Astron. Astrophys. 324, 395-409 (1997)

1. Introduction

Accretion of matter onto a central black hole is the most relevant process to power active galactic nuclei (Lynden-Bell 1969, Salpeter 1969, Rees 1984). However, the details of the conversion processes of gravitational energy into observable electromagnetic radiation are still largely unknown. The discovery of many blazar-type AGNs (Hartman et al. 1992, Fichtel et al. 1993) as sources of high-energy gamma-ray radiation dominating the apparent luminosity, has revealed that the formation of relativistic jets and the acceleration of energetic charged particles, which generate nonthermal radiation, are key processes to understand the energy conversion process. Emission from relativistically moving sources is required to overcome gamma-ray transparency problems implied by the measured large luminosities and short time variabilities (for review see Dermer & Gehrels 1995).

Repeated gamma-ray observations of AGN sources have indicated a typical duty cycle of gamma-ray hard blazars of about 5 percent, supporting a "2-phase" model for the central regions of AGNs (Achatz et al. 1990, Schlickeiser & Achatz 1992). According to the 2-phase model the central powerhouse of AGNs undergoes two repeating phases: in a "quiescent phase" over most of the time ( 95 percent) relativistic charged particles are efficiently accelerated in the central plasma near the black hole, whereas in a short and violent "flaring phase" the accelerated particles are ejected in the form of plasma blobs along an existing jet structure.

We consider the acceleration of charged particles during the quiescent phase. The central object accretes the surrounding matter. Associated with the accretion flow is low-frequency magnetohydrodynamic turbulence which is generated by various processes as e.g.:

(a) turbulence generated by the rotating accretion disk at large eddies and cascading to smaller scales (Galeev et al. 1979);

(b) stellar winds from solar-type stars in the central star cluster deliver plasma waves to the accretion flow;

(c) infalling neutral accretion matter becomes ionized by the ultraviolet and soft X-ray radiation of the disk. These pick-up ions in the accretion flow generate plasma waves by virtue of their streaming (Lee & Ip 1987);

(d) if standing shocks form in the neighbourhood of the central object they amplify any incoming upstream turbulence in the downstream accretion shock magnetosheath (McKenzie & Westphal 1969, Campeanu & Schlickeiser 1992).

These low-frequency MHD plasma waves from the accretion flow are the source of free energy and lead to stochastic acceleration of charged particles out of the thermal accretion plasma.

The dynamics of energetic charged particles (cosmic rays) in cosmic plasmas is determined by their mutual interaction and interactions with ambient electromagnetic, photon and matter fields. Among these by far quickest is the particle-wave interaction with electromagnetic fields, which very often can be separated into a leading field structure and superposed fluctuating fields . Theoretical descriptions of the transport and acceleration of cosmic rays in cosmic plasmas are usually based on transport equations which are derived from the Boltzmann-Vlasov equation into which the electromagnetic fields of the medium enter by the Lorentz force term. The quasilinear approach to wave-particle interaction is a second-order perturbation approach in the ratio and requires smallness of this ratio with respect to unity. In most cosmic plasmas this is well satisfied as has been established either by direct in-situ electromagnetic turbulence measurements in interplanetary plasmas, or by saturation effects in the growth of fluctuating fields. Nonlinear wave-wave interaction rates and/or nonlinear Landau damping set in only at appreciable levels of and thus limit the value of . We assume the AGN plasma to have very high conductivity so that any large-scale steady electric fields are absent. We then consider the behaviour of energetic charged particles in a uniform magnetic field with superposed small-amplitude plasma turbulence () by calculating the quasilinear cosmic ray particle acceleration rates and transport parameters. This is by no means trivial since especially for the interaction of non-relativistic charged particles with ion- and electron-cyclotron waves thermal resonance broadening effects are particularly important (Schlickeiser & Achatz 1993, Schlickeiser 1994). The acceleration rates and spatial transport parameters are then used in the kinetic diffusion-convection equation for the isotropic part of the phase space density of charged particles which for non-relativistic bulk speed reads

Here x denotes the spatial coordinate along the ordered magnetic field, p the cosmic ray particle momentum, is the spatial diffusion coefficient, A the momentum diffusion coefficient, and denotes the "Stossterm" describing the mutual interaction of the charged particles and their injection.

With respect to the generation of energetic charged particles, the basic transport Eq. (1) shows that stochastic acceleration of particles, characterized by the acceleration time scale , competes with continuous energy loss processes , characterized by energy loss time scales . Dermer et al. (1996) have recently inspected the acceleration of energetic electrons and protons in the central AGN plasma by comparing the time scales for stochastic acceleration with the relevant energy loss time scales. At small proton momenta the Coulomb loss time scale is extremely sensitive to the background plasma density and temperature, and for slight changes in the values of these parameters cosmic ray protons may not be accelerated above the Coulomb barrier. Although at small particle momenta the plasma wave's dissipation and the interaction with the cyclotron waves become decisive and might modify the acceleration time significantly, the results of Dermer et al. (1996) demonstrate that reasonable central AGN plasma parameter values are possible where the low-frequency turbulence energizes protons to TeV and PeV energies where photo-pair and photo-pion production are effective in halting the acceleration (Sikora et al. 1987, Mannheim & Biermann 1992). According to the results of Dermer et al. (1996) it takes about days for the protons to reach these energies, where is the mass of the central black hole in units of . The corresponding analysis for cosmic ray electrons shows that the external compactness provided by the accretion disk photons (Becker & Kafatos 1995) leads to heavy inverse Compton losses which suppress the acceleration of low-energy electrons beyond Lorentz factors of . It seems that due to their much smaller radiation loss rate cosmic ray protons are effectively accelerated during the quiescent phase in contrast to low energy electrons.

Now an important point has to be emphasized: once the accelerated protons reach the thresholds for photo-pair () and photo-pion production and the threshold for pion production in inelastic proton-matter collisions they will generate plenty of secondary electrons and positrons of ultrahigh energy which are now injected at high energies () into this acceleration scheme. eV denotes the mean accretion disk photon energy. It is now of considerable interest to follow the evolution of these injected secondary particles.

Although many details of this evolution are poorly understood, it is evident that the further fate of the secondary particles depends strongly on whether they find themselves in a compact environment set up by the external accretion disk, or not. As has been pointed out by Dermer & Schlickeiser (1993b) as well as Becker & Kafatos (1995) the size of the gamma-ray photosphere (where the compactness is greater unity so that any produced gamma-ray photon is pair-absorbed) is strongly photon energy dependent. The gamma-ray photosphere attains its largest size at photon energies GeV. Secondary particles within the photosphere having energies will initiate a rapid electromagnetic cascade which has been studied by e.g. Mastichiadis & Kirk (1995), which might even lead to runaway pair production and associated strong X-ray flares (Kirk & Mastichiadis 1992), and/or due to the violent effect of a pair catastrophy (Henri & Pelletier 1993) ultimately lead to an explosive event and the emergence of a relativistically moving component filled with energetic electron-positron pairs.

In contrast, if the secondary particles are generated outside the gamma-ray photosphere the secondary electrons and positrons will quickly cool by the strong inverse Compton losses generating plenty of gamma-ray emission up to TeV energies. As solutions of the electron and positron transport Eq. (1) for this case demonstrate (Schlickeiser 1984, Pohl et al. 1992) a cooling particle distribution () with a strong cutoff at low (but still relativistic) momentum develops, which grows with time as more and more protons hit the photo-pair and photo-pion thresholds. Such bump-on-tail particle distribution functions, which are inverted () below , are collectively unstable with respect to the excitation of electromagnetic and electrostatic waves such as oblique longitudinal Langmuir waves (Lesch et al. 1989). As described in detail by Lesch & Schlickeiser (1987) and Achatz et al. (1990), depending on the local plasma parameters (mainly the electron temperature of the background gas and the density ratio of relativistic electrons and positrons to thermal electrons) these Langmuir waves either (1) lead to quasilinear plateauing of the inverted distribution function and rapid collective thermalization of the electrons and positrons, or (2) are first damped by nonlinear Landau damping, but ultimately heat the background plasma strongly via the modulation instability once a critical energy density in Langmuir waves has been built up. Both relaxation mechanisms terminate the quiescent phase of the acceleration process. The almost instantaneous increase of the background gas entropy due to the rapid modulation instability heating again leads to an explosive outward motion of the plasma blob carrying the relativistic particles away from the central object.

As we have discussed, in both cases it is very likely that at the end of the quiescent phase an explosive event occurs that gives rise to the emergence of a new relativistically moving component filled with energetic electron-positron pairs. It marks the start of the flaring phase in gamma-ray blazars. The initial starting height of the emerging blob entering the calculation of the gamma-ray flux should be closely related to the size of the acceleration volume in the quiescent phase mainly determined by the maximum size of the gamma-ray photosphere. This scenario is supported by the measurements of Babadzanhanyants & Belokon (1985) that in 3C 345 and other quasars optical bursts are in close time correlation with the generation of compact radio jets. A similar behaviour has also been observed during the recent simultaneous multiwavelength campaign on 3C 279 (Hartman et al. 1996). Further corroborative evidence for this scenario is provided by the recent discovery of superluminal motion components in the gamma-ray blazars PKS 0528+134 (Pohl et al. 1995) and PKS 1633+382 (Barthel et al. 1995) that demonstrate a close physical connection between gamma-ray flaring and the ejection of new superluminal jet components in blazars.

It is the purpose of the present investigation to follow the time evolution of the relativistic electrons and positrons as the emerging relativistic blob moves out. Because of the very short radiative energy loss time scales of the radiating electrons and positrons it is important to treat the spectral evolution of the radiating particles self-consistently. In earlier work Dermer & Schlickeiser (1993a) and Dermer et al. (1997) have studied the spectral time evolution from a modified Kardashev (1962) approach by injecting instantaneously a power law electron and positron energy spectrum at height at the beginning of the flare, and calculating its modification with height in the relativistically outflowing blob due to the operation of various continous energy loss processes as inverse Compton scattering, synchrotron radiation, nonthermal bremsstrahlung emission and Coulomb energy losses. Here we generalize their approach by accounting for Klein-Nishina effects and including as well external inverse-Compton scattering as synchrotron self-Compton scattering self-consistently.

The acceleration scenario described above leads us to the following assumptions on the distribution of pairs inside a new jet component at the time of their injection: If the pairs are created by photo-pair production, their minimum Lorentz factor is expected to be in the range of the threshold value of the protons' Lorentz factor for photo-pair production. We use the standard accretion disk model by Shakura & Sunyaev (1973), which we will describe in more detail in the next section, to fix this threshold, determined by the average disk photon energy . The pair distribution above this cutoff basically reflects the acceleration spectrum of the protons, i. e. a power-law distribution with spectral index () which extends up to . This yields the initial pair distribution functions

with . The differential number of particles in the energy intervall per unit volume is then given by

where is a normalization factor related to the total particle density through .

The detection of TeV -rays from Mrk 421 suggests that such components must be produced/accelerated outisde the -ray photosphere for photons of energy TeV. The height of this photosphere due to the interaction of -rays with accretion disk radiation will be determined self-consistently in Sect. 4. Backscattering of accretion disk radiation by surrounding clouds is negligible in the case of BL Lac objects emitting TeV -rays (Böttcher & Dermer, 1995).

We point out that most of our basic conclusions are also valid if the pairs inside the blob are accelerated by other mechanisms, e. g. by a relativistic shock propagating through the jet.

Beginning at the injection height (henceforth denoted as ), we follow the further evolution of the pair distribution and calculate the emerging photon spectra.

Fig. 1. Model for the geometry of a relativistic AGN jet

The negligibility of pair absorption due to the interaction with the synchrotron and -ray emission from the jet is checked self-consistently during our calculations.

Interactions of the jet pair plasma with dilute surrounding material will cause turbulent Alfvén and Whistler waves. It has been shown by Achatz & Schlickeiser (1993) that a low-density, relativistic pair jet is rapidly disrupted as a consequence of the excitation of such waves. Thus, to insure stability of the beam over a sufficient length scale, we need that the density of pairs inside the jet exceeds the density of the surrounding material. In this case, pitch angle scattering on plasma wave turbulences leads to an efficient isotropization of the momenta of the pairs in the jet without destroying the jet structure. Thus, additional assumptions on our initial conditions are that the particle momenta are isotropically distributed in the rest frame of a new jet component (blob) and that the density of pairs in the jet where is the density of the background material.

This study is devided into two papers. In the first (present) paper we investigate the details of electron/positron cooling due to inverse-Compton scattering and follow the pair distribution and photon spectra evolution during the first phase in which the system is dominated by heavy radiative losses. In the second paper we will consider the later phase of the evolution where collisional effects (possibly leading to thermalization) and reacceleration become important, and a plausible model for MeV blazars which follows directly from our treatment will be presented (Böttcher, Pohl & Schlickeiser, in prep.).

In Sect. 2 of this first paper, we describe in detail how to calculate the energy-loss rates due to the various processes which we take into account and give useful approximative expressions for the inverse-Compton losses (as well scattering of accretion disk radiation as of synchrotron radiation), including all Klein-Nishina effects. In Sect. 3, we discuss the relative importance of the various processes. The location of the -ray photosphere for TeV -rays is briefly outlined in Sect. 4. The technique used to follow the evolution of the pair distributions is described in Sect. 5. In Sect. 6, we describe how to use the pair distributions resulting from our simulations in order to calculate the emanating -ray spectra, and in Sect. 7 we discuss general results of our simulations giving a prediction for GeV - TeV emission from -ray blazars which due to the lack of sensitivity of present-day instruments in this energy range could not be observed until now. Only two extragalactic objects have been detected as sources of TeV emission, namely Mrk 421 (Punch et al. 1992) and Mrk 501 (Quinn et al. 1995). In Sect. 8, we use our code to fit the observational results on the broadband emission during the TeV flare of Mrk 421 in May 1994 and on its quiescent flux. We summarize in Sect. 9.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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