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Astron. Astrophys. 358, 793-811 (2000)

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3. Scintillation

Even 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 [FORMULA]=[FORMULA], can be strong ([FORMULA][FORMULA]1) or weak ([FORMULA][FORMULA]1), where [FORMULA] is the Fresnel scale and [FORMULA] is the diffractive scale (e.g. Narayan 1992). The transition between these two regimes occurs near a transition frequency ([FORMULA]). For B1600+434 at a Galactic latitude of b=+48.6o, we find [FORMULA]=4.2 GHz (W98). At the observing frequency of 8.5 GHz, scattering should therefore be in the weak regime ([FORMULA][FORMULA]0.3). The transverse velocity of the ISM with respect to the line-of-sight to B1600+434 is determined by projecting the velocity vector of the earth's motion on the sky as function of time. We find a transverse velocity (v) between 20 and 40 km s-1. We will therefore adopt an average value of v=30 km s-1 throughout this paper. For lack of better knowledge, we assume any intrinsic transverse motion of the scattering medium to be zero.

3.1. Weak scattering

The modulation-index of a point source in the weak scattering regime is (Narayan 1992)

[EQUATION]

In the simplest case that B1600+434 is a point source smaller than the Fresnel scale of 3.9 µas (W98), we would expect a modulation-index around 35%. This is significantly larger than the observed modulation-indices of 2.8% for image A and 1.6% for image B. Hence, the total angular size of the images must be larger than the Fresnel scale. This does, however, not exclude that part of the source might still be compact.

The variability time scale for a point source is given by (Narayan 1992)

[EQUATION]

where [FORMULA] is the transverse velocity of the scintillation pattern with respect to the line-of-sight to the source in units of 30 km s-1. Inserting the values for the observing frequency ([FORMULA]), the transition frequency ([FORMULA]) and [FORMULA]=1.0, a time scale around 2 h is found, which is much smaller than the apparent variability time scales of days to weeks seen in a significant fraction of the light curves (Figs. 1-2).

If 5 to 10% of the flux density of the source is contained in a compact region ([FORMULA][FORMULA]), 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 [FORMULA]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 [FORMULA]2% over a [FORMULA]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 ([FORMULA][FORMULA]) 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 [FORMULA][FORMULA][FORMULA], both the modulation-index and variability time-scale change. The modulation-index decreases as follows (Narayan 1992)

[EQUATION]

whereas the time scale of variability increases as

[EQUATION]

Combining these two equations, using the transition and observing frequencies for B1600+434, gives the relation

[EQUATION]

for [FORMULA].

If the lensed source has an angular radius of about 40 µas, the modulation-index reduces to 2-3%, as observed. The variability time scale should then be around 1 day, still significantly smaller than the observed modulation timescale in a major part of the light curves. Although part of the very short-term ([FORMULA] few days) variability could be due to scintillation, many of the long-term variations seen in Figs. 1-2 certainly can not be explained this way.

The relatively nearby extragalactic radio source J1819+387 has a modulation-index m[FORMULA]0.5 and a variability time scale t[FORMULA]0.5-1 hour (Dennett-Thorpe & de Bruyn 2000). Using the relation between m and t (e.g. Eq. (5)), this translates to a variability time scale less than a day for a modulation-index of a few percent, in agreement with expectations from the Galactic ionized ISM model from TC93. Similarly, variations with a time scale of more than a week (Figs. 1-2) require a source size of 0.3 mas, reducing the modulation-index to 0.2%, which is well below the noise level in the VLA 8.5-GHz light curves.

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 ([FORMULA]10 kpc) or a persistently low transverse velocity ([FORMULA]few km s-1). From this, we conclude that weak scattering has great difficulties in accounting for the observed modulation-indices and longer variability time scales ([FORMULA]1 day) seen in the VLA 8.5-GHz light curves of the lensed images. However, it remains difficult to determine a reliable variability time scale from the image light curves, partly because of the relatively poor sampling (i.e. every 3.3 days). In any case, sigificant variability on short time scales ([FORMULA]12 h) is excluded (see above). For further discussion we refer to Sect. 3.2, where we present the structure functions of the observed variations.

[FIGURE] Fig. 5. The modulation-index as a function of the variability time scale for the weak scattering regime (Eq. (5)), using a transverse velocity v=40 km s-1 (solid line) or 20 km s-1 (dashed line). The short dashes perpendicular to the lines indicate different source sizes (i.e. 1, 2, 4,..., 64 in units of the first Fresnel zone, i.e. 3.9 µas). The skewed dashed regions in the upper-right corner indicate the strong scattering regime ([FORMULA]) for the two transverse velocities, using an equivalent phase-screen distance of 0.5 kpc. The gray region indicates a variability time scale of 1-30 days and a modulation-index range of 1.6-2.8%, as observed for the lensed images. The arrow indicates the observed 2.8% modulation-index and time scale [FORMULA]1 week as seen in the difference light curve (Fig. 2). Only for an equivalent phase screen distance [FORMULA]10 kpc (with v=30 km s-1) or a transverse velocity [FORMULA]few km s-1 (with D=0.5 kpc) does most of the gray region enter the weak scattering regime.

3.2. Strong scattering

In 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 ([FORMULA]), the observing frequency ([FORMULA]) and the scattering strength ([FORMULA]) (e.g. W98)

[EQUATION]

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

[EQUATION]

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

[EQUATION]

where [FORMULA] 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 [FORMULA] and [FORMULA] units of [FORMULA]. The distance to the equivalent phase screen (e.g. W98) is defined as

[EQUATION]

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 scintillation

Using 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

[EQUATION]

whereas the variability time scale is

[EQUATION]

We furthermore use [FORMULA]=0.5 (TC93), [FORMULA]=1.0 and [FORMULA]=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 ([FORMULA] kpc [FORMULA]) 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 [FORMULA] and [FORMULA], respectively, where [FORMULA] is the source size and [FORMULA] the size of the scattering disk (Narayan 1992). At 8.5 GHz, we find

[EQUATION]

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:

[EQUATION]

which is valid only if [FORMULA]. Inserting the usual numerical values for v, D and [FORMULA], we find

[EQUATION]

with [FORMULA] in units of days. We find that a scattering measure SM[FORMULA][FORMULA] is needed to explain modulations with a time scale of [FORMULA]1 week in image A. From TC93 we find that SM=[FORMULA] in the direction of B1600+434, corresponding to a time scale of one day. For deep modulations of about 1 month a scattering measure SM[FORMULA]0.5 is needed for image A. Both values are larger than can be expected on the basis of the ionized ISM model from TC93.

Differences in the scattering measure: A strong argument against refractive scattering is the large difference between the modulation-indices of images A and B. If this is due to a difference in the scattering measure, it requires [FORMULA][FORMULA][FORMULA][FORMULA]3.1 (Eqs. (13-14)), which is substantial over an angular scale of only 1.4 arcsec. The same factor is found for the weak-scattering regime.

The structure function: We have also calculated the structure function (SF; Simonetti et al. 1985):

[EQUATION]

following Blandford, Narayan & Romani (1986 [BNR86]), who investigated intensity fluctuations (i.e. "flickering") of extended radio sources, caused by refractive scattering. [FORMULA] 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 [FORMULA]=4, instead of a Kolmogorov slope of [FORMULA]=11/3.

Fitting the theoretical SF from BNR86 to the observed SFs 1 (Fig. 6) gives a scale length of L[FORMULA]0.9 kpc, which corresponds to an equivalent phase screen distance of 0.9[FORMULA]0.5 kpc (Eq. (9)). The `best-fit' image sizes are 62 and 108 µas, respectively, for images A and B. The saturation time scales (e.g. BNR86) found from these fits are [FORMULA]=2.5 days and [FORMULA]=4.4 days, even though there are clear variations with longer time scales in the light curves. The presence of variabilty with longer time scales ([FORMULA]week) has been confirmed by new multi-frequency VLA observations in 1999/2000 (Koopmans et al. in prep.).

[FIGURE] Fig. 6. Left: The structure functions of the normalized light curve of images A and B. The solid curves shows the expected structure functions, derived as in BNR86, assuming a scale length of 0.9 kpc for the Gaussian distribution of the ionized scattering medium and v=30 km s-1. Fits are only obtained for very different source sizes of 62 and 108 µas for images A and B, respectively. The 1-[FORMULA] error bars on the structure functions are derived from Monte-Carlo simulations. Right: The simulated structure functions of the normalized light curve of images A and B, replacing the observed normalized flux densities at each epoch with a random Gaussian distributed value with a 1-[FORMULA] scatter equal to the observed modulation-index in the light curves, i.e. 2.8% and 1.6% for images A and B, respectively

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-[FORMULA] 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 [FORMULA][FORMULA]4 days become highly unreliable.

Scatter-broadening:

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 [FORMULA]1 µas and cannot account for the apparent difference in the image size, derived from the SFs. This requires a scattering disk of [FORMULA]90 µas for image B at 8.5 GHz, if image A is not scatter-broadened, implying that [FORMULA][FORMULA]1 in the lens galaxy. If we take into account that scattering occurs at a frequency of 8.5[FORMULA][FORMULA]12.0 GHz, this implies a scattering disk of 3 mas at 1.7 GHz, which is nearly equal to the very conservative upper limit of [FORMULA]4 mas on the image sizes found from 1.7-GHz VLBA observations (Neal Jackson, private communication).

Recent polarization observations by Patnaik et al. (1999) gave rotation measures RM=40 rad m-2 and has RM=44 rad m-2 for images A and B, respectively. The difference of 4[FORMULA]4 rad m-2 is rather low and certainly does not support a high scattering measure in the disk/bulge of the lens galaxy.

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 scintillation

For 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,

[EQUATION]

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 ([FORMULA]), the modulation-index becomes [FORMULA], where [FORMULA] is the source size. We find (e.g. W98)

[EQUATION]

and for the point-source variability time-scale

[EQUATION]

where [FORMULA]=[FORMULA]=[FORMULA] sec. The time scale increases by [FORMULA], if the source size is larger than [FORMULA]. Combining the equations above, we find the relation

[EQUATION]

where m is the observed modulation-index. This relation is independent of source size, as long as the source is larger than the diffractive scale [FORMULA]. Using this equation, we immediately find that [FORMULA] for the deep modulations of [FORMULA]1 week, which is only true in the weak scattering regime. Thus, diffractive scattering offers no solution either, which comes as no surprise.

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Online publication: June 20, 2000
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