Self-Consistent Particle Acceleration in Active Galactic Nuc(6)

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

63.4. Synchrotron radiation 3.5. Synchrotron self-absorption

Classical synchrotron radiation is treated in several texts (e.g., Ginzburg& Syrovatskii 1965). Using our normalisation, the rate (x; ) at which a single electron electron emits photons into the frequency range dx is given by:

b (33) (x; )= 3 fx sin Rme c F 2x=(3b sin 2 )] h where b= B=Bc denotes the magnetic eld in units of the critical eld Bc= m2 c3=(eh)= 4 414 1013 G, f is the ne e structure constant, the particle's pitch angle and we have used the standard notation of Ginzburg and Syrovatski for the function F (x). Provided the energy of the emitted photon is much smaller than that of the emitting particle, the loss process can be considered continuous and represented by a rst-order di erential operator in the equation for electrons Eq. (11):

p

Le syn

1=@@ ne (; t) dx (x; ) 0 4= 3`B@@ 2 ne (; t):

Z

Low energy electrons interact strongly with the radiation eld if their Lorentz factor is such that the synchrotron photons they emit are reabsorbed within the source (McCray 1969, Ghisellini et al. 1988). The photon spectrum too is strongly affected below the self-absorption frequency (at which the optical depth equals unity). Such behaviour can be modelled using a second order derivative in momentum in the electron equation (Ghisellini et al. 1988). However, we do not wish to discuss in detail the heating and cooling processes of low energy electrons in this paper, nor are the details of the low energy spectrum important for our investigations. Consequently, we treat synchrotron self-absorption simply as a sink of soft photons. Starting from standard formulas for the absorption coefcient (e.g., Eq. (6.50) of Rybicki& Lightman 1979), and using the -function approximation Eq. (36) one can readily derive

Lssa(34)

R n (x; t)= 6 x 1=2 b 1=2 n (x; t)@@ f

ne2=(x=b)1=2

(40)

Here we have introduced the`magnetic compactness': U`B= m B 2 T R (35) ec in which UB= B 2=8 is the magnetic energy density and have replaced sin2 by its average (=2/3) for isotropic electr

ons. 2 Equation (34) contributes an amount 4`B minne ( min; t)=3 to e;cool for cool electrons. the source term Q The source term in the photon equation can be found using the` -function' approximation, originally introduced by Hoyle (1960) in which the emission of a single electron is approximated as monochromatic:

When only synchrotron emission and self-absorption are included, the photon kinetic equation (13) is:

@n (x; t)= 2` b@t 3B+ 6 bf

3=2 x 1=2 n (px=b; t) e 1=2 x 1=2 n

(x; t)@@

ne2

=(x=b)1=2

(41)

For an equilibrium electron distribution ne/ exp(=T ) (where T is the temperature in units of me c2=kB ) the stationary photon distribution is the Rayleigh-Jeans distribution nRJ= 4 R T (mc=h)3 Tx, whereas for a power-law electron distribution one obtains a stationary self-absorbed spectrum n/ x3=2 These results (x; ) q0 x0 (x x0 ); (36) treatment. of synchrotronagree with those found from an exact emission. and the quantities q0 and x0 must be determined by imposing requirements on the accuracy of the approximation. To repro- 3.6. Inverse Compton scattering duce the correct value for the total energy emitted by a single There is a close analogy between Compton scattering and synelectron we nd by integrating Eqs. (33) and (36) over x chrotron radiation, as has been pointed out by Felten& Morrison (1966). In the case of synchrotron radiation, the classical 4 (37) treatment, in which the energy of the emitted photon is small q0 x0= 3`B 2: compared to the energy of the electron, is adequate for our purOne further constraint could be obtained by requiring the pro- poses. This is not so for Compton scattering, because events duction rate of photons to be given precisely by the delta- in the\Klein-Nishina regime" turn out to be important. We function approximation. This leads to x0 0 46b 2 . However, therefore divide our treatment of scattering into two parts. For collisions occurring in the classical or Thomson regime we prefer the simple expression: we use a method closely similar to that used for synchrotron x0= b 2 (38) radiation. The electron loss term is: which corresponds formally to requiring that the approxima- Le (; t)= 4 UT@ 2 ne(; t) (42) R 3@ tion yields an accurate value for the moment dx (x; )x 0 37 . ics;T The appropriate photon source term follows simply from where Eqs. (36) and (37):

Qsyn

4`B 1 d n (; t) (x b 2 ) e 3b 1 p= 2`B b 3=2 x 1=2 ne ( x=b; t) 3

Z

UT=

Z xT

0

dx0 x0 n (x0; t)

(43)

is the photons the electron in(39) teractsenergy density ofCompton with whichin the Thomson via an inverse scattering