Self-Consistent Particle Acceleration in Active Galactic Nuc(8)

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

83.9. Electron-positron pair annihilation

This process is the inverse of photon-photon pair production described above. It acts as a sink of electrons and positrons and a source for photons. In this treatment we will consider only the\cosmic-ray" case discussed by Svensson (1982) and Aharonian et al. (1983) in which relativistic electrons/positrons annihilate only on the cool particles and not amongst themselves. With this assumption, the losses of electron-positron pairs can be written

Le ! (; t)= ee

TRann (

)ne (; t)

(58)

where Rann is given by Coppi& Blandford (1990) 1=2+ ln: (59) Rann= 83 Photons produced by pair annihilation have hxi= (+ 1)=2 '=2 so the photon term can be written

Qee! (x; t)= 2 T Rann (2 )ne (2; t):

(60)

Finally the rate of Eq. (58) integrated over energy gives us the rate of cooled electron removal due to electron-positron annihilation, it contributes thus to Le;cool in Eq. (12). Since this rate does not include the annihilation of cooled pairs among themselves, we use an extra term Le;cool (t)= 3 Necool(t) 2: (61) 4 Furthermore, we assume that the rate of emission of the 511 keV annihilation rate is given by the above equation, so we write 3 (62) Qee! (1; t)= 4 Necool(t) 2:

be at min= 100:1 . Electrons which cool through this boundary join the population of cool electrons. For the photons, the grid spacing in the logarithm of x is chosen to be twice as large as that in log for protons (or electrons). Because the -function approximation is used in the synchrotron source term, Eq. (39), this grid spacing eliminates the need for interpolation; each electron grid point is associated with a single photon grid point. Thus, since the softest photons are produced by electron synchrotron radiation, we 2 take xmin= b min. However, inclusion of the pion producing processes requires the photon grid to be extended above the x corresponding to the synchrotron radiation of electrons

of max, because of the very hard photons produced in 0 -decay. We therefore choose xmax= max and check that the photon bins adjacent to xmax remain practically empty throughout the run. Integrals over the photon and electron distributions are straightforwardly converted into summations using the trapezoidal rule, but care must be taken in discretizing the rst order derivatives with respect to . In the proton equation (6) and in the electron equation (11) these terms describe`continuous' acceleration and losses and in order to avoid a numerical instability, an upstream di erence must be taken. For particles undergoing losses`upstream' means larger, whereas for the accelerating particles it means smaller . Thus, for electrons, which experience no acceleration, the value of the distribution at max is held constant at a negligibly small value. The value at min is computed. On the other hand, protons experience only acceleration at min, so that np ( min) is held constant and is related to the e ective injection rate. At the upper end of the scale, losses always dominate for protons of max, so that, just as for electrons, np ( max ) is held constant as a boundary condition. Similarly, in order to take into account the rst order derivative with respect to the photon energy x (Eq. 47), we take an upstream di erence while n (xmax ) is held constant at a negligibly small value. In order to test both the code and the treatment of the physical processes, we have performed various runs for cases in which either an analytic solution is available, or in which we can compare our results with calculations in the literature. In the following sections, when appropriate, we will refer to the electron injection compactness, de ned in our normalisation by4.2. Performance Checks

4. The numerical methodWhen the physical process described in Sect. 3 are included, the system of kinetic equations (6), (11) and (13) is an integro-di erential set for the distributions np (; t), ne (; t) and n (x; t). Our approach to the numerical solution is to discretize the variables and x, and to integrate the resulting (sti ) set of coupled ordinary di erential equations forwards in time. Because we use crude but computationally rapid approximations to the physical processes we are able to integrate the equations using a standard NAG-library routine for sti systems. The grid chosen for discretization is equally spaced in the logarithm of, the same grid being used for both protons and electrons. In order to accommodate a large dynamic range of the distributions np ( max; t), ne ( max; t) and n (x;t) we use their logarithms in the numerical integration. In a typical run the resolution is 10 bins per decade for protons and electrons. This is a rather coarse grid; however, we found runs with higher resolutions to be excessively time consuming. The highest grid point max is adjusted so that the adjacent bins to ne ( max; t) and np ( max; t) remain negligibly small throughout

the run. We nd a value of max= 109 is adequate to ensure this for the models presented here. We choose the lowest grid point to4.1. Discretization

`e= 1 3

Z maxmin

d (s

1)Qe ( )

(63)

where Qe ( )= Qe;0=0

for min<< otherwise

max

(64)

is an external electron injection rate introduced for test purposes. 4.2.1. Proton acceleration For the rst test of the code we check the way the numerical method deals with proton acceleration. As stated above, protons are e ectively injected at a Lorentz factor inj= 100:1 . The solution of Eq. (1) in the case where Lp= 0 is given by