7 Discussion

The deviations between the flux density values of pulsars and the correlated WENSS and NVSS sources are mainly caused by scintillation effects. Small-scale inhomogeneities in the interstellar medium affect the travel path of radio waves and can amplify or weaken them. For reviews, see Rickett (1990) or Narayan (1992). Walker (1998) summarizes the involved equations and dependencies.

In the case of strong scintillation, the phase changes due to the scattering in a certain region are larger than the changes in phase due to normal geometry. The scintillation is called weak in the opposite case. The boundary between the two is dependent on the frequency of the radio waves and the distance to the pulsar. At 325 and 1400 MHz almost all pulsars are in the strong scintillation regime.

Two types of strong scintillation exist: diffractive and refractive. They differ in their typical timescale $\tau$, frequency bandwidth $\Delta \nu$ and size of the resulting flux density variation. This strength of the scintillation is usually quantified by the modulation index, i.e. the rms fractional flux density variation. In the case of strong scintillation waves from multiple locations in the scattering region interfere constructively (or destructively). Both the typical timescale and the frequency bandwidth are small. The modulation index equals one. The timescale is dependent on the relative velocities of the pulsar, the interstellar medium and the Earth. For the WENSS observations of pulsars typical timescales are 1 to 10 minutes and typical bandwidths are 10 Hz to 500 kHz. This means that for almost all pulsars any variations due to diffractive scintillation are averaged out over the 5 MHz WENSS bandwidth and when the $6 \times 18$ short observations spread over $6 \times 12$ hours are combined.

Figure 9: Greyscale WENSS map of PSR B0329+54. The plot is centered around the pulsar position. To improve the visibility of the spokes, the greyscale range has been adjusted (top panel). The grey oval in the lower left corner indicates the FWHM beam size. The spokes are caused by diffractive scintillation, the ring is caused by refractive scintillation

Only for PSR B0809+74 are the diffractive scintillation timescale and bandwidth large enough that some effect remains. From the equations given by Walker (1998) one finds a timescale of 12 minutes and a bandwidth of 500 kHz. The actual value of the typical scintillation timescale and bandwidth are even higher, since several authors have already shown that the Taylor-Cordes distance model (Taylor & Cordes 1993) gives too small predictions for this pulsar (e.g. Rickett et al. 2000). Diffractive scintillation can explain the WENSS image of this pulsar (Fig. 5). As 40 minutes is the time between two observations of a field in one mosaic, flux density variations on that time scale cause spokes in the map. Such spokes can also be seen in the complete map of PSR B0329+54 (Fig. 9).

Refractive scintillation is caused by the focussing effect of a large scattering region. The timescales and bandwidths involved are much larger than in the case of diffractive scintillation. The modulation index is also smaller. The scintillation bandwidth is of the order of the observing frequency. Refractive time scales for the pulsars detected in the WENSS vary from a couple of days to several years and the expected modulation index from 0.05 to 0.3.

If the refractive time scale is less than the time between observations of the same mosaic (couple of days to several years), any flux density variation will be averaged out when the mosaics are combined. However, stong flux density variations between 12 hour sessions cause a ring at the first grating ring of the synthesised beam. Since the mutual distances between the dishes are multiples of 72 m, the ring will have radius of 72 m/(c/325 MHz) radians, i.e. 44$^\prime$ in right ascension and 44$^\prime$  $\times~\mbox{$\mathrm{cosec}$ }~\delta$ in declination. The second and higher grating rings are not visible, since data far from the field center gets a low weighting factor when the final image is created.

It is hard to give good estimates for the expected diffractive and refractive scintillation timescales. They depend on the often poorly known pulsar velocity. For large pulsar velocities (compared to the velocity of the interstellar medium, being about 50 km s^-1) the dependence is as one over the square root of this velocity. Since some pulsar velocities might be up to several hundred kilometers per second, this cannot be neglected. I divided the pulsars that are detected in the WENSS in two groups, based on their expected refractive modulation index if their velocity is neglected. Both groups had similar relative deviations between their measured and expected flux densities.

The observed modulation index is about 0.4, much larger than the expected value of about 0.2. The WENSS pulsar flux densities vary more due to refractive scintillation than predicted by the equations. This has been observed before by several authors and is attributed to the assumption that the turbulence in the interstellar medium has a Kolmogorov spectrum (e.g. Blandford & Narayan 1985).

The NVSS flux densities are even more affected by scintillation effects. Each point in the NVSS maps is an average of about three snapshots. Two of these three are taken right after each other (snapshot series are taken at constant declination and increasing right ascension, see Fig. 7 in Condon et al. 1998) and very little averaging takes place. The expected refractive modulation index at 1400 MHz is larger than at the WENSS frequency. The expected diffractive frequency bandwidth is also larger at the NVSS frequency, even relative to the increased total bandwidth used in the NVSS. One therefore expects the differences between the pulsar flux densities in the NVSS and the flux densities reported by LYLG to be larger than the differences reported in this study. This is indeed observed: the modulation index of the WENSS sources in Table 2, excluding PSRs J0218+4232, B0329+54 and B2021+51 is 0.40, the modulation index of the NVSS sources in Table 1 of Han & Tian (1999) is 0.60.

De Breuck et al. (2000) show that the total spectral index distribution of WENSS sources that are correlated with a NVSS source ( $S_{\rm 325} > 50$ mJy and $\mid \!\! b \!\! \mid\ > 15^\circ$) has a mean of -0.80 and a standard deviation of 0.24. Pulsar have a mean spectral index of -1.6 (see LYLG). An increase of 0.5 due to scintillation will move the pulsar spectral index well into the distribution of normal sources, which are much more frequent. The spectrum of some point-like quasars will also be affected by scintillation and some of them will seem to have a much steeper spectrum than they really have. This effect should be taken into consideration if pulsar candidates are selected on the basis of their spectral index derived from the WENSS and NVSS.

Acknowledgements

I thank F. Verbunt and A.G. de Bruyn for discussion and comments. I thank B.W. Stappers for comments on the manuscript. I am supported by The Netherlands Research School for Astronomy (NOVA), a national association of astronomy departments at the Universities of Amsterdam, Groningen, Leiden and Utrecht.