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Astron. Astrophys. 335, 746-756 (1998)

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2. Governing equation and method of solution

The relevant transport equation to describe the evolution of gyrotropic velocity distribution functions [FORMULA] of pick-up ions in a background plasma moving at a velocity [FORMULA] can be written in the following general form (Skilling, (1971), Isenberg, (1997)

[EQUATION]

where [FORMULA] is the vector of spatial variables in an inertial frame (solar rest frame), v and [FORMULA] are the velocity and cosine of the particle pitch angle [FORMULA] in the solar wind rest frame, [FORMULA] is the unit vector of the large-scale magnetic field, Q is the local production rate of particles, and [FORMULA] is the scattering operator applied to the function f (see e.g. Schlickeiser, 1989)

[EQUATION]

In Eq. (3) [FORMULA], [FORMULA] and [FORMULA] are the respective Fokker-Plank diffusion coefficients which will be specified below.

Within the present paper we shall study the evolution of pick-up ion distribution functions in the solar wind environment under the following simplifying assumptions:

  1. The solar wind flow is assumed to be a stationary spherically-symmetric radial flow with U=const. Thus the latitudinal dependences of the solar wind velocity and number density are ignored here. Nevertheless the distribution function of pick-up ions is found as a function of latitude because of the latitudinal dependence of the interplanetary magnetic field, i.e its tilt angle [FORMULA] with respect to the radial direction as function of [FORMULA].

  2. The source term in Eq. (2) and hence the distribution function do not depend on longitude [FORMULA].

In a forthcoming study we shall also take into account these two abovementioned complications. The large-scale interplanetary magnetic field is assumed to have the standard Parker spiral configuration

[EQUATION]

where [FORMULA] = const (see e.g. ULYSSES data in Balogh et al., (1995); Smith & Balogh, (1995), [FORMULA], [FORMULA] AU, and [FORMULA] is the polar angle (colatitude) in the heliocentric spherical coordinate system ([FORMULA]). With these assumptions the general transport Eq. (3) can be written in the form

[EQUATION]

where [FORMULA] and [FORMULA] = arctg[FORMULA]. Here and in the following an outward pointing magnetic field is adopted. The second term on the left-hand side of Eq. (5) describes the convective motion of particles, the third term corresponds to the so-called adiabatic cooling, and the fourth one to adiabatic focusing.

Since Eq. (5) does not contain a derivative of f with respect to [FORMULA] it can be solved by specifying the latitude and integrating with respect to r. If in addition we introduce the differential number density of particles in phase space [FORMULA] with [FORMULA], Eq. (5) can then be rewritten in the conservation-law form as

[EQUATION]

where

[EQUATION]

[EQUATION]

[EQUATION]

Eq. (6) can either be solved directly by use of a finite-difference method or by an alternative approach calculating stochastic trajectories of particles in phase space. In this last case Eq. (6) is replaced by an equivalent system of stochastic differential equations (SDE's) (concerning the mathematical equivalence between Fokker-Plank equations and SDE's one may study e.g. Gardiner, (1990)). The SDE method has been used, for example, to model transport and acceleration of electrons (MacKinnon & Craig, (1991)), cosmic rays (Achterberg & Krülls, (1992); Krülls & Achterberg, (1994)), and pick-up ions (Chalov et al., (1995, 1997); Chalov & Fahr, (1996); Fichtner et al., (1996)). It can be shown that Eq. (6) is equivalent to the following system of SDE's (for more details see Chalov et al., (1997))

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

In Eqs. (11-12) [FORMULA] represents a two-variable Wiener process (see e.g. Gardiner, (1990)) describing uncorrelated variable fluctuations with Gaussian distributions, and [FORMULA] is a vector describing two statistically independent differential increments [FORMULA] and [FORMULA] satisfying a Gaussian probability distribution with an e-folding-width increasing with time dt according to

[EQUATION]

This distribution itself results as the solution of a Fokker-Planck type diffusion equation.

Each solution of Eqs. (10-12) is a stochastic trajectory in phase space. To obtain the differential number density of particles, N, we must simulate a statistically relevant set of stochastic trajectories and determine the corresponding density of these trajectories in phase space. To integrate Eqs. (10-12) numerically a simple Cauchy-Euler finite-difference method can be used.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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