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Astron. Astrophys. 336, 123-129 (1998)

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2. Gamma-ray production in the mirror model

According to Ghisellini & Madau model, a blob containing relativistic particles moves along the jet with the Lorentz factor [FORMULA] and velocity normalized to the speed of light [FORMULA]. The synchrotron radiation produced by electrons in the blob illuminates the BLR cloud(s) (the mirror), located at a distance l from the center of active galaxy. This radiation photoionizes the cloud(s) which re-emit isotropically broad line emission. The relativistic blob approaching the mirror will see its radiation significantly enhanced because of decreasing blob-mirror distance and relativistic effects. In order to understand the features of this model we start this analysis from a simple picture, i.e. a single blob with negligible longitudinal extent along the jet (in respect to l) and a one dimensional mirror located on the jet axis. Since this picture is not successful in explanation of the features of [FORMULA]-ray flare in 3C 279, we discuss more realistic case in which extended blob scatters radiation reflected by the two dimensional mirror.

2.1. A single thin blob and a one dimensional mirror

Let 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 l. The photons, re-emitted by the mirror (marked by 2), meet the blob again at a place marked by (B) which is at the distance s from location of the mirror. In (B) blob starts to produce [FORMULA]-rays (marked by 3). The production of [FORMULA]-rays stops when the blob passes through the mirror. The path, s, on which [FORMULA]-rays are produced by the blob, is given by

[EQUATION]

where [FORMULA] is the observed rise time of the [FORMULA]-ray flare, and c is the velocity of light. First [FORMULA]-rays are produced at the distance q from the base of the jet, which is equal to [FORMULA] and given by

[EQUATION]

[FIGURE] Fig. 1a and b. Schematic representation (not to scale) of the mirror model with a simple geometry, i.e. a single thin blob and a one dimensional mirror on the jet axis (a ), and with a realistic geometry, i.e. an extended blob and a two dimensional plane mirror (b ). The mirror (M) is located at the distance l from the base of the jet (A). We assume that first synchrotron photons (marked by 1) are produced by the blob in (A). First [FORMULA]-rays (marked by 3) are produced by ICS of soft photons (marked by 2) re-emitted by the mirror, at the distance s from the mirror (B) and the distance q from the base of the jet. In Fig. 1b the synchrotron photons illuminate the mirror at different locations (D) which are at a distance h from the jet axis. Different critical locations of the blob are marked by: (I) - the blob is at the base of the jet; (II) - the blob starts to produce first [FORMULA]-ray photons; (III) - the back of the blob crosses the mirror located at (C). The blob moves with velocity [FORMULA] towards the mirror and has longitudinal extent [FORMULA] in case b and negligible longitudinal extent in case a

The time lag, [FORMULA], between the beginning of synchrotron flare, which ionizes the cloud(s), and the beginning of [FORMULA]-ray flare is

[EQUATION]

Eqs. (2) and  (3) show that for the case of [FORMULA]-ray flares observed from 3C 279 (Kniffen et al. 1993, Wehrle et al. 1997) which has the rising time of a few days ([FORMULA] days in February 1996) and, the blob moving with typical Lorentz factor of the order of [FORMULA] ([FORMULA] for 3C 279, Wehrle et al. 1997), the distance from the base of the jet to the place of [FORMULA]-ray production should be of the order of a few hundred pc ([FORMULA] pc for 3C 279, see Eq. (2)). The corresponding time delay between synchrotron and [FORMULA]-ray flare should be of the order of a few years ([FORMULA] years for 3C 279). The distance q is about three orders of magnitudes larger than the typical dimension of the BLR (see GM). Therefore the rise time of [FORMULA]-ray flare observed in 3C 279 cannot be explained as a result of relativistic effects connected with the time of flight of a single thin blob in the radiation reflected by the BLR clouds. However a few day time scale of the flare might be connected with the longitudinal extension of the blob. Small inhomogeneities in such extended blob can be responsible for a short time scale variability of [FORMULA]-ray emission (flickering) as a result of kinematic effects discussed in this subsection. In terms of the mirror model we can estimate the flickering time of the [FORMULA]-ray emission during the rising time of the flare in the case of 3C 279 by reversing Eq. (2). Assuming [FORMULA] cm and [FORMULA], we estimate the time scale for the shortest possible flux variability caused by these effects as equal to [FORMULA] min.

2.2. An extended blob and a two dimensional mirror

Let us assume that the blob has longitudinal extent along the jet [FORMULA] (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 l from the place of first injection of electrons (assumed at the base of the jet). The mirror is two-dimensional with negligible thickness and extends in perpendicular direction in respect to the jet axis. Note that Ghisellini & Madau considered the spherical mirror. However our assumption on the plane mirror simplifies the formulas derived below, because of simpler geometrical relations (rectangle triangles) and does not introduce any additional artifact features since only the part of the mirror located close to the jet axis is important. For simplicity we assume that the observer is located on the jet axis. The synchrotron photons (marked by 1, in Fig. 1a and b b), which are produced by electrons at the place marked by (A), illuminate the mirror (M) at any place (D). The photons reprocessed by the mirror (marked by 2) meet at the first time the blob at the distance s from the mirror,

[EQUATION]

At this place first [FORMULA]-rays (marked by 3) are produced by the blob and the [FORMULA]-ray flare begins to develop. The [FORMULA]-ray emission increases very fast up to the moment when the front of the blob meets the mirror. This happens at the time

[EQUATION]

measured from the beginning of the [FORMULA]-ray flare. The flare finishes at the time

[EQUATION]

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 [FORMULA] and the distance l of the mirror from the base of the jet. For relativistic blob ([FORMULA]) and the [FORMULA]-ray flares occurring on a time scale of days (as observed in blazars) the dependence of [FORMULA] on l is not important (see Eqs. (5) and (6)). The full [FORMULA]-ray flare is then produced on a distance, [FORMULA], measured from the mirror, which is given by

[EQUATION]

Eq. (6) shows that duration of the [FORMULA]-ray flare can be consistent with the mirror model for reasonable dimensions of the blob. However the question arises if the observed [FORMULA]-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 blob

In order to determine the evolution of [FORMULA]-ray power emitted in time t by the blob, the following formula has to be integrated

[EQUATION]

[EQUATION]

where [FORMULA], and [FORMULA] is the angle CBD defined in Fig. 1b. The first integral has to be performed over distances of [FORMULA]-ray emission region (part of the blob) from the mirror x. The second integral is over different paths (defined by the cosine angle µ) which has to be passed by synchrotron photons and reprocessed photons in order to produce [FORMULA]-ray photon found in time t at the location of the mirror. This integral is equivalent to the integration over contributions from different scattering centers of the mirror defined by the height h (see Fig. 1a and b b). Therefore the limits of integration over µ can be changed to integration over h by using

[EQUATION]

The third integral has to be performed over the regions in the blob, r (measured from the front of the blob), emitting synchrotron photons which can produce reprocessed photons serving next as a target for relativistic electrons at x.

At a point defined by h, the mirror is illuminated by the synchrotron radiation from the blob with the flux (see GM)

[EQUATION]

where [FORMULA] is the distance between the regions of synchrotron emission and the scattering centers on the mirror, z is the distance of the place of production of synchrotron photons from the mirror, and [FORMULA]. The synchrotron luminosity in the blob frame can be expressed by

[EQUATION]

[FORMULA] describes the synchrotron power emitted by average relativistic electron, and [FORMULA] is the electron density in the blob as a function of r at the moment [FORMULA]. For distances d smaller than [FORMULA], we take in Eq. (10) [FORMULA], since the synchrotron luminosity cannot exceed the maximum possible value determined by the dimension of the blob [FORMULA].

The points on the mirror at the distance, h, re-emits a part a of incident flux [FORMULA] isotropically (GM),

[EQUATION]

The relativistic electrons with density [FORMULA], responsible for production of [FORMULA]-ray photons at the time t (Eq. 8), has to be counted at the moment [FORMULA] and at place in the blob measured from its front [FORMULA].

The limits of integration over distances of parts of the blob from the mirror x, which produce [FORMULA]-rays observed at time t at the location of the mirror, can be found from the analysis of propagation of the front and the back of the blob. The lower limit is

[EQUATION]

where [FORMULA], and the upper limit is

[EQUATION]

where [FORMULA].

Only photons reprocessed by the part of the mirror at a distance from the jet axis smaller than [FORMULA] can contribute to the [FORMULA]-ray production by the parts of the blob located at the distance x from the mirror. This maximum height [FORMULA] can be found by analysing the time of flights of photons and the blob (Fig 1a and bb) and depends on l and x. It has to fulfil the following equation,

[EQUATION]

which has the solution

[EQUATION]

where [FORMULA]. [FORMULA] takes the maximum possible value, [FORMULA], if the synchrotron photons, produced on the front of the blob at the distance l from the mirror, excite parts of the mirror which produce reprocessed photons serving next as a target for production of [FORMULA]-rays by electrons at the moment when the back of the blob crosses the location of the mirror. For this constraint, following condition has to be fulfilled

[EQUATION]

The above equation has the solution

[EQUATION]

For the relativistic blob ([FORMULA]) and [FORMULA],

[EQUATION]

Hence for the parameters of the [FORMULA]-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 [FORMULA]-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 x and h, we determine the part of the blob (its longitudinal extent r) which emits synchrotron photons. These photons initiate next the production of [FORMULA]-ray photons observed at the moment t. As before we analyse the time of flights of photons and the blob and obtain the lower limit on the longitudinal extent of the blob in Eq. (8)

[EQUATION]

where [FORMULA], and [FORMULA], and the upper limit

[EQUATION]

where [FORMULA]. Finally we find the distance z of the blob from the mirror at the moment of emission of synchrotron photons which initiate the production of [FORMULA]-ray photons at the time t. For given x, h and r, we determine z from the following condition

[EQUATION]

with [FORMULA]. The solution of this equation is

[EQUATION]

where [FORMULA].

Since we want to know the relative change of the [FORMULA]-ray flux with time, the computations of the [FORMULA]-ray light curves have been performed assuming that the parameters describing the reflection, and [FORMULA]-ray and synchrotron efficiencies of a single relativistic electron in the blob are [FORMULA]. In principle the values of [FORMULA] and [FORMULA] 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 [FORMULA]-ray light curve on the longitudinal distribution of electrons in the blob, [FORMULA]. In general [FORMULA] may depend on the blob geometry and electron density as a function of r. The results of computations of the [FORMULA]-ray light curves for a few different cases are shown in Fig. 2a and b a. The dashed curve in this figure shows the [FORMULA]-ray light curve in the case of cumulative distribution of electrons in the blob (integrated over perpendicular extent of the blob), [FORMULA], corresponding to the homogeneous, spherical blob with longitudinal extent [FORMULA] cm, moving with the Lorentz factor [FORMULA]. The mirror is located at the distance [FORMULA] cm from the base of the jet. The expected light curve in this case is almost symmetrical with the maximum corresponding to the center of the blob. The full curve shows the light curve for the homogeneous, cylindrical blob with other parameters of the model as in previous case. The dot-dashed curve shows the [FORMULA]-ray light curve produced by the blob with cylindrical geometry but with exponential decrease of density of relativistic electrons. We apply the following distribution

[EQUATION]

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 r, e.g. according to

[EQUATION]

We consider the cases with [FORMULA] and [FORMULA], for which the [FORMULA]-ray light curves are shown in Fig. 2a and b a by the dotted and long-dashed curves, respectively.

[FIGURE] Fig. 2a and b. Gamma-ray light curves produced by a blob with different geometries and distributions of relativistic electrons. It is assumed that the mirror is located at the distance [FORMULA] cm from the place where first synchrotron photons are produced by the blob (close to the base of the jet). The blob has longitudinal extent [FORMULA] cm and moves along the jet with the Lorentz factor [FORMULA]. a Different curves show the [FORMULA]-ray light curves in the case of: a spherical, homogeneous blob (dashed curve), a cylindrical, homogeneous blob (full curve), and inhomogeneous blobs for the distribution of electron densities given by Eq. (24) (dot-dashed curve), and Eq. (25) with [FORMULA] (dotted curve) and [FORMULA] (long-dashed curve). b The [FORMULA]-ray light curves are shown by the thick curves for a cylindrical, homogeneous blob in which the density of electrons depends on the distance x from the mirror according to: Eq. (26) (dotted curve), Eq. (27) (dashed curve) and [FORMULA] (full curve). The corresponding synchrotron flares are marked by the thin curves.

Fig. 2a and b b shows the [FORMULA]-ray and corresponding synchrotron light curves assuming that the density of electrons in the blob depends on the distance x from the mirror but is homogeneous inside the blob. These light curves are normalized to the flux at their maximum. The dotted curves corresponds to the continuous increase of density of relativistic electrons in the blob according to

[EQUATION]

and the dashed curves to the case when the electron density decreases according to

[EQUATION]

These [FORMULA]-ray light curves are very similar to the [FORMULA]-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 [FORMULA]-ray light curves are due to the fact that the blob reaches the mirror after very short time [FORMULA] measured from the beginning of the [FORMULA]-ray flare. For [FORMULA], the radiation field seen by relativistic electrons do not change significantly. Note that the production of [FORMULA]-ray photons in the blob occurs at small distance from the mirror (given by Eq. (7)) in comparison to the distance l of the mirror from the base of the jet. The beginning of the [FORMULA]-ray flare is delayed in respect to the synchrotron flare by [FORMULA]. For the parameters considered in Fig. 2b this delay is of the order of [FORMULA] day.

In all discussed above cases the [FORMULA]-ray flux increases initially on a very short time scale (given by Eq. (5)). For the parameters applied above this time is [FORMULA] min. The [FORMULA]-ray flare finishes at time [FORMULA] given by Eq. (6), which for these parameters is [FORMULA] days.

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© European Southern Observatory (ESO) 1998

Online publication: July 7, 1998
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