4. Synchrotron emission of the jet
4.1. Energy losses of the relativistic electrons
We assume that the shock passing through the jet material accelerates a population of relativistic electrons and/or positrons. During the acceleration process and afterwards these relativistic particles are subject to energy losses due to the approximately adiabatic expansion of the jet material and synchrotron radiation. To determine the exact form of the energy spectrum of the relativistic particles in a given jet region the kinetic equation including acceleration and energy terms must be solved. Heavens & Meisenheimer (1987) present analytic and numerical solutions for some simplified cases. They find that the energy spectrum follows a power law with a high energy cut-off. The cut-off occurs at the energy for which energy gains due to shock acceleration balance the synchrotron energy losses (e.g. Drury 1983). The cut-off becomes steeper further downstream from the shock. For simplicity we assume that the relativistic particles in a given jet region are initially accelerated at a time during a short time interval to a power law spectrum with a sharp high energy cut-off at . In terms of the number of particles this can be expressed by
Here is the rate at which relativistic particles are accelerated in the jet by the shock at time . The normalisation of this energy spectrum and the position of the high energy cut-off depend on the local conditions for diffusion in the jet (e.g. Drury 1983). These are not straightforward to estimate and we therefore assume for simplicity that the initial high energy cut-off of the relativistic particles freshly accelerated at time is independent of . In the internal shock model for GRB the shock is caused by the collision of shells of jet material moving at different velocities. For GRB it is implicitly assumed that the collision energy is dissipated very close to instantaneously. In the case of microquasars the propagation of the shock is resolved in time. For the normalisation of Eq. (5) we therefore assume
where is the position of the shock at time and a model parameter. This implies that the rate at which the collisional energy is dissipated and partly conferred to the relativistic particles is almost constant for and decreases exponentially at larger distances. The onset of the exponential behaviour then signifies the point at which almost all of the collisional energy has been dissipated and the shock starts to weaken significantly. This would coincide with the time at which the two colliding shells have practically merged into one. Alternatively, in only intermittently active sources the exponential decrease may be caused by the shock, and therefore the fast shell causing the shock, reaching the end of the jet.
After the passage of the shock the relativistic particles continue to loose energy. The rate of change of the Lorentz factor of these particles due to the nearly adiabatic expansion of the jet is given by (e.g. Longair 1981)
where is the volume of the jet region the particles are located in. We assume that the bulk velocity of the shocked jet material is constant and this implies that the jet is only expanding perpendicular to the jet axis, i.e. . Changing variables from to then yields
where is the Thompson cross section and the rest mass of an electron. By summing Eqs. (8) and (9) and integrating we find the Lorentz factor at time of those electrons which had a Lorentz factor at time (see also Kaiser et al. 1997)
Note here that the high energy cut-off, , also evolves according to Eq. (10).
4.2. Synchrotron emission
where is the synchrotron emission spectrum of a single electron with Lorentz factor . Here we assume that the magnetic field in the jet is tangled on scales smaller than the radius of the jet. This then implies that , where is one of the synchrotron integrals normalised to give (e.g. Shu 1991). To get the total emission of the jet behind the jet shock we have to sum the contributions of all the regions labeled with their shock times within the jet. From Eq. (13) we see that this implies integrating Eq. (15) over . This integration must be performed numerically. Finally, to compare the model results with the observations we have to transform to the rest frame of the observer, . Note here that the luminosity inferred from the observations at frequency was emitted in the gas rest frame at a frequency .
As mentioned above, the steepening of the radio spectrum of the superluminal jet components observed during outbursts of microquasars is caused by the integrated emission from an extended region of the jet. The further away from the shock the emission is created in the jet, the lower the cut-off in the energy spectrum of the relativistic electrons will be. However, if the jet region contributing to the total emission is not too large the properties of the energy spectrum will not change dramatically within this region. In this case we may estimate the approximate location of the break in the radio spectrum beyond which the spectrum steepens significantly. Most of the energy lost by relativistic particles is radiated at their critical frequency, , where is the Larmor frequency. Defining the break frequency, , as the critical frequency of the most energetic particles just behind the jet shock we get
where q is the elementary charge and is the magnetic permeability of the vacuum. The steepening of the radio spectrum of the observed outbursts of microquasars strongly suggest that the observing frequency, , is close to the break frequency, i.e. . Therefore, if most of the observed emission comes from the region just behind the shock, we expect from Eq. (16) that . This implies that and are not independent parameters of the model but that they are correlated.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000