta) = \Delta \cdot {\rm e}^{-\Delta^2/2},$$ (3)

which follows after a conversion to polar coordinates. This distribution is also plotted in Fig. 2. A Kolmogorov-Smirnov test assigns a 15 procent probability that our sample is drawn from the distribution (3).

There are two objects with positional differences between 3 and 10 $\sigma$. These are in confused regions and will be discussed in Sect. 4.

  

Table 1: J2000 positions of pulsars and correlated WENSS sources. Offsets are the roots of the sum of the squared differences in right ascension and declination in $^{\prime \prime }$ (absolute) and in $\sigma$ (relative)

$$\begin{tabular}{lc@{ }c@{ }c@{.}c@{ $\pm$ }r@{.}l c@{ }c@{ }c... ... & 36 & 7 & 7 & 94 & 1 & 19 \\ \\ \hline\hline \end{tabular}$$

   
3.2 Flux densities

Table 2 lists the observed flux densities of the correlated WENSS sources and their uncertainties. The flux densities can be compared with known pulsar flux densities, but these are usually measured at frequencies of 400 MHz and higher. By assuming a power law with constant spectral index, these flux density data can be extrapolated. I have used flux density data from Lorimer et al. (1995), hereafter LYLG. These flux densities are averaged over many observations spread over years and their uncertainties include variations due to scintillation. LYLG provide data for 24 of the 25 pulsars in Table 1. Flux densities for PSR J0218+4232 are taken from its discovery paper (Navarro et al. 1995).

Pulsar flux densities usually obey a power law with a negative exponent ( $S \propto \nu^\alpha$ with $\alpha < 0$) in the frequency range from 325 to 1400 MHz. I have fitted the logarithms of the flux densities with a straight line. These lines are plotted in Fig. 3. From this fit a flux density at 325 MHz is estimated. This estimate is plotted against the flux density of the WENSS counterpart in Fig. 4. It is known that some pulsars have a low frequency turnover, usually located around 100 MHz (Malofeev et al. 1994). However, some pulsars exhibit a turnover at a higher frequency: PSR B0329+54 around 300 MHz (Lyne & Rickett 1968) and PSR B2021+51 around 400 MHz. The spectrum of PSR B2319+60 is flat between 200 and 600 MHz (Malofeev et al. 1994). PSRs B1946+35 and B2154+40 also have a turnover, but it is not clear whether this located between 325 and 400 MHz (Malofeev et al. 1994). In these cases the assumption of a constant power law is not correct and this results in an overestimated flux density at 325 MHz. Navarro et al. (1995) discovered that the flux density of PSR J0218+4232 has a non-pulsed component. They find that the continuum flux density at 325 MHz varies between 100 and 200 mJy. I used this flux density estimate for a comparison with the WENSS flux density. An estimate from the pulsed flux density was not derived.

Figure 3: Flux densities against frequency for 24 pulsars with a WENSS counterpart (PSR J0218+4232 is excluded, see text). Filled circles mark flux densities measured by Lorimer et al. (1995). The dotted line is a fitted power law with a constant spectral index. Flux densities from the corresponding WENSS sources are indicated by a cross. Flux densities from the corresponding NVSS sources are marked with a plus. The errors in the WENSS and NVSS source flux densities are smaller than or of the order of the size of the symbol and do not include any uncertainty caused by scintillation

3.3 Source maps

Figure 5 shows maps of the WENSS sources which are correlated with a pulsar. Most sources are consistent with a point source convolved with the beam shape. The shapes of PSR B0329+54 and PSR B0809+74 are affected by scintillation (see Sect. 7). The integrated and peak flux densities of PSR J0218+4232 differ by 20 percent. This suggests that the source is extended. However, a two-dimensional fit shows that its shape is not significantly different from the beam shape.

  

Table 2: List of estimated pulsar flux densities (PSR S₃₂₅) and WENSS source peak flux densities (WENSS S₃₂₅) for the 25 correlated objects. Navarro et al. (1995) observed that the continuum flux density for PSR J0218+4232 varies significantly between 100 and 200 mJy

$$\begin{tabular}{lr@{.}l@{ $\pm$ }r@{.}lr@{ $\pm$ }rr} \hline\... ...\,\,}}}{8}& & 81 & 16 & 11.9\\ ~\\ \hline\hline \end{tabular}$$