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

2. The Fokker-Planck equation: positron spectrum

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

where , is the positron Lorentz factor, is the dynamical friction (energy loss rate), is the energy dispersion rate, and are the annihilation and the particle escape rates, respectively, and 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,

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

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 . We are thus looking for the solution of Eq. (1) in the form , which gives an equation for the unknown function (Moskalenko 1995)

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

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

which is always fulfilled if the source function has the form and . A regular singular point 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 , that is the number of positrons with Lorentz factor 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 is the solution obtained after the i -th iteration, then the equation

with the initial condition ¹ allows us to get the next approximation of the solution. For Eq. (4), the condition could be taken. To start the iteration procedure one can use the Maxwell-Boltzmann distribution , 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) 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 . The combination of functions , where , on the place of in the right side allows sometimes to get a convergence faster.

© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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