Astron. Astrophys. 324, 461-470 (1997)

1. Introduction

There is general consensus that active galactic nuclei (AGN) consist of accreting supermassive black holes (BH) (up to ) surrounded by accretion disks. The disk consists of thermal plasma which is heated via viscous accretion up to temperatures of about . The existence of thermal plasma in this temperature range has been established beyond any doubt by optical observations which exhibit line widths up to and by UV observations which clearly show spectra compatible with the thermal spectrum of an accretion disk with .

There are at least two pieces of evidence for the presence of relativistic electrons in AGN, first the clear detection of hard X-ray and -ray spectra and second the radio observations of superluminous motions on VLBI scales, which describe the existence of a plasma with relativistic bulk motions. Furthermore, the most successful models of X-ray emission in AGN imply the copious production of pairs via interactions, preprocessing by inverse Compton scattering of the soft photons emitted by the accretion disk (Svensson 1987; Lightman and Zdiarski 1987; Done and Fabian 1989; Svensson 1990). The main finding of those studies is that for monoenergetic injection of at large Lorentz factors () and for the range , the X-ray spectral index is , which is in general agreement with AGN observations (Svensson 1990). Here l denotes the compactness parameter, given by

Here, is the Eddington luminosity, is the Thomson cross-section, and are the electron and proton masses and is the Schwarzschild radius.

Dermer and Schlickeiser (1993) have shown that the -radiation observed from quasars and blazars may originate in a distance R of gravitational radii from the central engine. At that distance, the compactness parameter , i.e. no pair production appears. Instead, the relativistic leptons scatter via the inverse Compton process the UV radiation within a relativistically moving jet.

It is the aim of this contribution to consider the problem of rapid charged particle acceleration in the presence of an intense UV radiation field characteristic for the environment of the accretion disk in active galactic nuclei. The UV photons scatter the particles via inverse Compton scattering. The Compton cooling rate of a relativistic electron scattering with a low-energy photon () is

where is the radiation energy density. As can be related to the luminosity L of a source of size R through one can write the Compton cooling time scale as

This means that for a compactness all electrons cool before they can escape (). To avoid a pileup of cool particles reacceleration (or annihilation, if the particles come in pairs) is necessary.

The "standard" acceleration mechanisms - diffusive shock wave acceleration and resonant acceleration by magnetohydrodynamical (MHD) turbulence (also known as Fermi I and Fermi II-process) seem to fail since they cannot provide Lorentz factors less than . This means that particles accelerated via shocks or turbulence have to be preaccelerated - this is also known as the injection problem (Blandford 1994; Melrose 1994).

Let us shortly summarize the arguments which lead to the result that particles need before they can be picked up by the shock wave acceleration process or by resonant acceleration by MHD turbulence:

The threshold is associated with the requirement of effective damping of the Alfvén waves by the relativistic electrons. The maximum frequency of Alfvén waves is the ion cyclotron frequency . For effective electron acceleration, the relativistic electron gyro frequency should be comparable to (Kuijpers 1996). Therefore we have a minimum Lorentz factor for the electrons in a hydrogen plasma of .

If acceleration proceeds at a collisionless shock of width , electrons will not see the shock as a discontinuity unless their scattering mean free-path . Both by observations and numerical simulations (Formisano et al 1975; Winske et al. 1985) one finds that a quasi-parallel collisionless shock has a width of a few times , where is the ion plasma frequency. If scattering is due to strong magnetic fluctuations one has . Electrons will not see the shock as a discontinuity unless ( is the Alfvén velocity). When the subshock is erased by the diffusive action of accelerated protons, the shockwidth will exceed , which is the gyroradius of a proton with momentum . Electrons need to have at least the same momentum before shock acceleration becomes efficient. For shock acceleration to operate a shock has to be super-Alfvénic (). Consequently, this requirement proves stronger than the first one.

In all cases some mechanism is needed to preaccelerate electrons to energies exceeding the typical "thermal" energy per particle in the downstream flow.

It was proposed that Whistler waves with frequencies between the ion and electron-gyrofrequency may solve the injection problem, or at least relax the injection condition down to , since the waves can be in resonance with electrons of this energy (Levinson 1992).

However, there is a significant difference between MHD-turbulence and Whistler turbulence. The MHD-turbulence originates in macroscopic turbulent fluid motions (like winds, jets-etc..). The source scale of the turbulence is much larger then ion gyroradii. The turbulent energy is transferred from large scales via dissipationless cascading down to the small dissipation scale which is approximately the ion gyroradius (e.g. Biskamp 1993). There the particles can be accelerated by the resonant interaction with the MHD-waves. Obviously the large energy reservoir for particle acceleration by MHD turbulence easily explains the high energies in relativistic particles. This is an enormous advantage of the Fermi-processes because in every acceleration model the energy of the accelerator should be larger then the final energy in the accelerated particles.

Such a condition is difficult to achieve for the excitation of Whistler waves since they are driven by phase space anisotropies of the particle distribution function (e.g. Krall and Trivelpiece 1973). To accelerate particles the anisotropy of the distribution has to be enormous in order to excite strong enough Whistler turbulence. The energy density in the waves has to be high enough in order to accelerate particles. Since it is the anisotropy either of distribution function of the background plasma or of a high energy particle beam which is responsible for the excitation of the waves, the required anisotropy means an extreme deviation from a Maxwellian distribution. In other words, the presence of intense Whistler turbulence would mean already a strong preacceleration of electrons in order to get the anisotropic distribution function needed. Because of this requirements we think that Whistler waves are not a real solution of the injection problem.

In this contribution we investigate the possible role of magnetic field-aligned electric fields in the injection problem context. Such fields are related to magnetic field-aligned potential drops, which are known to occur, e.g. in the Earth's magnetosphere at several altitudes above auroral arcs. For terrestrial observers auroral acceleration regions are the closest realization of a powerful cosmic acceleration process, relatively easy to probe and indeed documented by a wealth of data (cf. Block and Fälthammar 1990; Lysak 1990). The more one should be surprised that few attempts have been made to transfer the knowledge gained from the Earth's aurora to other cosmical situations. Acceleration in field-aligned electric fields has received much attention in context of pulsar magnetospheres (e.g. Goldreich and Julian 1969; Ruderman and Sutherland 1975) and is thought to occur in solar flares (e.g. Sakai and Ohsawa 1987; Litvinenko 1996). An acceleration process so common to the benign plasma environment of the Earth can be rightly suspected to exist in any cosmic magnetic field that is sufficiently agitated.

Field-aligned potential drops as a product of intense field-aligned electric currents imply the existence of significantly sheared magnetic fields. In the following section we describe how magnetic fields are sheared in the corona of an accretion disk in the vicinity of a black hole. The forces creating the magnetic shear are the ultimate cause of the related acceleration process and supplier of the energy provided for particle acceleration. A prominent feature of this acceleration process is that energy supply and energy conversion (into relativistic electron beams) are physically separated. There is one plasma regime which we identify with the accretion disk, mostly characterized by a high value of the plasma beta, i.e. the ratio of gas and magnetic pressure (). The acting of mechanical forces, like friction, pressure gradients, or the inertial force, can not be fully balanced by magnetic forces but require the transport of shear stresses out of the interaction volume into another plasma regime, with low where the forces are eventually balanced. The transport of shear stresses is effected by field-aligned currents. We will identify the low- region with the corona of the accretion disk. In Sect. 3 we describe the acceleration region making use of an analytical kinematic model introduced by Schindler et al. (1991). In Sect 4. we introduce a simple dynamical numerical model for the formation of localized regions of significant field-aligned electric fields and describe how electrons are accelerated and under which conditions they can reach large Lorentz-factors in the vicinity of the excessively UV radiating innermost parts of an accretion disk around a massive black hole. A discussion of our findings is given in Sect. 5.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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