Astron. Astrophys. 323, 271-285 (1997)

3. Weak turbulence theory in the non relativistic regime

First, we consider the case where the pair plasma is not dense enough to be energetically dominant. The streaming of the relativistic pairs propelled by the radiation field triggers micro-instabilities by amplifying waves of the ambient plasma. The anisotropy due to the longitudinal (along the jet axis) magnetic field will favor the destabilization of Langmuir, and Alfvén waves with wave vectors having direction close to the magnetic field (and jet axis). For this reason, only this two kind of micro-instabilities will be studied in the following sections. We do not consider other kind of instabilities with much lower growth rates. The pair beam, close to the soft photon source, can be efficiently heated by strong Langmuir turbulence. Because the beam is hot in the sense it has high internal energy (and a mean Lorentz factor) , these instabilities have a broad band. Consequently, they excite broad waves spectra which promote the weak turbulence theory. Indeed, a broad band spectrum allows fast phase mixing in the resonant interaction between particles and waves.

In the case of Langmuir turbulence, interactions occur at Landau- erenkov resonance (), and the phase mixing time (correlation time ) is such that

where , is the plasma frequency, and is the ambient electron density.

In the case of Alfvén turbulence, interactions occur at Landau-synchrotron resonance (), and the corresponding correlation time is such that

where for a particle of Lorentz factor , and a mean magnetic field B, , is the plasma gyro-frequency.

The condition of a short correlation time compared to the time scale of the distribution evolution and to the growth time of the instability are the usual condition to apply the so-called quasi linear theory. However, we need to go beyond quasi linear theory in the situation we deal with, because the origin of the instability is maintained externally (by the anisotropy of the incident soft radiation field), so that a quasi linear evolution cannot remove it. Therefore we come to next order of the theory of weak turbulence to calculate the saturation spectrum. Furthermore the broad band character of the spectrum allows to validate the random phase approximation and to obtain a low level of turbulence, which justifies the perturbative expansion. In the Langmuir waves, the turbulent energy density W verifies

The quantity refers to the cold electron temperature.

In the Alfvén waves, the turbulent fluctuations verifies

The nonlinear effects couple unstable modes with damped modes which redistribute energy to the particles; however a part of the energy is carried away by an inertial cascade. Acceleration of particles comes from the damped modes (especially backward modes) that are continuously supplied by their coupling with ever-unstable modes. Thus Compton losses ensure a stationary state by balancing the stochastic acceleration of the particles. So the evolution of distribution function is governed by a Fokker-Planck equation that contains a drag term describing the Inverse Compton process and a diffusion tensor describing the effect of the turbulence on the particles, namely pitch angle scattering and stochastic acceleration. The Fokker-Planck description is valid as long as and are much shorter than the diffusion time; which is well satisfied for broad band spectra (see Eqs. (11) and (12)) and low turbulence level (Eq. (13)).

Despite the pitch angle scattering, the distribution will keep a strong anisotropy because of the Compton radiation field. We recall the important point that the hotter the pairs, the stronger the Compton boost (see HP). This is the key point to understand that strong boosting can exist in presence of intense IC emission since the latter is balanced by strong heating.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

helpdesk.link@springer.de