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Astron. Astrophys. 323, 271-285 (1997)

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2. Anisotropic Compton losses

As made precise in the introduction the acceleration of the relativistic particles strongly competes with the Inverse Compton (IC) cooling process of these pairs on soft photons coming from an accretion disk or from synchrotron radiation. We consider here only the former case as developed in MHP. The reader should refer to this paper for a complete treatment of the anisotropic Inverse Compton process in the Thomson regime. However, we sum up here the main physical results.

In the Thomson regime, in an incident soft radiation field given by [FORMULA], an electron with a Lorentz factor [FORMULA] must emit a power (Blumenthal & Gould (1970)):

[EQUATION]

Where [FORMULA] is the direction between the electron and the soft photon. The solid angle subtended by an unit vector in the direction of the electron (of the photon) is [FORMULA] ([FORMULA]). The factor [FORMULA] is the soft photon energy in [FORMULA] unit. Whatever the source of soft photons is, we can characterize the soft radiation field by its Eddington parameters:

[EQUATION]

For a relativistic particle the IC power per unit of solid angle is emitted in a sharp cone of opening angle [FORMULA], and takes the form

[EQUATION]

We have defined [FORMULA] (and [FORMULA]) the cosine of the angle between the electron (the soft photon) direction and the jet axis.

If the forward direction corresponds to [FORMULA], then the anisotropy of the incident radiation field favors the emission in the backward direction. The particles moving forward are submitted to softer radiation losses, and the pair plasma will then be accelerated along the jet axis. This is the so-called Compton rocket effect (O'Dell (1981)) which expresses the transfer of stochastic internal energy of the plasma in bulk motion along the jet axis. The transfer to the momentum of the relativistic particle is given by

[EQUATION]

Where [FORMULA] and [FORMULA] are respectively the photon and the particle directions. We can define a particular frame moving with the velocity [FORMULA] to the respect of the observer frame where the mean particle speed [FORMULA]. In this frame the relativistic particle distribution is supposed to be isotropic. In the soft photon source frame a saturation velocity [FORMULA] of the plasma can be obtained by the cancellation of the parallel IC force integrated over the particle distribution (O'Dell (1981)). Namely,

[EQUATION]

or

[EQUATION]

and where [FORMULA].

For [FORMULA] and [FORMULA], where [FORMULA], the saturation Lorentz factor is at the first order in [FORMULA], and [FORMULA]

[EQUATION]

The bulk Lorentz factor [FORMULA] will then closely follow [FORMULA] until the relaxation time of [FORMULA] to [FORMULA] equals the evolution time of [FORMULA]. The evolution of [FORMULA] with the distance above the source, and the final value of the bulk Lorentz factor both depends on the kind of the soft photon source (see MHP).

Let us now consider Eq. (3). The anisotropy leads to an IC emitted power strongly reduced in the inner part of a cone of opening angle [FORMULA]. The minimal emitted power is obtained for [FORMULA], namely

[EQUATION]

For [FORMULA] the emitted power is twice [FORMULA], and we can rewrite Eq. (3) as

[EQUATION]

where, at the first order, we have put [FORMULA] in the second member of (9).

The angle [FORMULA] is then given by

[EQUATION]

We can then defined a cone of opening angle [FORMULA] where the IC losses are strongly reduced such that the internal and the external IC emitted power verify [FORMULA]. This result is independent of the nature of the soft photon source.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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