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

3 cles escape the region containing these centres. The aim is to describe the particle distribution using a kinetic equation in which not just acceleration, but also losses su ered as a result of interactions with ambient photons can be included. However, one important question about the nature of an AGN, which we cannot answer a priori is that of how large the acceleration region is compared to the size of the`source region', i.e., that region which contains the relativistic particles responsible for the observed nonthermal emission. Two possibilities represent opposite extremes: 1. acceleration occurs throughout the whole of the source region, and 2. acceleration takes place in only very small parts of the source. An example of case 1 is the acceleration of particles by a smooth, unshocked accretion ow (Payne& Blandford 1981, Cowsik& Lee 1982, Webb& Bogdan 1987, Schneider& Bogdan 1989). Particles may escape from the acceleration region in this case (perhaps by being accreted into the black hole) but if they do, they cease to emit observable radiation and, by de nition, have left the source. An example of case 2 is the acceleration of particles at a shock front in the accretion ow (Protheroe& Kazanas 1983, Sikora et al. 1987). Provided the mean free path of the particle is small compared to the source, only those particles in the immediate vicinity of the shock undergo acceleration. Once particles have been swept out of this region, they are highly unlikely to return to it, but may still be energetic enough to produce observable radiation whilst cooling on the ambient photons, and so cannot be considered to have left the source. In this paper we deal exclusively with case 1, in which scattering centres are distributed throughout the source region. Denoting by np (p)dp the di erential number density of pro^ tons of momentum p in the interval dp, the kinetic equation for protons within the source region can be written (e.g., Schlickeiser 1984, Kirk et al. 1994)^@ np (p; t)+@ p np (p; t)+ np (p; t)^^^esc@t@p tacc^ t^p (^ p; p; t)= Qinj (p pinj)+ L n (1) where Qinj is the number of particles injected into the acceleration process per second per unit volume with momentum pinj^ and Lp (which can be a di erential and/or integral operator acting on np ) denotes the losses su ered

by energetic protons.^ The second term in Eq. (1) provides a continuous energy input by the rst-order Fermi process into those protons which remain in the acceleration region, whereas the third allows for^ escape, the average residence time being tesc . If we ignore for a moment the losses, and take the sim^ ple case of constant Qinj,^acc, and tesc, the solution to Eq. (1) t which satis es the boundary condition np (p; 0)= 0 is (Ax^ ford 1981)^ from pinj up to a cut-o at pmax (t)= pinj et=tacc which increases with time. Below the cut-o i.e., for p< pmax (t) the solution is independent of time. The power-law index is determined by the relative strengths of the acceleration term and escape term.^^ In the following we will usually assume tacc= tesc, in which case np/ p 2, such as is expected, for example, of particles^ accelerated at a strong shock front in a gas of adiabatic index 5=3. Interesting conclusions can be drawn in this special case about the energy given to and extracted from the protons by building the moment of Eq. (1) with the kinetic energy of a proton: ( 1)mp c2, where is the Lorentz factor p (= 1+ p2=(mp c)2). Denoting the total energy (minus the rest mass) contained in accelerated protons by E, we nd after integrating by parts: dE= V Q m c2 ( inj p inj 1) Lloss dt Z 1 n+V mp c2 dp ( 1)^ p (p; t); (3)^acc t 0

where inj= 1+ p2=(mp c)2, Lloss is the rate at which eninj ergy is lost by protons (to the processes of pair production and pion production) and V is the source volume. The loss processes discussed in the following section operate e ectively only on relativistic protons, so we can expect that if a steady state is set up in which E= constant, the spectrum will be given by the loss-free solution Eq. (2) all the way from the injection momentum up into the relativistic regime. However, the integral in the third term on the right-hand side of Eq. (3) is dominated by the nonrelativistic and transrelativistic regimes, provided the particle density does not diverge at large p. Consequently, we make only a small error by using the loss-free distribution in this integral. Provided pinj mp c, we nd in the steady state:

p

Lloss= V cpinjQinj:

(4)

According to this equation, the rate at which protons put energy into pair production and pion production during the acceleration process (which is in this model the entire nonthermal luminosity of the AGN) is determined in the steady state solely by the rate at which they are injected into the acceleration process at low momentum. In particular, it is independent of quantities connected with the actual loss process itself, such as the background photon or matter density and the maximum Lorentz factor to which particles can be accelerated, even though these may depend nonlinearly on the density itself. In connection with Eq. (4), we note that the rate Linj at which energy is injected at momentum pinj is small compared to the nonthermal luminosity: Linj=Lloss= pi

nj=2mp c 1, so the nonthermal emission stems not from the unknown injection process, but from the rst-order Fermi mechanism we are modelling. In this study of AGNs, we will be concerned only with relativistic protons, since it is in this regime that the accelera^^ 1 tacc=tesc^ tion and loss processes can compete with each other. It is then np (p; t)= taccQinj pp^ pinj more convenie …… 此处隐藏：4435字，全部文档内容请下载后查看。喜欢就下载吧 ……