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

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4. Radio microlensing: theory

Microlensing is unlikely for bright ([FORMULA]1 Jy) flat-spectrum radio sources at low frequencies ([FORMULA]few GHz), which have typical angular sizes of [FORMULA]1 mas, determined by the inverse Compton limit on their surface brightness (e.g. Kellerman & Pauliny-Toth 1969). This angular size is much larger than the typical separation of a few µas between caustics in the magnification pattern, thereby reducing any significant microlensing variability (see Sect. 5).

High-frequency ([FORMULA]10 GHz) sources with flux densities less than a few tens of mJy, however, can be as small as several tens of µas. If part of the source is moving with relativistic velocities ([FORMULA]), Doppler boosting allows an even smaller angular size. In those cases, microlensing can start to contribute significantly to the short-term variability seen in these sources (e.g. Gopal-Krishna & Subramanian 1991).

This is in stark contrast to optical microlensing, where the variability time scales are dominated by the transverse velocity ([FORMULA]) of the galaxy with respect to the line-of-sight to the static source, as in the case of Q2237+0305 (e.g. Wyithe, Webster & Turner 1999). Microlensing time scales between strong caustic crossings in the optical waveband are therefore several orders of magnitude (i.e. [FORMULA][FORMULA][FORMULA]) larger than in the radio waveband. This makes superluminal radio sources the perfect probes to study microlensing by compact objects in strong gravitational lens galaxies, using relatively short ([FORMULA]1 year) monitoring campaigns.

4.1. Relativistic jet-components

If the lensed source consists of a static core and a synchrotron self-absorbed jet-component, which moves away from the core with a velocity [FORMULA]=[FORMULA], the Doppler factor of this jet-component is given by


where [FORMULA]=[FORMULA] and [FORMULA] is the direction of the observer (e.g. Blandford & Königl 1979).

The observed flux density of a circular-symmetric radio source with an observed angular radius [FORMULA] is


where [FORMULA] is the observing frequency and [FORMULA] the observed brightness temperature of the source. We assume the source has a constant surface brightness. However, due to the Doppler boosting, the apparent brightness temperature of a relativistic jet-component moving towards the observer can be significantly brighter than the inverse Compton limit of about [FORMULA][FORMULA][FORMULA] K (e.g. Kellerman & Pauliny-Toth 1969). The true comoving surface brightness temperature of a flat-spectrum radio source is related to the observed surface brightness temperature ([FORMULA]) through


where z is the redshift of the source (e.g. Blandford & Königl 1979). If we substitute Eq. (22) into Eq. (21), we find that the flux density of the relativistically moving jet-component is


with [FORMULA]=10[FORMULA][FORMULA]. Inverting this equation, we find an approximate angular radius of the jet-component


Given the observed frequency, the redshift of the jet-component and the Doppler boosting [FORMULA], we can subsequently set a limit on the angular radius of the jet-component.

In the case of B1600+434, the redshift of the lensed quasar is z=1.59 (Fassnacht & Cohen 1998). The observing frequency is [FORMULA]=0.85 and [FORMULA][FORMULA][FORMULA], where f is the fraction of the total average source flux density in the relativistic jet-component, [FORMULA][FORMULA][FORMULA] mJy (KBXF00) and [FORMULA] is the average magnification at the image position. From the singular isothermal ellipsoidal (SIE) mass model (Kormann, Schneider & Bartelmann 1994), we find [FORMULA][FORMULA]1.7 ([FORMULA]=0.2) and [FORMULA][FORMULA]1.3 ([FORMULA]=0.9). We use [FORMULA][FORMULA]1.5 as a typical value.

After inserting all the known observables into Eq. (24) and adopting [FORMULA]=0.5, we find an approximate relation between the fraction of the total observed flux-density of B1600+434 contained in the jet-component and its angular size in the source plane


This equation can be used to put constraints on the light-curve fluctuations seen in B1600+434, and decide whether they are the result of microlensing of a single relativistic jet-component.

4.2. Microlensing time scales

The typical time scale which one would expect between relatively strong microlensing events is the angular separation between strong caustic crossings divided by the angular velocity of the jet-component in the source plane.

In case the source is not lensed, the apparent velocity (in units of c) of the jet-component is


where [FORMULA] is the angle between the jet and the line-of-sight and [FORMULA] the bulk velocity of the jet-component (e.g. Blandford & Königl 1979). The apparent angular velocity (in vector notation) of the jet-component becomes


where [FORMULA] and [FORMULA] are the redshift and the angular diameter distance to the stationary core, respectively. If the jet-component moves with superluminal velocities ([FORMULA]), one expects angular velocities in the order of several tenths of [FORMULA] per week. Because the source is lensed by the foreground galaxy, its observed angular velocity (in the lens plane) becomes




is the local transformation matrix of the source plane to the lens plane, with [FORMULA] and [FORMULA] being the local convergence and shear (e.g. Gopal-Krishna & Subramanian 1991; Schneider et al. 1992).

We calculate the source and caustic structure in the source plane, however. Thus, the angular velocity and source structure undergo the inverse transformation of the angular velocity in the source plane (Eq. (28)). The angular velocity that we need to use, is therefore given by Eq. (27). Using the observed redshift z=1.59 (Fassnacht & Cohen 1998) of B1600+434 the angular velocity in the source plane reduces to


where we assume a flat Friedmann-Robertson-Walker universe with [FORMULA]=1 and H0=65 km s-1 Mpc-1.

4.3. Microlensing modulation-indices

The normalized modulation-index ([FORMULA]) of a superluminal jet component, due to microlensing, is


where µ=[FORMULA] is the microlensing light curve of the jet component. We determine [FORMULA] as function of the angular size of the source by averaging over randomly-oriented simulated microlensing light curves and find that [FORMULA] can be well approximated by the analytical function


for [FORMULA][FORMULA]20µas, where [FORMULA] is the turnover size of the source after which the modulation-index [FORMULA] decreases linearly with source size and [FORMULA] is the asymptotic modulation-index for a source with [FORMULA][FORMULA]0. We fit this function to the numerical results to obtain both [FORMULA] and [FORMULA].

We subsequently combine Eqs. (25) and (32) with the fact that the observed modulation-index in the lensed images is [FORMULA] and find


where [FORMULA] is in units of µas.

Many jets consist of more than a single jet-component. If we assume that (i) the jet consists of N similar jet-components, each containing a fraction [FORMULA]=[FORMULA] of the total flux-density and (ii) the magnification curves of the jet-components to be uncorrelated, we expect the modulation-index of the combined jet-components to decrease roughly as [FORMULA][FORMULA]. Hence, we find that [FORMULA][FORMULA][FORMULA], or


where f=[FORMULA][FORMULA]1. Hence, multiple jet-components will in general decrease the modulation-index. However, if the individual jet-components are very compact - i.e. are much smaller than the typical separation between strong caustics -, the microlensing variability will be dominated by single caustic crossings, creating strong isolated peaks in the light curve. We then expect Eq. (34) to break down, such that the factor [FORMULA] in the numerator can be removed.

In the case of scintillation, N compact jet-components ([FORMULA][FORMULA][FORMULA]) always vary independently, such that the observed modulation-index roughly decreases as [FORMULA]. However, in the case of microlensing, the modulation-index, caused by the same compact jet-components moving over a magnification pattern, can be independent from N or even increase as [FORMULA].

4.4. Source constraints

At this point, we give a qualitative recipe to obtain constraints on the jet-component parameters from the observed light curves.

  1. First, the observed modulation-indices of both lensed images can be used to solve for the fraction of flux density in the jet-component (f), as well as its Doppler factor ([FORMULA]), by comparing them to those found from numerical simulations for different mass functions (MFs), which fix both [FORMULA] and [FORMULA]). One obtains a set of two equations (i.e. Eq. (33)) with two constraints ([FORMULA] and [FORMULA]) and two unknown variables ([FORMULA] and f), which in some cases can be uniquely solved for. The combinations of MFs, which do not give a consistent solution, can be excluded, thereby putting constraints on the allowed MFs in the line-of-sight towards the lensed images.

  2. Second, combining the typical observed variability time scale between strong microlensing events and the angular separation of these from the numerical simulations, one can obtain a value for [FORMULA] (Eq. (27)). Combining Eqs. (20) and (26), one then also readily solves for both [FORMULA] and [FORMULA].

Thus, given (i) a mass model of the lens galaxy found from (macroscopic) lens modeling, (ii) some plausible range of MFs near the lensed images and (iii) the observed modulation-indices and variability time scales in the light curves of the lensed images, one can in principle solve for several parameters of the simple jet/jet-component structure: [FORMULA], f, [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. However, one should keep in mind that some of the parameters might be degenerate and that we also do not know intrinsic brightness temperature of the components.

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Online publication: June 20, 2000