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

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7. Relativistic regime

The heating of the pair plasma is leading to the situation where most of the energy is supported by the relativistic pair plasma. Here again the instability generating mechanism is the streaming of a beam in the plasma. The unusual streaming of a cold beam (the ambient medium) in a hot pair plasma needs a particular treatment of both growth rates and non-linear transfer. In a first step, we derive the dispersion relation of electrostatic, and electro-magnetic waves, and consider their destabilization by the backward streaming cold electron-proton beam.

All quantities are derived in the relativistic plasma frame where, for convenience, the primes are omitted.

7.1. Pair plasma electrostatic waves

The general dispersion relation for longitudinal modes in a magneto-active plasma is given by:

[EQUATION]

For a mono-energetic isotropic distribution function

[EQUATION]

adding an imaginary part to [FORMULA] to account for the Landau damping we easily obtain after integrating over µ

[EQUATION]

where [FORMULA] is an arbitrary small positive quantity introduced to treat the pole according to causality. This quantity describes the treatment of the resonant pole in Eq. (73).

Mainly we have three distinct regimes: [FORMULA], [FORMULA], and [FORMULA].

i. [FORMULA]:
By expanding to the lowest order in [FORMULA] the dispersion relation becomes for [FORMULA]

[EQUATION]

The beam instability implies [FORMULA]. Thus the resonant wave number [FORMULA] implies [FORMULA] and then no allowed resonance for such [FORMULA] values.

ii. [FORMULA]:
Always by expanding to the lowest order in [FORMULA] the dispersion relation becomes

[EQUATION]

The waves are evanescent in this [FORMULA] regime.

iii. [FORMULA]:
In his case we obtain

[EQUATION]

which leads to

[EQUATION]

For this regime, electrostatic waves can spontaneously propagate in the relativistic plasma. The cold perturbative medium contributes to the dispersion relation and may destabilize the waves. Injecting a mono-energetic distribution

[EQUATION]

where [FORMULA], and [FORMULA] is the beam density in the pair plasma frame we obtain

[EQUATION]

For the waves propagating backward with respect to the direction of motion of the beam, the solutions can be search with [FORMULA] where Z is complex and [FORMULA]. By expanding to the lowest order in Z this leads to a second order equation with a positive discriminant. No waves are destabilized in this direction.

For forward propagating waves [FORMULA], the dispersion relation is of the third degree in Z. Neglecting the factors with [FORMULA], the ratio [FORMULA] of the cold beam to the pair density [FORMULA] must verify the following condition to give rise to an instability:

[EQUATION]

For [FORMULA] the condition is not relevant. Even in this case there is no way for destabilizing electrostatic Langmuir modes.

This may be explain by the fact that the phase speed of electrostatic modes generated in a relativistic electron-positron beam is [FORMULA], ruling out any resonance with non-relativistic particles.

7.2. Pair plasma electro-magnetic waves

7.2.1. Dispersion relation and absorption

In a magneto-active plasma, the general form of the dispersion relation of both right and left electro-magnetic modes is given for any distribution function [FORMULA]:

[EQUATION]

a denotes the different species present in the plasma.

[EQUATION]

with [FORMULA] sgn [FORMULA].

For Alfvén modes [FORMULA]. The dispersion relation for the relativistic medium expressed in spherical coordinates in momentum space is

[EQUATION]

Where [FORMULA]. After integrating over µ, for both electron and positron contributions, the principal value leads to the dispersion relation

[EQUATION]

Which leads to the modified Alfvén speed of the relativistic medium:

[EQUATION]

The treatment of the poles allows to examine the absorption of these waves. It depends very sensitively to the following characteristic wave number [FORMULA]. At [FORMULA], the waves are over-damped by synchrotron absorption. However, the damping rate vanishes rapidly when k is sufficiently smaller than [FORMULA] (exponentially for an exponentially decreasing distribution, like a power law for a power law distribution). Thus, the dissipation range is restricted to k smaller but close to [FORMULA].

7.2.2. The proton back-streaming instability

We now turn to the contribution of the cold [FORMULA] beam to the dispersion relation. For [FORMULA]

[EQUATION]

The general dispersion relation regardless the Landau damping effects is:

[EQUATION]

Frequency regimes with [FORMULA] give a fortiori [FORMULA], and [FORMULA] can be neglected in the numerator of the cold plasma dispersion relation.

The left polarized modes are destabilized by the ions. In this case the resonant wave number is [FORMULA] for [FORMULA]. Then a backward propagating mode with [FORMULA] ([FORMULA] real and [FORMULA]) and with a frequency [FORMULA] (Z complex, and [FORMULA]) leads to a second order equation

[EQUATION]

A negative discriminant can be obtained for [FORMULA], as long as

[EQUATION]

in the relativistic case [FORMULA] and [FORMULA] leads to [FORMULA]. Therefore an instability is found for Alfvén waves that resonate with the proton back-stream. The maximum growth rate of pair Alfvén waves is

[EQUATION]

This instability exists only if [FORMULA], which puts a threshold that we will discuss later on.

The backward propagating right polarized Alfvén modes are not destabilized by the cold electron beam, since the discriminant of the corresponding second order equation is always positive.

The above mentioned condition [FORMULA] leads to an upper limit for the mean Lorentz factor of the relativistic population [FORMULA]. The factor [FORMULA] is a saturation value for the internal energy of the pair distribution due to their interaction with the cold protonic plasma. Below [FORMULA] the interaction with the cold ions contributes to increase the internal energy of the pair population, while above this value the pairs cool by Compton radiation.

7.3. Non-linear transfer

This particular instability does not evolve according to the quasi linear relaxation followed by random phase mode coupling. Indeed, the excited waves have a narrow band spectrum [FORMULA] about [FORMULA], namely [FORMULA] (Eq. (91)). A quasi monochromatic wave can grow and can probably trap the protons. In fact whatever the protons are trapped or not, a self modulation instability of the monochromatic wave will develop building non-linear wave packets (possibly solitons). The shortest wavelength components of these wave packets will undergo Landau-synchrotron damping on the pair plasma. Therefore, the ambient protons couple to the pair plasma through this non-linear process whose theory would deserve detailed investigations. This process should necessarily lead to a driving of the ambient protons and to the heating (by Landau-synchrotron absorption) of the pair plasma. As long as no other process is at work, a regulation mechanism brings the internal energy of the pair plasma to a saturation value. When this internal energy is smaller, the interaction with the ambient protons heats the pair plasma, and when the internal energy is larger, the interaction no more holds and the radiation losses cool the pair plasma. Further heating can be provided by the turbulence carried by the MHD jet (see HP).

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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