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

4 and sources of photons. Because of the normalisation used, the R quantity dx xn is simply related to the photon compactness np (; t)d= T Rnp (p; t)dp^ (5) (Guilbert et al. 1983) given by and nd, in the relativistic regime: (14)`= 4 L T c3: Rme i h np (; t)= Lp (n;; t): (6)@np (; t)+@ p@t@ tacc np (;t)+ tesc For a spherical source we have Z^^ where t is now dimensionless, tacc= ctacc=R and tesc= ctesc=R.`= 1 (15) The nonrelativistic part of the acceleration process can be 3t esc dx xn: avoided by using the loss-free solution as a boundary condition 3. The Physical Processes mensionless density np (; t) by

1+ (7) np ( 0; t)= n0= 0 tacc=tesc; with^ n0= T Rtacc Qinj(pinj=mp c)tacc=tesc; (8) where 0 is the lowest value of the Lorentz factor to be considered. For tacc= tesc, the AGN luminosity in a steady state can be expressed in terms of the compactness loss (9)`tot= 4LRm T3: ec Setting the source volume V= 4 R3 in Eq. (4) leads to 3`tot= 3n0 mp: (10) tacc me Using the same normalisation, we complement Eq. (6) by writing for the relativistic electrons and positrons:@ne (; t)= Qe (ne;; t)+ Le (ne;; t) (11)@t Here Le denotes the various electron loss terms while Qe are the injection terms. Note that there is no acceleration or escape term included in the electron equation, since we assume the loss terms to be much larger. In our numerical treatment, we impose a lower limit min on the Lorentz factor of the relativistic electron population and assume particles which cool through this boundary join a population of cold electrons whose number density Necool (t) (i.e., the number in a volume TR) is determined by the equation

We proceed now to discuss the various contributions to the kinetic equations (6), (11), (12) and (13) of the physical processes by which the components of the system interact with each other and with the magnetic eld.3.1. Proton-proton interactions

Inelastic proton-proton collisions act as an energy loss mechanism for relativistic protons. They also inject -rays, relativistic elect

rons, and neutrinos resulting from the decay of the produced neutral and charged pions. While one can calculate in detail the spectra of the products once the relativistic proton distribution is given (e.g., Dermer 1986, Mastichiadis& Protheroe 1990), for the present calculation it sufces to use the functional approximation to the di erential cross-section for production of energetic pions d (E )=dE= 0 0 0:15 mp c2 ) where pp ' 0:06 is the protonpp T (E proton cross section (e.g., Atoyan 1992a). Thus, to treat the proton losses we write

L p (; t)= pp

0 targ pp np

np ( 0; t) np (; t)

(16)

dNecool(t)= Qe;cool (n; t)+ Le;cool (n; t) (12) e e dt Contributions to the source term Qe;cool (ne; t) arise from synchrotron and Compton cooling, while the contributions to the sink term Le;cool (ne; t) come mainly from electron-positron annihilation. Finally, the spatially averaged photon equation reads:@n (x; t)+ n (x;t)= Q (n; x; t)+ L (n; x; t); (13)@t t esc where x is the dimensionless photon frequency: x= h=(me c2 ). Photons leave the source on the timescale t esc (measured in units of the light crossing time tcross= R=c) and, rather than being advected into the black hole, are assumed to propagate freely after escape. The terms L and Q denote the sinks

where ntarg is the density of target protons and we have asp sumed that each proton-proton collision removes protons from the energy bin and+ d but injects them there from higher energy bins of energy 0==(1 kpp ). Here kpp is the proton inelasticity and it is taken to be kpp=:45 Since the mean energy per gamma-photon produced at the decay of the 0 -meson is 0:5E= 0:075 mp c2, the photon production spectrum, in this approximation, can be taken as 2 0 (17) Qpp (x; t)= 0:075 ppntarg np ( 1; t) p where 1= xme=(0:075mp ). The above approximation essentially injects photons with a slope equal to the relativistic proton spectrum at energies greater than ' 70 MeV. It does not treat correctly injection at energies below this value. However, detailed calculations (e.g., Dermer 1986) have shown that the photon spectrum attens considerably at energies below 100 MeV, so that the approximation introduced above is adequate for our purposes. We turn next to the injected electrons (or positrons). Since the electron (positron) carries o, on average, 26% of the initial pion energy, it follows that hEe i 0:039 Ep (Atoyan 1992a) and the corresponding injection spectrum resulting from the decay of charged pions is given by 2 0 (18) Qe (; t)= 0:039 ppntarg np ( 2; t); p pp