Since our observations were made at very high radio frequencies, they should be representative of the weak scintillation regime (see Malofeev et al. 1996), where the flux modulation is expected to be small. Unexpectedly, we have observed substantial variations in the measured flux-densities. In fact averaging the observations (neglecting different integration times) we obtain mean values for the modulation indices of , , and . It is significant that the variations in each pulsar are present whenever it is observed. Interestingly, the data plotted in Fig. 2 suggest an anti-correlation of the observed modulation index with the dispersion measure. However, the number (i.e. four) of pulsars is too small to test the significance of this result. Moreover, while the most distant pulsar B0355+54 shows only moderate modulations around a mean value of the flux-density, the other sources exhibit flux variations as large as 100% of the average value, sometimes even resulting in a non-measurable flux density during a single sub-integration. Such behaviour is opposite to that expected for either weak scintillation or RISS (e.g. Sieber 1982, Kaspi & Stinebring 1992, and Fig. 8b of Malofeev et al. 1996).
In order to extrapolate from low frequency ISS observations to predict ISS at our high frequencies, we assume the Kolmogorov model for the ISM density spectrum:
where q is the spatial wavenumber and is proportional to the mean squared electron density fluctuation along the line-of-sight (e.g. Cordes et al. 1985). Although the actual value of the power index, , is still under discussion (see e.g. Armstrong et al. 1995, Malofeev et al. 1996), it is usually assumed that the fluctuations behave like other forms of turbulence, i.e. energy fed into large scale irregularities cascades down into smaller units of scale where energy is dissipated, thus heating the plasma. The resulting spectrum is that of a Kolmogorov power law with .
We follow Rickett (1990) and introduce a parameter, u, characterizing the strength of the ISS. At typical pulsar observing frequencies the scintillations are strong and we can define
where is the decorrelation bandwidth of the DISS, i.e. the typical frequency range over which the observed intensity is more than 50% correlated. With the Kolmogorov spectrum model, we find . In strong scintillation , and the decorrelation bandwidth is related to another measurable quantity, viz the pulse broadening time constant, . As scattered rays propagate different rays experience slightly different path lengths, resulting in a broadening of the pulse. The observed pulse is then a convolution of the emitted pulse by an exponentially decaying waveform of time scale . It follows that = 1 (Cordes et al. 1985). Taking observed values for scaled to 1 GHz (Taylor et al. 1993), we can calculate the strength parameter u and scale it to the observing frequency for our observations. In all cases we obtain , (i.e. weak scintillation), in which case Eq. (3) does not apply and .
For weak scintillation, the modulation index is approximately given by (Rickett 1990):
These predicted values are listed in Table 2 and plotted in Fig. 2. We have also compared the values predicted by this simple theoretical formula with extrapolations based on data obtained for these pulsars between 4.75 GHz and 10.55 GHz by Malofeev et al. (1996). They measured weak scintillation for these pulsars and found modulation indices consistent with or below the predictions based on Eq. (4). We note that the more complicated scintillation model proposed by Malofeev et al. (1996) predicts values that are about 20% greater than from Eq. (4), but still small compared to our observed modulation indices (see Table 2). Only for B0355+54 are the observed modulations comparable to those predicted for scintillations. If the apparent inverse correlation with dispersion measure (Fig. 2) were true, it would mean a yet unexpected propagation effect. However, given the low statistical significance of the apparent inverse correlation, being based on only four pulsars, such an effect is presently a matter for conjecture.
A further comparison can be made between the observed time scales and ISS predictions. In all four pulsars the time scale for a substantial flux variation estimated from Fig. 1 is 10-20 minutes. For B2021+51, however, variations at a 2 -level on time scales smaller than 5 min are visible, indicating that in some cases flux density variations have not been resolved. We have listed a predicted weak ISS time scale in Table 2, using equation 2.2 of Rickett (1990). The values range from 17 minutes for B1929+10 to 68 minutes for B0355+54. Whereas the observed time scales are comparable with the shortest of these, they do not show any increase with distance.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998