Self-Consistent Particle Acceleration in Active Galactic Nuc(7)

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

7 regime i.e., photons of energy x0 less than xT 3=(4 ). Equa- inside the source, we follow Lightman& Zdziarski (1987) and tion (42) gives rise to an injection of cooled electrons at a rate write for the photon escape time 2 4UT minne ( min; t)=3. The photon source term in the Thomson T regime is given by: (49) t esc= 1+ 3 f (x); 1 x 0:1 (44) f (x)= (1 x)=0:9 0:1< x< 1 (50) 0 x 1:

Photons with x> xT will interact with electrons of energy in Klein-Nishina scatterings. In order to treat this much more Therefore, when only the e ects of photon downcomptonizacomplicated case, we assume that an electron loses all its en- tion are included, the photon kinetic equation (13) becomes ergy in one such collision and joins the population of cooled particles. Approximating the cross-section by (x0 ) ' T=x0@n (x; t)+ n 1(x;t)= T@ (h xi n (x; t)): (51)@t@x 1+ 3 Tf (x) where x0= x, the electron loss term can be writtenp x0 1=2 x 1=2 ne ( 3x=4x0; t)n (x0; t)

p Z min 3=(4x);3x=4] Qics;T (x;t)= 43 dx0 0

where

(

Le;KN ( ics

For x 1, Eq. (47) is no photons (45) large fraction of their energylonger valid as theComptonlose a in each collision. scattering can then be considered as a catastrophic loss process and Because the scattered photons emerge with the energy of the to deal with this case, we follow Svensson (1987) and write incoming electron, the photon source term is simply Lcs (x;t)= T n (x; t)=x: (52) Z 1 e 0 (xx0 ) 1 n (x0; t) Qics;KN (x; t)= ne (x; t) dx (46) The corresponding injection of (relativistic) electrons is 3=4x e (53) Note that these are only approximate expressions since the Qcs (; t)= Tn (; t)=: logarithmic energy dependence of the cross section has been This injection comes from the upscattering of cooled electrons, neglected. Nonetheless, this does not a ect our results because and when integrated over all energies contributes to Le;cool in rates in the Klein-Nishina regime are in any case suppressed Eq. (12). by the proportionality of the cross section to the inverse of the energy (the\Klein-Nishina" cut-o ). 3.8. Photon-photon pair production3.7. Photon downscattering on cold electrons Inverse Compton scattering describes the upscattering of pho-

Z ne(; t) 1 dx0 n (x0; t);t)= x0 xT

tons by relativistic electrons. In addition, cool electrons ( 1), downscatter hard photons. The Thomson optical depth of cool electrons is in our normalisation simply T= Necool. The rate of photon downscattering in the Thomson regime can then be found from the Kompaneets equation, assuming the temperature of cool electrons is zero and neglecting the stimulated scattering term:@ (47) Lcs (x; t)= T@x h xi n (x; t)]: h xi is the average energy shift of a photon with energy x colliding with an electron at rest. This quantity was calculated from the relation

This process acts as a sink of high energy photons, as well as an injection term of electrons. To calculate the loss rate of photons we follow Coppi and Blandford (1990): Photons of energy x are lost at a rate

L !ee (x; t)= n (x; t)

Z

1

where R is a t to the reaction rate given by2 R (!)= 0 652 ! !3 1 ln(!)H (! 1) (55) where H (y) is the Heaviside function. Photon-photon pair production is also responsible for the injection of relativistic electrons and positrons. Assuming, as in the case of Bethe-Heitler pair production, that the electron and p

ositron emerge with equal energy, and noting that the photon-photon pair production process requires at least one hard photon of energy x> 1, which interacts predominantly with a soft photon of energy around 1=x, we nd from conservation of energy 1 2= x+ x x: (56) The injection term for electrons is then

0

dx0 n (x0; t)R (xx0 )

(54)

h xi=

R x0 max

x0 min

dx0 (x x0 )d (x; x0 )=d; R xmax dx0 d (x; x0 )=d x0 0

(48)

scattering (see, for example, Akhiezer& Berestetskii 1969) and the limits of the integration x0max= x and x0min= x=(2x+ 1) are determined from kinematical constraints. For x 1, h xi= x2=(2x+ 1) ' x2, which when substituted in Eq. (47) yields the usual Kompaneets equation (under the simpli caZ 1 tions assumed here). For x 1, h xi departs from the x2 dependence, as relativistic e ects start becoming important. Qe !ee (; t)= 4n (2; t) dx0 n (x0; t)R (2 x0 ): 0 In order to take into account the spatial di usion of photons

where d (x; x0 )=d is the di erential cross section for Compton

min

(57)