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

6. Stochastic particle acceleration

The interactions between particles and turbulent fluctuations are characterized by an exchange of energy which can lead to the acceleration of the particles. This exchange is described by a diffusion equation of the particles in momentum space.

The quasi-linear coefficients describe scattering effects and can be evaluated with the stationary turbulent spectra. For the Langmuir waves:

In magnetized plasma the relevant diffusion coefficient is parallel to the magnetic field direction.

The Alfvén waves contribute essentially to the relaxation of the anisotropic part of the distribution function and tends to isotropize the particle distribution function with a pitch-angle scattering frequency:

The acceleration by Alfvén waves is of the second order in with respect to the pitch-angle frequency.

6.1. Derivation of the Fokker-Planck equation

In strong field regime, with an isotropic turbulence, we derive the diffusion equation in spherical coordinates. Therefore, for Langmuir turbulence we express the momentum diffusion coefficient and the pitch-angle scattering frequency in term of the previous parallel coefficient, namely for the Langmuir turbulence

For Alfvén turbulence we introduce an acceleration coefficient

The diffusion is balanced by the Inverse Compton losses given by the Eq. (3), which can be written as ; describes the losses for a Lorentz factor

where , and .
we obtain,

where .

6.2. Characteristic times

The previous stationary turbulent spectra straightly give the diffusion coefficients and the pitch-angle frequency.

For the Langmuir waves injecting Eq. (37) in Eq. (50) we obtain

The pitch-angle µ is in the range . The exact expression of the pitch-angle frequency is unknown since the core of the non-linear three waves interaction cannot be derived analytically. Thus, approximatively is only derived from the growth rate and the Eq. (51)

The ratio of the Langmuir acceleration time to the Alfvén angular diffusion time is

For a non relativistic thermal plasma with a kinetic temperature of order of the Compton temperature , the thermal speed of the ambient electrons is . The ratio of the cyclotron to the plasma frequency cannot be greater than in standard astrophysical conditions. The Langmuir turbulence is then inefficient at high energy, and all angular scattering and acceleration processes are dominated by Alfvén turbulence, even if the Lorentz factor (or a momentum ) which balances the two rates tends to increase with an increasing angle. Anyway, Langmuir turbulence is very efficient to accelerate particles up to Lorentz factors of order of ten to one hundred for hotter thermal electrons; which prevents an accumulation of low energy pairs (the so called "dead-end" problem) and ensures an injection above the threshold for resonant interaction with Alfvén waves.

As described in Sect. 2, the IC anisotropic emission of a relativistic particle in a soft photon field is strongly reduced in a small cone of opening angle . The particle distribution can then be divided in two different contributions depending on the pitch-angle of the particle.

The pitch-angle scattering by Alfvén turbulence is the fastest process inside the cone. Thus, the particle distribution function is weakly dependent of , and is obtained by balancing, in the Fokker-Planck equation, the IC cooling and the acceleration contributions. Note that at low energies, the distribution function is also angle independent because the angular diffusion is dominated by Langmuir waves whose pitch-angle frequency .

Outside the cone the IC cooling is the dominant energetical process and controls the shape of the distribution function.

6.3. Stationary solutions of the Fokker-Planck equation

6.3.1. Inside the cone

For , the distribution function in the disk frame can be described by

where .

At high energies, and for , is solution of

The frequency obtained from Eq. (8) is

For momenta in the absorption range , and from Eqs. (53) and (57) scales like . Here corresponds to the resonance with the mode of wave number , and . The acceleration time scales as p like the cooling time, leading to a power-law stationary solution with an index given by

For , particles resonate with the flatter transfer spectrum and thus the diffusion coefficient varies as with . This kind of spectrum can be associated with an external source of Alfvén turbulence from the accretion disk, at large scales (), and loaded by the MHD jet (see HP). In this range, the acceleration rate does not follow the variation of the IC cooling rate and the distribution function drops exponentially with p.

The maximum momentum can be estimated by the balance between the IC cooling and the acceleration times. Moreover, as these times have the same energy dependence this balance is verified for all momenta , and the spectral index . Thus, the main effect of the Langmuir turbulence at low energy and Alfvén turbulence at higher energies is to built a flat energy distribution function inside the cone, dominated by high energy pairs with a maximum Lorentz factor . The high energy particles are submitted to a strong pitch-angle scattering and are injected in the outer part of the acceleration cone. The angular diffusion by the Langmuir turbulence () dominates at small energy the angular diffusion by Alfvén waves (). The low energy pairs are re-injected in the inner part of the cone leading to a self-consistent mechanism (see Fig. 2).

Fig. 2. Scheme of the acceleration mechanism.

6.3.2. Outside the cone

For , the injection process described above can be treated as sources terms in the Fokker-Planck equation. We have neglected the width of the acceleration cone, so the injection occurs at .

The high energy particles are injected outside the cone at , the cooled low energy pairs are re-injected in the cone with a momentum . Therefore, we introduce a source term for the population outside the cone:

where is the source flux.

The inhomogeneous Fokker-Planck equation (Eq. (56)) can be written as

The external cooling frequency is of order of . Thereafter all the coefficients are replaced by there average value over µ; , , and . Expanding the distribution function in Legendre polynomials, , we can solve the exact solution of (64) for ; which gives

The spectral index is the positive root of a second degree algebraic equation. Since the acceleration rate is smaller than the IC cooling rate in this angular region, we get the following estimate:

The fundamental component (n = 0) dominates at high energy and thus we obtain a power-law distribution in . We therefore obtain a power-law distribution similar to that usually expected from shocks.

In a more general way, whatever are the momentum dependence of the diffusion coefficient and the pitch-angle scattering frequency, the Inverse Compton cooling process imposes the form of to be in .

In the general case keeping the acceleration coefficient in Eq. (64), the homogeneous solution is a pile-up distribution function

For a more general acceleration coefficient , the solution is

where . The index must be lower than 3, in order to have an IC cooling time () dominant in the outer part of the cone.

Moreover we can develop the source term as

The inhomogeneous distribution function can be cast into the form

where the heaviside function for and, for .

For , using the new variable , the stationary solution is a power-law function with an index equals to 4. Namely,

We can write the explicit value of the zero order coefficient of the previous expansion as

Therefore, for , the particle energy density distribution varies as, .

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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