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

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5. Radio microlensing: results

In this section, we examine the observed variability in the light curves of B1600+434 A and B in terms of microlensing and, following the procedure delineated in Sect. 4.4, derive constraints on the properties of the jet-component, as well as on the MFs near both lensed images. In Sect. 7, we will use these results to determine the microlensing modulation-index as function of frequency and compare this to the results from independent WSRT 1.4 and 5-GHz monitoring data of B1600+434 (Sect. 2). An overview of the combined microlensing/scintillation situation in B1600+434 is given in Fig. 8, which might act as guide to the overall situation.

From now on, we will (i) use a source structure consisting of a static core plus a single relativistically moving spherically-symmetric jet-component, (ii) assume that the core and jet parameters do not change appreciably over the time-span of the observations, (iii) assume that all short-term variability in the image light curves (Fig. 1) is dominated by microlensing and that scatter-broadening is negligible. The static core does not contribute to the short-term variability, because its velocity with respect to the magnification pattern of the lens is much smaller than that of the jet-component (i.e. [FORMULA][FORMULA][FORMULA]). At this moment, we feel that a more detailed model is not warranted. One should, however, keep in mind that the conclusions drawn below depend on these assumptions !

5.1. Numerical simulations

From KBJ98, we find for B1600+434 that the local convergence and shear are close to [FORMULA]=[FORMULA]=0.2 for image A, using a singular isothermal ellipsoidal (SIE) mass distribution for the lens galaxy. Similarly for image B, we find [FORMULA]=[FORMULA]=0.9. Using these input parameters, we simulate the magnification pattern of a [FORMULA] field for different MFs, where [FORMULA] is the Einstein radius of a 1-[FORMULA] star projected on the source plane. We use the microlensing code developed by Wambsganss (1999). Using the lens redshift [FORMULA], the source redshift [FORMULA]=1.59 and H0=65 km s-1 Mpc-1 in a flat ([FORMULA]=1) FRW universe, we find that [FORMULA]=[FORMULA]as. The magnification pattern is calculated on a grid of 1000[FORMULA]1000 pixels. Each pixel has a size of 0.107 µas by 0.107 µas.

We simulate 100 randomly-oriented light curves on this grid, for a range of jet-component sizes ([FORMULA]=0.125, 0.25,..., 16.0 µas) and MFs for the compact objects (see Sect. 5.2). Each simulated light curve is 54 µas long. For each step (5 pixels) on the light curve, we calculate the magnification of a circular-symmetric jet-component with a constant surface-brightness and a radius [FORMULA], by convolving the magnification pattern with its surface-brightness distribution (e.g. Wambsganss 1999).

We also assume that the surface density near image B is dominated by stellar objects in disk and bulge of the lens galaxy, even though the halo does contribute to the line-of-sight surface density. If we assume that the halo surface density near image B is equal to that near image A ([FORMULA]=0.2), the density of the caustics in the resulting magnification pattern remains completely dominated by the significantly higher number density of compact objects in the disk and bulge ([FORMULA]=0.7). We have tested this by modifying the microlensing code to allow multiple mass functions. The rms variabilities calculated from these modified magnification patterns are the same within a few percent from those without the halo contribution (for the range of mass functions and surface densities that we used in this paper), which justifies this simplification.

5.2. The mass function of compact objects in the lens galaxy of B1600+434

We use a range of different MFs, subdivided in two classes: (i) power-law MFs and (ii) single-mass MFs.

5.2.1. The power-law MF

In the solar neighborhood it appears that the stellar MF can be represented by a single or segmented power-law of the form [FORMULA] (e.g. Salpeter 1955; Miller & Scalo 1979). Recent observations towards the Galactic bulge (e.g. Holtzman et al. 1998) suggest a break in the MF around 0.5-0.7 [FORMULA], with a shallower slope at lower masses. A lower-mass cutoff is not well constrained, although the break in the MF suggests it might lie around a few tenths of a solar mass. Evolution of the MF from z=0.4 to z=0.0 affects the upper mass cutoff only and has no significant influence on microlensing, which is dominated by the mass concentrated around the lower mass cutoff.

  1. First, we simulate the magnification pattern, using a power-law MF with [FORMULA] (Salpeter 1955) and a mass range between 0.01-1.0 [FORMULA].

  2. Second, we also investigate the slopes -2.85 and -1.85 for image B - going through the bulge/disk - assuming again a mass range of 0.01-1 [FORMULA].

  3. Third, we use an MF mass range of 0.1-10 [FORMULA] near image B, with a slope of -2.35. This gives an average stellar mass of about 0.3 [FORMULA], more in line with observations of the bulge of our Galaxy (e.g. Holtzman et al. 1998).

We use the power-law MFs for image B only, even though it is clearly a rough approximation of the true MF. A similar power-law MF for image A, passing through the halo, seems unlikely, especially if the halo consists of stellar remnants (e.g. Timmes, Woosley & Weaver 1996).

The results from these simulations - [FORMULA] as a function of [FORMULA] - are fitted by Eq. (32). The values for [FORMULA] and [FORMULA] are listed in Table 2 (models BP1-4).


Table 2. Summary of the modulation-index caused by microlensing as a function of source size for the power-law MFs (BP1-4) near image B and the single-mass MFs (AS1-9 and BS1-7) near images A and B

5.2.2. The single-mass MF

In steep ([FORMULA][FORMULA]-1) MFs most of the mass is concentrated close to the lower-mass cutoff in the MF. It therefore seems appropriate to approximate the MF by a delta-function. We simulated single-mass MFs for both image A and B, for 0.1, 0.2, 0.4, 0.6, 0.8, 1.0, 1.5 [FORMULA] and additionally for 2.5 and 5.0-[FORMULA] for image A. The results are listed in Table 2.

From these simulation, we notice several things: First, [FORMULA] is almost independent from the average mass of the compact objects, for a given surface density. Second, there appears to be a strong correlation between the average mass of the compact objects and the turnover angle ([FORMULA]), which means that for a given surface density, shear and source size, the modulation-index is larger for a higher average mass of the compact objects. The results from our simulations are consistent with the results in Deguchi & Watson (1987) and Refsdal & Stabell (1991, 1997).

5.3. Microlensing in B1600+434?

As we can see from Table 2, the modulation-index in image A can be explained by a relatively small jet-component with a moderate boosting factor. However, one might expect image B to show similar, if not stronger variability for a similar MF. This is not the case, however. Before we examine this in terms of a different mass function in the disk/bulge and halo of the lens galaxy, we first explore two alternative explanations:

  • What if the dimensionless surface density near image B is near unity (i.e. [FORMULA][FORMULA]1)? In that case, the magnification pattern can become very dense and suppress the modulation-index (e.g. Deguchi & Watson 1987; Refsdal & Stabell 1991, 1997). For B1600+434 we used [FORMULA][FORMULA][FORMULA] (Sect. 5.1), in which case the density of the caustics remains relatively small, in contrast with the case where [FORMULA][FORMULA]1 and [FORMULA][FORMULA]0 (e.g. Refsdal & Stabell 1997). This is supported by simulations with [FORMULA] near image B, which give nearly the same results as for [FORMULA].

  • In Sect. 3.2 we showed that scatter-broadening of image B can suppress scintillation caused by the Galactic ionized ISM. Similarly, is can suppress variability due to microlensing. An overview of the situation is given in Fig. 8. Image A, however, is seen through the galaxy halo at about 4 [FORMULA] kpc above the disk (KBJ98). It is therefore extremely unlikely to pass through a region with a high scattering measure (Sect. 3.2). In the remainder of the paper, however, we assume that both images A and B are not affected by scatter-broadening (situation 2 in Fig. 8).

We will now explore the only other plausible solution; a very different MF of compact objects near images A and B.

5.3.1. Limits on the MF and source structure

In Table 3, we have listed all combinations between the simulated MFs for images A and B (Table 2) that reproduce the observed short-term modulation-indices of 2.8% (A) and 1.6% (B) for a consistent set of parameters f and [FORMULA] (Sect. 4.4). Several examples of simulated light curves are shown in Fig. 7.

[FIGURE] Fig. 7. Left: Eight arbitrary simulated light curves for a single compact 5-µas source, moving over the magnification pattern near image A. The values f and [FORMULA] from Table 3 were used. The simulated curves were subsequently scaled by f. A 1.5-[FORMULA] single-mass MF was used. The light curves were created through a convolution of the source surface-brightness distribution with the magnification pattern. The source was moved over a 500-pixels path (i.e. 214 µas), randomly placed on the magnification pattern. Right: Idem, for image B, using a 0.2-[FORMULA] single-mass MF. The ratio between the modulation-index for the simulated light curves, is approximately equal to the observed ratio of modulation-indices in the observed light curves of B1600+434 A and B (Fig. 1). The scale, in this random example, would correspond to 35 weeks, the duration of the monitoring campaign of B1600+434 ([FORMULA][FORMULA]18).

[FIGURE] Fig. 8. The overall microlensing and scattering situation in B1600+434, when seen `side-ways'. The inclination of the disk has been slightly exaggerated (i=75o). Situation 1: Light coming from image A passes predominantly through the halo. Light coming from image B first passes through the bulge/disk, where the image might be scatter-broadened by the lens-galaxy ionized ISM (Sect. 3.2). The image subsequently subtends a larger solid angle, suppressing microlensing in the lens galaxy and scintillation caused by the Galactic ionized ISM. Situation 2: Only microlensing in the bulge/disk and halo in the lens galaxy occurs. Subsequent scatter-broadening in our galaxy does not suppress microlensing variability.


Table 3. All combinations of the MFs (Table 2) near image A and B that give a consistent solution of the parameters: f[FORMULA]. Given these two parameters, one reproduces the observed modulation indices for both lensed images, using Eq. (33) and Table 3.

Constraints on the MFs in B1600+434: From Table 3 one finds that a significantly higher average mass of compact objects in the halo is needed than in the bulge/disk to explain the modulation indices of both images. No consistent solutions are found for an average mass of compact objects in the halo [FORMULA]1 [FORMULA] for the range of MFs that we investigated. If we furthermore put a conservative upper limit of [FORMULA]0.5 [FORMULA] on the average mass of compact objects in the bulge/disk of B1600+434 - which lies around the break in the Galactic bulge MF (e.g. Holtzman et al. 1998) - only MFs BS1-3 and BP1-4 (Table 2) remain viable MFs for the bulge/disk of B1600+434.

Constraints on [FORMULA],fand [FORMULA]: Using the MFs assumed viable above, we find from Table 3: 0.05[FORMULA]f[FORMULA]0.11 and 1.1[FORMULA][FORMULA][FORMULA]4.0. Using Eq. (25) and the values of f and [FORMULA] listed in Table 3, the jet-component size then lies between 2[FORMULA][FORMULA][FORMULA]5 µas.

Constraints on [FORMULA]: To estimate a time scale for strong microlensing variability, we calculate the average power spectrum of the 100 light curves for each MF, for the source size of 2 and 4 µas. The power-spectrum is typically relatively flat at low frequencies, smearing out the long-period modes in the light-curves. At higher frequencies the power drops rapidly. We therefore expect the strongest Fourier modes in the light curves to lie around the turn-over frequency, where the power drops to about 50%. Consequently, we define the typical angular scale of variability ([FORMULA]) to correspond with the half-power frequency in the power-spectrum. In Table 4, we listed the results for those MFs that give a consistent solution (Table 3).


Table 4. The typical angular scale ([FORMULA]) in µas between strong microlensing events in the simulated light curves, as defined through the power spectrum (see text). Listed are the values for two source sizes, 2 and 4 µas, approximately corresponding to the range of jet-component sizes that reproduce the observed modulation-index in the light curves (see text).

If we now take a separation of [FORMULA]2 weeks as indicative for the separation of strong modulations in light curve of image A (see days 80-140 in Figs. 1-2), we find an angular velocity of the jet-component in the source plane between 3 and 9 µas/week. Using Eq. (30), we then derive 9[FORMULA][FORMULA][FORMULA]26. This range strongly depends on the local structure of the magnification pattern, which can differ strongly from place to place (e.g. Wambsganss 1990).

Constraints on [FORMULA] and [FORMULA]: Using the constraints of the Doppler factor ([FORMULA]) and angular velocity of the jet-component ([FORMULA]), one also obtains constraints on the angle between the jet-component direction with respect to the line-of-sight to the observer and the bulk velocity of the jet-component ([FORMULA]). From Fig. 9, we subsequently find: [FORMULA][FORMULA][FORMULA][FORMULA][FORMULA] and [FORMULA][FORMULA]0.995, for the allowed ranges of these parameters.

[FIGURE] Fig. 9. The bulk velocity ([FORMULA]) and angle with respect to the the line-of-sight to the observer ([FORMULA]) of a relativistic jet-component, as a function of its Doppler boosting ([FORMULA]) and apparent velocity ([FORMULA]), calculated using Eqs. (20) and (26).

Thus, several combinations of MFs for images A and B (Table 3) give solutions that reproduce the observed modulation-indices of both images for a consistent, although not unique, set of jet-component parameters. The derived constraints on the jet-component, however, do agree with observations of confirmed superluminal sources (e.g. Vermeulen & Cohen 1994).

5.3.2. Microlensing by compact halo objects

It appears we have found a lower limit of [FORMULA]1 [FORMULA] on the mass of compact objects in the halo around the lens galaxy, under the assumptions that all variability we see is due to microlensing, the jet is dominated by a single component and there is no scatter-broadening. Let us now explore the implications of this in more detail, first mentioning several potential problems.

  • Could microlensing be due to a globular cluster (GC) in the halo of the lens galaxy? It is easy to show that the probability of seeing a lensed image through a GC surface density [FORMULA] is


    We take a population of N[FORMULA]150 GCs inside the Einstein radius of the lens galaxy (LG), with a velocity dispersion of [FORMULA]=7 km s-1. These values are typical of those found for our galaxy (e.g. Binney & Merrifield 1999). Using [FORMULA][FORMULA]200 km s-1 (KBJ98), [FORMULA]=[FORMULA]=0.2, we then find a probability [FORMULA][FORMULA][FORMULA] that the microlensing optical depth ([FORMULA][FORMULA][FORMULA]) of the GC exceeds that of the dark matter halo. The probability that [FORMULA] exceeds [FORMULA]=0.9, thereby causing similar or larger microlensing variability, is [FORMULA]. Hence, it is very unlikely that a GC in the line-of-sight to lensed image A could enhance the microlensing optical depth significantly.
  • What is the influence of binary systems on the mass limit of a compact object in the dark matter halo? We know that a large fraction of stars in the bulge is locked up in binary systems (e.g Holtzman et al. 1998). In the case of high microlensing optical depths, one can consider a binary as a single microlensing object, with a mass equal to the sum of the individual masses.

    From Table 2 we see that a higher mass of compact objects gives a higher microlensing modulation-index for a given source size. Hence, if we know the typical stellar mass of objects in the bulge/disk and assume they are all single objects, not in a binary system, we underestimate its modulation-index. One has to take this effect into account.

Having dealt with these possible complications, let us explore the lower mass limit of the compact objects in more detail. If we assume that (i) all compact object in the bulge/disk are not in binary systems and (ii) all compact objects are in binary systems and (iii) use the lowest average mass of objects in the bulge/disk (Tables 2-3), we find a very conservative lower limit of 0.5 [FORMULA] on the mass of individual compact objects in the halo. In the more realistic case where most of the stars in the bulge/disk are in binaries and most compact objects are probably not, a lower limit of 1.5 [FORMULA] is found, assuming that the average bulge/disk stellar mass in the halo is [FORMULA]0.1 [FORMULA]. If the bulge/disk stars have average masses somewhere between 0.1 and 0.3 [FORMULA] and are foremost in binaries, the lower limit increases to [FORMULA]2.5 [FORMULA].

As in the case of scintillation, scatter-broadening of image B also suppresses microlensing (Sect. 5.3). If this happens, one would underestimate the true microlensing modulation-index of image B. This would give one more freedom to decrease f and/or increase [FORMULA] for the microlensed jet-component, thereby changing the required average mass of compact objects in the halo. It would, however, never eliminate the need for them.


Table 5. Summary of constraints on the jet-component parameters, derived within the context of the microlensing hypothesis. We assume T[FORMULA]=0.5 and a flat FRW universe with [FORMULA]=1 and H0=65 km s-1 Mpc-1.

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Online publication: June 20, 2000