3. Discussion of the elementary processes
From Fig. 5 we can deduce some estimates about which of the various energy loss and exchange processes discussed above play an important role in different energy regimes of particles.
First, we note that the energy transfer rates due to collisional effects (elastic scattering, bremsstrahlung emission) and the pair annihilation rate scale linearly with particle density n. The synchrotron and SSC losses basically scale as (the SSC losses, additionally, ), whereas the external inverse-Compton losses basically scale as . The SSC losses also strongly depend on the steepness of the initial distribution functions since a harder particle spectrum yields a harder synchrotron spectrum which, in turn, leads to higher inverse-Compton losses.
The effect of elastic scattering is of minor importance for ultrarelativistic particles. We find the time-scale for thermalization to be given by
for which increases rapidly towards higher particle energies. Here, is the particle density in units of cm-3.
The distribution functions' high-energy tail will be most heavily influenced by one of the radiative energy-loss processes discussed above. Of course, the relative importance of the different radiation processes is strongly dependent on the exact values and variation of parameters, especially the magnetic field strength and the luminosity of the central photon source.
The time-scales for the respective processes can be estimated as
- where - for the synchrotron-self-Compton process,
for the synchrotron process,
where , and for the external inverse-Compton losses in the Thomson regime, and
In the beginning phase of the jet evolution, the effect of pair annihilation makes a negligible contribution to the distribution functions' evolution. The annihilation timescale for particles near the lower cutoff can be estimated as
which indicates that pair annihilation and bremsstrahlung are of minor importance to ultrarelativistic pair plasmas of density cm-3.
These considerations and the investigation of the resulting photon spectra lead us to some general simplifications in the treatment of the particle spectra evolution as long as the plasma is ultrarelativistic:
(a) Pair annihilation is negligible.
(b) Elastic (Moller and Bhabha scattering) and inelastic scattering (bremsstrahlung emission) are only important for relatively low-energetic particles () which do not exist in our model system during the phase we are dealing with in this paper.
(c) The effect of energy dispersion is neglected.
(d) Bremsstrahlung emission is negligible for pair densities cm-3.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998