Astron. Astrophys. 356, 975-988 (2000)

## 3. Dynamics of the jet

We follow Marscher & Gear (1985) and Ghisellini et al. (1985) in the assumption that all relevant physical quantities of the jet and the jet material are simple power law functions of the unprojected distance from the source centre, R. The shock traveling along the jet will compress the jet material but we will assume that it does not change the behaviour of the physical quantities as a function of the distance from the source centre. The radius of the cross section of the jet is assumed to follow . For a freely expanding, conical jet . The energy density of the magnetic field as measured in the rest frame of the jet material is then given by , where energy flux conservation would require . Here and in the following dashes denote quantities measured in the rest frame of the jet material moving at relativistic speeds while quantities measured in the frame of the observer are undashed.

The two frames of reference, the rest frame of the observer and that of the shocked jet material, are defined such that the origins of both coincide when the radio outbursts starts, i.e. when the shocks are formed in the centre of the source and start traveling outwards. Consider a section of the jet after the shock has passed through it. At time t this section is located at

in the rest frame of the observer. Here, is the angle of the jet to the line of sight and R is the unprojected distance of the jet section from the centre of the source, i.e. the origin of the observer's rest frame. is the deprojected velocity of the shock as measured in the rest frame of the observer and is the deprojected velocity of the shocked jet material in this frame in units of the speed of light. The expression in brackets in Eq. (1) takes account of the Doppler shifted time measurements taken in the observer's frame caused by the receding (+) or approaching (-) component of the motion of the jet material (e.g. Rybicki & Lightman 1979). Transforming t and R to the frame comoving with the jet material we find

where is the Lorentz factor corresponding to the velocity of the shocked jet material, . For the origin of the comoving rest frame, , we recover from Eqs. (2) and (3) the well-known result with the usual relativistic Doppler factor . Since the shock moves along the jet, most of the observed emission is not produced at the origin of the comoving frame but at . Suppose the section of the jet introduced above was passed by the shock at time as measured in the frame comoving with the shocked gas. Since this section is at rest in this frame, it is subsequently located at , where is the velocity of the shock as measured in the shocked gas' frame. Using this and substituting R from Eq. (3) in Eq. (2) yields

This expression illustrates the fact that the emission we observe at a given time t from different parts of the jet was not produced simultaneously in the frame of the shocked gas.

We assume that only those parts of the jet contribute to the total emission which have been passed by the shock. The emission regions within the jet can therefore be labeled with their `shock time', , and are observed at the intrinsic time given by Eq. (4). Since , this implies . For convenience we introduce the ratio , where we have used Eqs. (2) and (3).

© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000