## 2. Governing equation and method of solutionThe relevant transport equation to describe the evolution of gyrotropic velocity distribution functions of pick-up ions in a background plasma moving at a velocity can be written in the following general form (Skilling, (1971), Isenberg, (1997) where is the vector of spatial variables in
an inertial frame (solar rest frame), In Eq. (3) , and 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: -
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 with respect to the radial direction as function of . -
The source term in Eq. (2) and hence the distribution function do not depend on longitude .
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 where = const (see e.g. ULYSSES data in Balogh et al., (1995); Smith & Balogh, (1995), , AU, and is the polar angle (colatitude) in the heliocentric spherical coordinate system (). With these assumptions the general transport Eq. (3) can be written in the form where and = arctg. 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 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)) In Eqs. (11-12) represents a two-variable
Wiener process (see e.g. Gardiner, (1990)) describing uncorrelated
variable fluctuations with Gaussian distributions, and
is a vector describing two statistically
independent differential increments and
satisfying a Gaussian probability distribution
with an e-folding-width increasing with time 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,
© European Southern Observatory (ESO) 1998 Online publication: June 18, 1998 |