| |
*Astron. Astrophys. 321, 513-518 (1997)*
## 4. Discussion
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
helpdesk.link@springer.de |