## 3. ScintillationEven though the difference in modulation-index between images A and B seems to require a considerable change in the properties of the Galactic ionized ISM over an angular scale of 1.4 arcsec (Sect. 3.2), we still proceed to investigate whether the short-term variability, superposed on the gradual and presumably intrinsic long-term decrease of the flux density of the lensed images, can be the result of scintillation. We will follow the prescription of Narayan (1992) and its numerical implementation by Walker (1998; [W98]) for the Galactic ionized ISM model from TC93. This assumes that the inhomogeneities of the ionized ISM can be described by a Kolmogorov power-law spectrum (e.g. Rickett 1977; Rickett 1990) and that the ionized ISM model from TC93 is approximately valid. Support for the approximate validity of the TC93 model in the direction to B1600+434 is given by the dispersion and scattering measures of nearby pulsars, showing no apparent deviations from this model. Also, no evidence is found in the low-frequency (327-MHz) WENSS catalogue (e.g. Rengelink et al. 1997) for diffuse HII emission or SN remnants that could introduce small-scale perturbations in the Galactic ionized ISM. Depending on the line-of-sight through the galaxy and the observing
frequency, the scattering of radio waves, expressed in the scattering
strength
=,
can be strong
(1)
or weak
(1),
where is the Fresnel scale and
is the diffractive scale (e.g.
Narayan 1992). The transition between these two regimes occurs near a
transition frequency (). For
B1600+434 at a Galactic latitude of ## 3.1. Weak scatteringThe modulation-index of a point source in the weak scattering regime is (Narayan 1992) In the simplest case that B1600+434 is a point source smaller than
the Fresnel scale of 3.9 The variability time scale for a point source is given by (Narayan 1992) where is the transverse velocity
of the scintillation pattern with respect to the line-of-sight to the
source in units of 30 km s If 5 to 10% of the flux density of the source is contained in a compact region (), the rms fluctuations decreases to the observed modulation-index of 2-3% for B1600+434-A and B. The variability time scale would still remain 2 h. We have observed B1600+434 with the WSRT at 5 GHz during several 12 h periods and find no evidence for short-term variability 2% over a 12 h time scale (Koopmans et al. in prep.). This excludes the posibility that the longer time-scale variations are purely the result of undersampled light curves. Hence, a simple compact source structure, embedded in a more extended non-scintillating region of emission, can not explain the observed variability. A more extended source () is therefore required, if we want to explain the observed modulation-index in terms of scintillation. In case the source is extended, with a size , both the modulation-index and variability time-scale change. The modulation-index decreases as follows (Narayan 1992) whereas the time scale of variability increases as Combining these two equations, using the transition and observing frequencies for B1600+434, gives the relation for . If the lensed source has an angular radius of about
40 The relatively nearby extragalactic radio source J1819+387 has a
modulation-index In Fig. 5, we have summarized the weak and strong scattering
regimes, as functions of the modulation-index, the variability time
scale and the source size. The variability seen in image A (but
also image B) is especially hard to explain by weak scattering
without either invoking unlikely high values for the equivalent
distance of the phase screen (10 kpc)
or a persistently low transverse velocity
(few km s
## 3.2. Strong scatteringIn the strong scattering regime, we can not use the numerical results derived from the TC93 model, from which one expects B1600+434 to be in the weak-scattering regime at 8.5 GHz. We therefore make direct use of the relation between the scattering measure (SM), the distance to the equivalent phase screen (), the observing frequency () and the scattering strength () (e.g. W98) The scattering strength and the Fresnel scale determine both the modulation-index and variability time scale of a source, given the source size. The Fresnel scale, given by specifies the angular distance from the source over which there is about one radian phase difference between rays, due to the difference in path length. The scattering measure (e.g. TC93) for an extra-galactic source is defined as where is the structure constant normalizing the Kolmogorov power-law spectrum of the ionized ISM inhomogeneities (e.g. Cordes, Weisberg & Boriakoff 1985). From now on, we assume that SM has units of kpc and units of . The distance to the equivalent phase screen (e.g. W98) is defined as Despite the fact that the difference in modulation-index of the lens images seems to require very different properties of the Galactic ionized ISM on a scale of 1.4 arcsec, we will investigate the two distinct strong scattering regimes, i.e. refractive and diffractive (e.g. Rickett 1990; Narayan 1992), in more detail in the next two sections. ## 3.2.1. Refractive scintillationUsing Eq. (6) and the scaling laws from Narayan (1992), one finds for a point source in the strong scattering regime that the modulation-index is whereas the variability time scale is We furthermore use =0.5 (TC93), =1.0 and =8.5 throughout this section. From Eqs. (10-11) it is immediately obvious that for a point source in the refractive regime, an extremely high value for SM is needed ( kpc ) to obtain the modulation-index of images A and B. The time scale of variability would be around 15 years. Clearly the point-source approximation is not valid. For extended sources, the modulation-index and time scale of variability scale as and , respectively, where is the source size and the size of the scattering disk (Narayan 1992). At 8.5 GHz, we find which is independent of the distance to the equivalent phase screen. If we subsequently use the scaling laws, combined with Eqs. (10-11), we find a relation between the time scale of variability and the modulation-index: which is valid only if . Inserting
the usual numerical values for with in units of days. We find that a scattering measure SM is needed to explain modulations with a time scale of 1 week in image A. From TC93 we find that SM= in the direction of B1600+434, corresponding to a time scale of one day. For deep modulations of about 1 month a scattering measure SM0.5 is needed for image A. Both values are larger than can be expected on the basis of the ionized ISM model from TC93.
following Blandford, Narayan & Romani (1986 [BNR86]), who investigated intensity fluctuations (i.e. "flickering") of extended radio sources, caused by refractive scattering. is the normalized light curve as shown in Fig. 1. BNR86 take a slightly steeper spectrum of the phase fluctuations with a power-spectrum slope =4, instead of a Kolmogorov slope of =11/3. Fitting the theoretical SF from BNR86 to the observed
SFs
To test the reliability of these saturation time scales, we replaced the normalized flux densities at each epoch in Fig. 1 by Gaussian-distributed values with a 1- scatter equal to the observed modulation-index of the light curve. In Fig. 6 the result is shown, from which it is immediately clear that the light curves are undersampled such that the SFs and the saturation time scales for time lags 4 days become highly unreliable.
The difference between the modulation-indices of images A and B can be explained by a difference in the scattering measure of the Galactic ionized ISM towards both images, as well as by a difference in their respective image sizes (see above). However, image B has a smaller magnification due to the lensing potential and should therefore be smaller than image A. Consequently, image B should show stronger variability than image A, whereas it does not. The only viable alternative to obtain a larger size for image B is through scatter-broadening by the ionized ISM in the lens galaxy. The expected scattering disk at 8.5 GHz due to the Galactic ionized
ISM is 1 Recent polarization observations by Patnaik et al. (1999) gave
rotation measures RM=40 rad m Hence, although scatter-broadening cannot be excluded, to fully explain the observed difference between the modulation-indices of images A and B in terms of galactic scintillation, one would require an extremely high scattering measure in the lens galaxy. ## 3.2.2. Diffractive scintillationFor diffractive scintillation at 8.5 GHz to be at work, one requires both a very high scattering measure and a very small source, neither of which seems plausible. However, to be complete we briefly discuss this possibility. The modulation-index is unity for a point source, much larger than seen in both lensed images. However, for a source larger than the scale on which there are phase changes of about 1 radian (), the modulation-index becomes , where is the source size. We find (e.g. W98) and for the point-source variability time-scale where == sec. The time scale increases by , if the source size is larger than . Combining the equations above, we find the relation where © European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |