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Astron. Astrophys. 325, 401-413 (1997)

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2. The Fokker-Planck equation: positron spectrum

Assuming the isotropy of the positron energy distribution function [FORMULA], the steady-state Fokker-Planck equation takes the form

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA] is the positron Lorentz factor, [FORMULA] is the dynamical friction (energy loss rate), [FORMULA] is the energy dispersion rate, [FORMULA] and [FORMULA] are the annihilation and the particle escape rates, respectively, and [FORMULA] is the positron injection term.

In the steady-state regime, without sources and sinks, the kinetic coefficients obey the equation (Lifshitz & Pitaevskii 1979) which results from the absence of the flux density in the energy space,

[EQUATION]

where [FORMULA] is to be a Maxwellian distribution [FORMULA] (kT is the dimensionless plasma temperature). This equation fixes a relation between the coefficients

[EQUATION]

We emphasize that the coefficients of the Fokker-Planck equation have an additive property. They represent the sum of coefficients for various processes which have to be evaluated separately.

Although the plasma cloud serves as a thermostat with true Maxwellian distribution, annihilation and sinks distort the distribution [FORMULA]. We are thus looking for the solution of Eq. (1) in the form [FORMULA], which gives an equation for the unknown function [FORMULA] (Moskalenko 1995)

[EQUATION]

[EQUATION]

Eliminating [FORMULA] in favour of [FORMULA] yields the integro-differential equation for the distorted function

[EQUATION]

[EQUATION]

while [FORMULA] at [FORMULA] was assumed (cf. Eq. (2)). The last term in Eq. (5) follows simply from conservation of the total number of positrons

[EQUATION]

which is always fulfilled if the source function has the form [FORMULA] [FORMULA] and [FORMULA]. A regular singular point [FORMULA] in Eq. (5) does not lead to any singularity of the solution, which is Maxwellian-like at the low-energy part. Equation (6) gives also an idea of physical meaning of term [FORMULA], that is the number of positrons with Lorentz factor [FORMULA] escaping from the plasma volume per 1 sec. The approach can be easily generalized to include inelastic processes, stochastic acceleration etc.

Eq. (4) or (5) can be resolved numerically with an algorithm which reduces it to a first-order differential equation. Let [FORMULA] is the solution obtained after the i -th iteration, then the equation

[EQUATION]

[EQUATION]

with the initial condition 1 [FORMULA] allows us to get the next approximation [FORMULA] of the solution. For Eq. (4), the condition [FORMULA] could be taken. To start the iteration procedure one can use the Maxwell-Boltzmann distribution [FORMULA], although, in some cases, when a solution of Eq. (7) deviates strongly from Maxwellian, that causes a deviation in normalization during first iterations. Since a solution of Eq. (7) [FORMULA] multiplied by a constant would be also a solution, it has to be normalized in the end of iteration process. This algorithm converges quickly and gives a good approximation of the solution already after several iterations. The actual signature of the convergence could be an equality [FORMULA]. The combination of functions [FORMULA], where [FORMULA], on the place of [FORMULA] in the right side allows sometimes to get a convergence faster.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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