Self-Consistent Particle Acceleration in Active Galactic Nuc(2)

Adopting the hypothesis that the nonthermal emission of Active Galactic Nuclei (AGN) is primarily due to the acceleration of protons, we construct a simple model in which the interplay of acceleration and losses can be studied together with the formation o

2 neutrons (i.e., after their decay into protons and electrons{ Giovanoni& Kazanas 1990). Finally, the ux of escaping neutrons has been used to calculate the contribution of AGNs to the di use -ray background (Johnson et al. 1994). Neutrinos, on the other hand, may be produced at a rate which could be detected with an experiment such as DUMAND (Stecker et al. 1991, Biermann 1992, Sikora& Begelman 1992, Szabo& Protheroe 1992, 1994). In these models the relativistic protons also inject a population of electrons and positrons, which cool by synchrotron

and/or inverse Compton radiation and initiate intense pair cascades. However, the resulting nonthermal radiation (which is, at least partly, responsible for the saturation of the acceleration) was speci ed a priori and not calculated selfconsistently. In this sense, the hadronic models include a model of particle acceleration and injection but lack a self-consistent treatment of the electromagnetic cascades. A rst attempt to create a synthesis of these models was made by Stern et al. (1991), Stern& Svensson (1991) and Stern et al. (1992). These authors followed the electromagnetic cascades resulting from the injection of electrons by relativistic protons and included the feedback of the photons on the relativistic protons. They found that this system showed limit cycles much like a predator-prey system. However, the Monte Carlo approach which they use complicates the interpretation of the results in terms of a speci c feedback mechanism. Such a feedback e ect was found analytically by Kirk& Mastichiadis (1992{ henceforth\KM92") using the kinetic equation approach. They showed that relativistic protons become unstable to a combination of proton-photon pair production and electron synchrotron radiation once their number density and energy exceeds a certain critical value. This feedback leads eventually to rapid energy losses for the relativistic protons. However, the analysis of KM92 employs a stationary proton distribution and uses a linearised set of equations which exclude electromagnetic cascades, making it impossible to follow the evolution of the system into the saturation phase. Motivated by this shortcoming we present here a model capable of describing time-dependent e ects in the production of the nonthermal spectra of AGNs, albeit under a set of highly simplifying assumptions. Our method is to describe the three basic components of the central region of an AGN{ protons, electrons and photons{ by a system of three spatially averaged kinetic equations. The aim is to incorporate in an approximate manner all the important processes acting on these constituents and to use numerical methods to integrate the system forwards in time in the manner described by Fabian et al. (1986) and Coppi (1992) for photons and electrons. A time-dependent method is also essential if one is to account for the highly nonlinear coupling of the acceleration process with losses caused by the associated photons. We propose to use this technique to gain a better understanding of the origin and properties of variability in AGNs as well as the way in which their photon spectra are formed. Our method should also provide better information about quantities such as the expected luminosity in high energy neutrinos. In the present paper, however, we restrict ourselves to a detailed description of the method together with a discussion of its strengths and weaknesses. We do not attempt a systematic investigation of the parameter regime appropriate to AGNs, but present a

sample set of results obtained using the full code. Preliminary results of this work have been presented by Mastichiadis& Kirk (1992). The contents of the paper are organised as follows: in Sect. 2 we describe the way in which we model the acceleration process. Only protons are assumed to undergo acceleration{ it being tacitly assumed that the relativistic electrons are dominated by rapid loss processes. We choose a model in which a rst-order partial di erential equation is used to describe the rst-order Fermi process and a`loss' or`escape' probability is introduced to account for the possibility that protons might leave the emission region, for example by accretion into the black hole. In such a model, acceleration occurs homogeneously throughout the emission region, as might be expected, for example, if a converging accretion ow provides the acceleration (Schneider& Bogdan 1989). Apart from acceleration, the microscopic processes of importance in the system of kinetic equations are relatively well understood. For the protons these include proton-proton and proton-photon collisions (Mannheim& Biermann 1989, Begelman et al. 1990) whereas the electrons and photons (in addition to the source terms provided by the proton related processes) experience synchrotron radiation, Compton scattering, photon-photon pair production, electron-positron annihilation and Compton downscattering on cooled electrons (Coppi& Blandford 1990). These processes and the approximations we employ to describe them are discussed in Sect. 3. The method used to convert the integro-di erential kinetic equations into a system of ordinary di erential equations suitable for integration by standard numerical methods is presented in Sect. 4, and this is followed by a series of tests which check the behaviour of our approximation schemes in circumstances in which either analytic solutions are available e.g., when only synchrotron cooling or inverse Compton cooling are present, or in which there exist calculations in the literature with which to compare, e.g., the spectra of stationary electromagnetic cascades in the Thomson regime (Lightman& Zdziarski 1987). As an example of the application of the full code, Sect. 5 presents time-dependent results for one particular set of parameters which are appropriate for an AGN. One of the features of this run is the development of the pair productionsynchrotron instability (KM92) once the marginal stability threshold is crossed. The X-ray spectrum of power-law index= 5=3 which is predicted in the linear phase of the instability is seen to persevere well into the nonlinear phase and a stationary state is achieved in which the photons produced by accelerated protons are responsible for saturating the acceleration process. To conclude, we brie y summarise the main advantages and limitations inherent in our approach in Sect. 6.

2. The Particle Acceleration ModelAccording to the theory of particle acceleration by the rstorder Fermi mechani

sm (see, for example, Kirk et al. 1994) stochastic encounters of particles with so-called`scattering centres' occur within an`acceleration region', such that each interaction results in a small increase in the particle's energy. At the same time, a particle has a nite probability of escaping from the acceleration region, and the combination of these two e ects leads to a distribution of accelerated particles which, under certain circumstances, is of the power-law type. We shall adopt this theory here, and make the simplest possible assumptions concerning the rate at which energy is gained from the scattering centres as well as the rate at which parti-