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. ). 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 ==0.2 for image A, using a singular isothermal ellipsoidal (SIE) mass distribution for the lens galaxy. Similarly for image B, we find ==0.9. Using these input parameters, we simulate the magnification pattern of a field for different MFs, where is the Einstein radius of a 1- star projected on the source plane. We use the microlensing code developed by Wambsganss (1999). Using the lens redshift , the source redshift =1.59 and H0=65 km s-1 Mpc-1 in a flat (=1) FRW universe, we find that =as. The magnification pattern is calculated on a grid of 10001000 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 (=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 , 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 (=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 (=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 (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 , 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.
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 - as a function of - are fitted by Eq. (32). The values for and 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 (-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 and additionally for 2.5 and 5.0- for image A. The results are listed in Table 2.
From these simulation, we notice several things: First, 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 (), 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:
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 (Sect. 4.4). Several examples of simulated light curves are shown in Fig. 7.
Table 3. All combinations of the MFs (Table 2) near image A and B that give a consistent solution of the parameters: f, . 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 1 for the range of MFs that we investigated. If we furthermore put a conservative upper limit of 0.5 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 ,fand : Using the MFs assumed viable above, we find from Table 3: 0.05f0.11 and 1.14.0. Using Eq. (25) and the values of f and listed in Table 3, the jet-component size then lies between 25 µas.
Constraints on : 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 () 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 () 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 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 926. 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 and : Using the constraints of the Doppler factor () and angular velocity of the jet-component (), 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 (). From Fig. 9, we subsequently find: and 0.995, for the allowed ranges of these parameters.
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 1 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.
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 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 is found, assuming that the average bulge/disk stellar mass in the halo is 0.1 . If the bulge/disk stars have average masses somewhere between 0.1 and 0.3 and are foremost in binaries, the lower limit increases to 2.5 .
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 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=0.5 and a flat FRW universe with =1 and H0=65 km s-1 Mpc-1.
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000