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Astron. Astrophys. 324, 395-409 (1997)

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2. Energy-loss rates

We first consider in detail the various processes through which the pairs inside a new AGN jet component lose energy. The blob is assumed to move outward perpendicularly to the accretion disk plane with velocity [FORMULA] where [FORMULA] is the Lorentz factor of the bulk motion. The mechanisms that we take into account are inverse-Compton scattering of accretion disk photons, synchrotron and synchrotron-self-Compton (SSC) losses. It is well-known that energy exchange/loss due to elastic (Moller and Bhabha scattering) and inelastic scattering (pair bremsstrahlung emission) do not contribute significantly for ultrarelativistic particles. We consider them in detail in the second paper of this series, dealing with mildly-relativistic pair plasmas. The same is true for pair annihilation losses.

2.1. External inverse-Compton losses

We are now considering the single-particle energy loss rate due to inverse-Compton scattering of external photons coming directly from a central source, which we assume to be an accretion disk. We use the accretion disk model of Shakura & Sunyaev (1973) predicting, for a central black hole of [FORMULA] - [FORMULA], a blackbody spectrum according to a temperature distribution [FORMULA] given by

[EQUATION]

[EQUATION]

In general, the energy loss rate due to inverse-Compton scattering is given by the manifold integral

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA],

[EQUATION]

[EQUATION]

Here, primed quantities refer to the rest frame of the electron, and the subscript 's' denotes quantities of the scattered photon. The definition of the angles is illustrated in Fig. 2. Now, let the superscript ' [FORMULA] ' denote quantities measured in the rest frame of the accretion disk. Then, the differential number of accretion disk photons in the blob frame is

[EQUATION]

[EQUATION]

Using this photon number and the full Klein-Nishina cross section, Eq. (5) becomes

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

and [FORMULA] and [FORMULA] are the radius of the inner and outer edge of the accretion disk, respectively.

If all scattering occurs in the Thomson regime ([FORMULA]), we find (neglecting terms of order [FORMULA]):

[EQUATION]

[EQUATION]

where [FORMULA]. Fig. 3 shows the energy loss rate computed using the full Klein-Nishina cross-section compared to the calculation in the Thomson limit as quoted above as well as the calculation in the Thomson limit, combined with a point-source approximation for the accretion disk (e. g. Dermer & Schlickeiser 1993).

[FIGURE] Fig. 2. Definition of the angles for calculation of the inverse-Compton losses. 'L' in the right panel denotes the motion of the labor frame with respect to the electron rest frame. The subscript `ph' refers to quantities of the photon
[FIGURE] Fig. 3. Energy loss rates of a test electron/positron due to inverse-Compton scattering of accretion disk photons. Total disk luminosity: [FORMULA] erg s-1, height: [FORMULA] pc

For electron energies of [FORMULA], Eq. (12) is a very good approach to the Inverse-Compton losses. In the case of small distances to the accretion disk ([FORMULA] pc in the case of a disk luminosity [FORMULA] erg s-1 ; [FORMULA] pc for [FORMULA] erg s-1) the point-source approximation is not an appropriate choice.

A very useful approximation for all electron energies is based on replacing the integration [FORMULA] by setting [FORMULA]. Furthermore, one can approximate the thermal spectrum, emitted by each radius of the disk, by a [FORMULA] function in energy, [FORMULA] where

[EQUATION]

With these simplifications, and neglecting terms of order [FORMULA], Eq. (9) becomes

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

and

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The integral I can be solved analytically, yielding

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

For small values of E, one should use the Taylor expansion of Eq. (15), namely

[EQUATION]

[EQUATION]

This result (using Eq. [16] for [FORMULA]) is illustrated by the dot-dashed curve in Fig. 3. In Fig. 5, the inverse-Compton losses (according to Eq. [13]) for set of parameters which is assumed to be typical for BL Lac objects ([FORMULA], [FORMULA] pc, [FORMULA]) are compared to the synchrotron and SSC energy losses derived in the following subsections.

2.2. Synchrotron and synchrotron-self-Compton losses

If, for the synchrotron (sy) losses, we neglect inhomogeneities and effects of anisotropy, we have

[EQUATION]

[EQUATION]

(e. g., Rybicki & Lightmann 1979) where [FORMULA].

Evaluating the single-particle energy loss rate due to the SSC process, we restrict ourselves to regarding only single scattering events. If the particle distribution functions are given by power-laws, the synchrotron photon spectrum is also described by a power-law. However, since we are intrested in the detailed shape of the distributions, deviating from a simple power-law, we have to calculate the synchrotron spectrum in more detail.

In an optically thin source, the differential number of synchrotron photons in the energy interval [FORMULA] is given by

[EQUATION]

[EQUATION]

The summation symbol denotes the sum of the contributions from electrons and positrons to the synchrotron spectrum. Averaging over all pitch-angles of the electrons, the spectral emissivity [FORMULA] of a single electron of Lorentz factor [FORMULA] can be expressed as

[EQUATION]

(Crusius & Schlickeiser 1986) where [FORMULA] is the finestructure constant,

[EQUATION]

[FORMULA] is the non-relativistic electron/positron gyrofrequency, [FORMULA] is the plasma frequency of the relativistic pair plasma, i. e.

[EQUATION]

[EQUATION]

and [FORMULA] are the Whittaker functions.

If the particle density exceeds [FORMULA] cm-3 synchrotron self absorption (SSA) is negligible for the following two reasons: The optical depth due to synchrotron self absorption for an initial power-law distribution [FORMULA] can be estimated as

[EQUATION]

(Schlickeiser & Crusius 1989). This optical depth becomes unity for

[EQUATION]

where [FORMULA] is the lower cutoff Lorentz factor in units of [FORMULA], [FORMULA] is the magnetic field in units of 0.1 G, [FORMULA] is the pair density in units of [FORMULA] cm-3 and [FORMULA] is the blob radius in units of [FORMULA] cm. The frequency [FORMULA] is of the same order as the Razin-Tsytovich frequency

[EQUATION]

where the luminosity is strongly suppressed due to plasma effects. Here, [FORMULA] denotes the average electron Lorentz factor. This demonstrates that for pair densities [FORMULA] cm-3 synchrotron self absorption can be neglected. For lower densities we include it in our calculations, evaluating

[EQUATION]

(e. g., Rybicki & Lightman, 1979) self-consistently.

From the point of view of reacceleration, synchrotron self absorption can be neglected since particles resonating with photons at the lower cut-off of the synchrotron spectrum should have Lorentz factors of

[EQUATION]

which is lower than the particle Lorentz factors we deal with in the simulations carried out in this paper.

If all scattering occurs in the Thomson limit, [FORMULA], the first order SSC energy loss rate is easily determined by

[EQUATION]

where the synchrotron luminosity can be calculated as

[EQUATION]

yielding for the initial power-law distribution functions

[EQUATION]

[EQUATION]

Discarding the Thomson approximation and using the notation of the previous subsection, the exact expression for the SSC energy losses is

[EQUATION]

[EQUATION]

In this case of an isotropic radiation field in the blob frame, the integrations over [FORMULA] and [FORMULA] can be solved analytically if we neglect terms of order [FORMULA] and of order [FORMULA]. This yields

[EQUATION]

where [FORMULA] cm-3 is the energy loss rate of an electron of energy [FORMULA] scattering an isotropic, monochromatic radiation field of photon energy [FORMULA] and photon density 1 cm-3 and

[EQUATION]

[EQUATION]

[EQUATION]

For small values of [FORMULA] (i. e. the Thomson limit) the Taylor expansion

[EQUATION]

to lowest order reduces Eq. (32) to Eq. (28) for [FORMULA]. In Fig. 4, the analytic solution (32) is compared to the Thomson scattering result (Eq. [27]).

[FIGURE] Fig. 4. Energy loss rates of a test electron/positron due to inverse-Compton scattering of synchrotron photons. [FORMULA] G, [FORMULA] cm, [FORMULA] cm-3. Thomson-scattering (dotted curves) and exact Klein-Nishina cross-section (solid curves)

It should be noted that for such an ultrarelativistic particle distribution Compton scattering its own synchrotron photons, Klein-Nishina corrections do not lead to a break in the energy dependence of the energy-loss rate, but due to the fact that with increasing energy a decreasing fraction of the scattering events occurs in the Thomson regime, lead to a much smoother flattening of the energy depencence than what is obtained from the relatively sharp soft photon distribution coming from the accretion disk. In the energy regime considered here, the energy-loss rate scales as [FORMULA] with [FORMULA] for the spectral index [FORMULA] of the particle distributions [FORMULA]. Clearly, this effect depends strongly on the temporal particle distribution which determines the synchrotron spectrum.

The result of Eq. (32) (using Eq. [34] for [FORMULA]) is included in Fig. 5 for a magnetic field strength of [FORMULA]  G and a blob radius of [FORMULA] cm. Higher-order SSC scattering (up to n th order in the n th time step) is incorporated in our simulations by replacing [FORMULA] by [FORMULA] in Eq. (32).

[FIGURE] Fig. 5. Energy loss rates of a test particle (Lorentz factor [FORMULA]) due to inverse-Compton scattering of external (accretion disk) photons (EIC; disk luminosity [FORMULA], [FORMULA], [FORMULA] pc, [FORMULA]) and of synchrotron photons (SSC; [FORMULA] G, [FORMULA] cm, [FORMULA] cm-3, [FORMULA]), and to synchrotron emission ([FORMULA])
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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