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

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1. Introduction

Since its launch in 1991, CGRO (Compton Gamma-Ray Observatory) has observed numerous extragalactic radio sources above 100 MeV (see von Montigny et al. (1995)). Among the sources detected more than 40 are classified as radio-loud objects (BL Lac objects and flat radio spectrum Quasars) all with the same characteristics: intense high energy emission, strong variability in all wavelengths, and superluminal motions. More recently, galactic sources exhibiting multi-wavelength Quasar spectra, superluminal motions at small scales, and called for these reasons "micro-Quasars", have been detected by SIGMA (Finoguenov et al. (1994), Gilvanov et al. (1994)) and CGRO (Harmon et al. (1995)).

To explain the strong and highly variable high energy emission of these objects many authors have considered a well-collimated relativistic jet in bulk motion characterized by Lorentz factors [FORMULA] between 2 and 20. In this hypothesis, the high energy emission is beamed in the forward direction into a cone of opening angle [FORMULA]. Such approach removes the important problem of the gamma-ray absorption by pair creation in a static and homogeneous source (see Maraschi et al. (1992) for discussions).

The origin of the high energy emission is mainly considered in two different ways. A first class of models involve a cascade of gamma-rays due to meson production by energetic protons (with an energy [FORMULA]) as proposed by Mannheim & Bierman (1992), or associated with energetic neutrons (Mastichiadis & Protheroe 1990). These models are referred to be hadronic as opposed to the leptonic models in which gamma-rays are mostly produced by Inverse Compton (IC) scattering of low energy photons by relativistic electrons. The soft photons are coming either from synchrotron radiation (Synchrotron self-Compton process), see for example Ghisellini et al. (1991), or from an accretion disk. In the last case the disk radiation can either be reprocessed by surrounding clouds as proposed by Sikora et al. (1994), Blandford & Levinson (1995), and Levinson & Blandford (1996) for "micro-Quasars" objects, or interact directly with the relativistic particles (Dermer & Schlickeiser (1993)).

This is also our viewpoint (Henri & Pelletier (1991) thereafter: HP), and Henri, Pelletier & Roland (1993)). We have developed a model treating the high energy emission of radio-loud sources (Marcowith, Henri & Pelletier (1995); MHP thereafter). The aim of the MHP paper was to reproduce the global feature of radio-loud Quasar spectra, and particularly the spectral break observed in the MeV region by COMPTEL. For that purpose we first compute the anisotropic IC emitted power by an ultra-relativistic electron and the resulting anisotropic IC spectrum by a population of relativistic particles. With these tools, we were able to compute a self-consistent formation of the relativistic pair plasma, including both pair creation and annihilation. Note that the anisotropy of the high energy emission is a consequence of both disk radiation and particle distribution anisotropy in the disk frame. In this picture the spectral break is a consequence of a differential energy absorption of gamma-rays along the jet due to pair production. Such multiple emission zone model first proposed by Pelletier, Henri & Roland (1992), and Blandford (1993) allow us to obtain a good fit over ten decades of energy of the two best quasi-simultaneous observations of Radio-loud Quasar and Blazar: 3C273 (Lichti et al. (1995)) and 3C279 (Hartmann et al. (1992), Collmar et al. (1992)). In this mostly hydro-dynamical model, we focussed on the treatment of the radiation transfer along the jet, assuming a power law relativistic pair distribution continuously re-heated to balance IC losses. But a kinetic theory is necessary to constrain the acceleration mechanisms at the origin of the particle distribution. In a first approach HP have proposed the formation of a relativistic beam by pair cascade on the photons coming from an accretion disk, and argue that a turbulent heating by Alfvén waves could easily explain final bulk Lorentz factors of order of 10 observed in VLBI. In this model, all the turbulence is supported by a sub-relativistic electron-proton jet launched by an accretion disk. The relativistic beam confined by the magnetohydrodynamical accretion ejection flow can then be re-accelerated up to distances where the Compton drag becomes inefficient. This two flow configuration first proposed by Pelletier (1985) is presented in more details in Sol et al. (1989).

Other detailed theories of the particle heating mechanism in the inner region of compact objects have only been recently considered. Mastichiadis & Kirk (1995) have developed a diffusive shock acceleration model for a relativistic proton population in the accreting motion in the vicinity of a massive black hole. Dermer et al. (1996) have also considered a stochastic acceleration process, where both proton and electron populations are heated by plasma turbulence generated in black hole magnetospheres.

Accounting for all these astrophysical problems, the present paper tackles in a formal way the kinetic theory of the acceleration mechanism of a relativistic pair plasma coupled with a sub-relativistic MHD plasma and moving with a relativistic bulk speed [FORMULA] in an intense soft photon radiation field. The pair plasma is submitted to strong anisotropic Inverse Compton losses in the soft photon source frame, and in turn produces X and gamma-rays.

Two distinct regimes may appear depending on whether the ambient plasma is the more massive, and supports the turbulence or not. In the first stage, the Compton radiation field accelerates a tenuous beam of electron-positron that pervades the cold electron-proton ambient plasma. Here we understand "cold' as non relativistic temperature ([FORMULA]); so we call this stage "non relativistic" regime. A second stage takes place when the electron-positron beam has been so heated and densified by pair creation that it becomes more massive than the ambient one. It is therefore convenient to analyze the micro-turbulence in its rest frame, the ambient medium being considered as a perturbing back-stream. So we called this second stage the "relativistic" regime. Each regime has a specific kinetic theory that provides a self consistent calculation of the internal energy and bulk Lorentz factors.

This general picture can be applied to astrophysical objects emitting high energy radiations (X and gamma-ray): active galactic nuclei, galactic microquasars, gamma-ray bursters, and perhaps X-ray binaries. The internal energy and the bulk Lorentz factor of the pair beam can be derived in term of the soft photon source compactness.

This paper is organized as follows. The Sect. 2 recalls the main effects of the anisotropic Compton emission from the relativistic particles. Sections 3 to 6 present the non relativistic regime. In Sect. 3 we explain why the theory of weak turbulence applies, why we need a non-linear theory to calculate the turbulent spectrum, and how the spectrum is used to calculate the coefficients of the Fokker-Planck equation that governs the evolution of the pair distribution. The Sect. 4 is devoted to the contribution of Langmuir turbulence, whereas Sect. 5 is devoted to the contribution of the Alfvén turbulence: linear growth rates, and main non-linear mode couplings. The stationary solutions of a Fokker-Planck equation are derived in Sect. 6. A power-law solution appears naturally as a consequence of the cooling of high energy particles in the intense soft radiation field.

The next section (Sect. 7) is devoted to the linear kinetic theory of the unusual physical situation in the relativistic regime. In particular, an interesting micro-instability due to cold proton streaming in a relativistic plasma is derived. The non-linear effects are also discussed. The Sect. 8 is devoted to the astrophysical applications of the kinetic theory to high energy extragalactic and galactic sources. In each non relativistic and relativistic regime the mean Lorentz factor and the bulk Lorentz factor of the pair plasma are estimated before concluding in Sect. 9.

Thereafter the prim denotes quantities expressed in the relativistic pair frame.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998