## 2. Gamma-ray production in the mirror modelAccording to Ghisellini & Madau model, a blob containing
relativistic particles moves along the jet with the Lorentz factor
and velocity normalized to the speed of light
. The synchrotron radiation produced by electrons
in the blob illuminates the BLR cloud(s) (the mirror), located at a
distance ## 2.1. A single thin blob and a one dimensional mirrorLet us discuss the simplest possible case in which the blob, the
mirror, and the observer are located on the jet axis (see Fig. 1a
and b a). We assume that the blob has negligible dimensions in respect
to the other dimensions of considered system. First synchrotron
photons are emitted by the blob at the distance marked by (A) which
has been chosen as located at the base of the jet. These photons
(marked by 1) excite the mirror (marked by (M)), which is at a
distance where is the observed rise time of the
-ray flare, and
The time lag, , between the beginning of synchrotron flare, which ionizes the cloud(s), and the beginning of -ray flare is Eqs. (2) and (3) show that for the case of
-ray flares observed from 3C 279 (Kniffen et al.
1993, Wehrle et al. 1997) which has the rising time of a few days
( days in February 1996) and, the blob moving
with typical Lorentz factor of the order of
( for 3C 279, Wehrle et al. 1997), the distance
from the base of the jet to the place of -ray
production should be of the order of a few hundred pc
( pc for 3C 279, see Eq. (2)). The
corresponding time delay between synchrotron and
-ray flare should be of the order of a few years
( years for 3C 279). The distance ## 2.2. An extended blob and a two dimensional mirrorLet us assume that the blob has longitudinal extent along the jet
(see Fig. 1a and b b), and negligible
perpendicular extent. Its perpendicular dimension do not introduce
interesting effects if the observer is located at small angles to the
jet axis. As in the picture considered above, electrons produce
synchrotron radiation which is reflected by the mirror located at a
distance At this place first -rays (marked by 3) are produced by the blob and the -ray flare begins to develop. The -ray emission increases very fast up to the moment when the front of the blob meets the mirror. This happens at the time measured from the beginning of the -ray flare. The flare finishes at the time when the back of the extended blob crosses the place of location of
the mirror. This equation simply relates the expected time scale of
the flare to the length of the extended blob and
the distance Eq. (6) shows that duration of the -ray flare can be consistent with the mirror model for reasonable dimensions of the blob. However the question arises if the observed -ray light curves of flares in blazars can be explained in such a model. Below we analyse this problem assuming different geometries of the blob with different density distributions of relativistic electrons. ## 2.2.1. Gamma-ray light curve produced by extended blobIn order to determine the evolution of -ray
power emitted in time where , and is the
angle CBD defined in Fig. 1b. The first integral has to be
performed over distances of -ray emission region
(part of the blob) from the mirror The third integral has to be performed over the regions in the
blob, At a point defined by where is the distance between the regions of
synchrotron emission and the scattering centers on the mirror,
describes the synchrotron power emitted by
average relativistic electron, and is the
electron density in the blob as a function of The points on the mirror at the distance, The relativistic electrons with density ,
responsible for production of -ray photons at the
time The limits of integration over distances of parts of the blob from
the mirror where , and the upper limit is where . Only photons reprocessed by the part of the mirror at a distance
from the jet axis smaller than can contribute
to the -ray production by the parts of the blob
located at the distance where . takes the
maximum possible value, , if the synchrotron
photons, produced on the front of the blob at the distance The above equation has the solution For the relativistic blob () and , Hence for the parameters of the -ray flares observed in 3C 279, only parts of the mirror close to the jet axis (laying mainly inside the jet) can re-emit soft photons which serve as a target for production of -rays. Therefore the limits of integrations in Eq. (21) of the paper by Ghisellini & Madau (GM) are not correct because they do not take into account the dynamics of the blob. From this reason, the energy densities of photons re-emitted by the mirror, but observed in the blob frame, are time independent and overestimated in that paper. For given where , and , and the upper limit where . Finally we find the distance with . The solution of this equation is where . Since we want to know the relative change of the
-ray flux with time, the computations of the
-ray light curves have been performed assuming
that the parameters describing the reflection, and
-ray and synchrotron efficiencies of a single
relativistic electron in the blob are . In
principle the values of and
may depend on the blob propagation, e.g. if the
spectrum of electrons in the blob depends on its propagation along the
jet. We do not consider such cases in order not to complicate the
model too much. First we investigate the dependence of the
-ray light curve on the longitudinal distribution
of electrons in the blob, . In general
may depend on the blob geometry and electron
density as a function of which might correspond to the distribution of electrons, produced
by the relativistic plain shock, with the maximum on the front of the
cylindrical blob and exponentially decreasing tail towards the end of
the blob. In contrary, the density of electrons could increase
exponentially with We consider the cases with and , for which the -ray light curves are shown in Fig. 2a and b a by the dotted and long-dashed curves, respectively.
Fig. 2a and b b shows the -ray and
corresponding synchrotron light curves assuming that the density of
electrons in the blob depends on the distance and the dashed curves to the case when the electron density decreases according to These -ray light curves are very similar to
the -ray light curve (full curve) obtained in the
case with constant electron density in the blob during its propagation
in the jet. Small differences between these -ray
light curves are due to the fact that the blob reaches the mirror
after very short time measured from the
beginning of the -ray flare. For
, the radiation field seen by relativistic
electrons do not change significantly. Note that the production of
-ray photons in the blob occurs at small distance
from the mirror (given by Eq. (7)) in comparison to the distance
In all discussed above cases the -ray flux increases initially on a very short time scale (given by Eq. (5)). For the parameters applied above this time is min. The -ray flare finishes at time given by Eq. (6), which for these parameters is days. © European Southern Observatory (ESO) 1998 Online publication: July 7, 1998 |