## 4. Radio microlensing: theoryMicrolensing is unlikely for bright
(1 Jy) flat-spectrum radio sources at
low frequencies (few GHz), which
have typical angular sizes of 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 High-frequency (10 GHz) sources
with flux densities less than a few tens of mJy, however, can be as
small as several tens of This is in stark contrast to optical microlensing, where the variability time scales are dominated by the transverse velocity () 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. ) 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 (1 year) monitoring campaigns. ## 4.1. Relativistic jet-componentsIf the lensed source consists of a static core and a synchrotron self-absorbed jet-component, which moves away from the core with a velocity =, the Doppler factor of this jet-component is given by where = and 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 is where is the observing frequency and 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 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 () through where with =10. 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 , 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
After inserting all the known observables into Eq. (24) and adopting =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 scalesThe 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 where is the angle between the jet and the line-of-sight and 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 and are the redshift and the angular diameter distance to the stationary core, respectively. If the jet-component moves with superluminal velocities (), one expects angular velocities in the order of several tenths of per week. Because the source is lensed by the foreground galaxy, its observed angular velocity (in the lens plane) becomes where is the local transformation matrix of the source plane to the lens plane, with and 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 where we assume a flat Friedmann-Robertson-Walker universe with
=1 and
H ## 4.3. Microlensing modulation-indicesThe normalized modulation-index () of a superluminal jet component, due to microlensing, is where for
20 We subsequently combine Eqs. (25) and (32) with the fact that the observed modulation-index in the lensed images is and find where is in units of
Many jets consist of more than a single jet-component. If we assume
that (i) the jet consists of where
In the case of scintillation, ## 4.4. Source constraintsAt this point, we give a qualitative recipe to obtain constraints on the jet-component parameters from the observed light curves. -
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 (), by comparing them to those found from numerical simulations for different mass functions (MFs), which fix both and ). One obtains a set of two equations (i.e. Eq. (33)) with two constraints ( and ) and two unknown variables ( 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. -
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 (Eq. (27)). Combining Eqs. (20) and (26), one then also readily solves for both and .
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: ,
© European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |