## 5. Numerical resultsIn this first paper on pitch-angle and velocity diffusions of
pick-up ions in the solar wind we consider the time evolution of
particles injected at on a spherical shell
with radius . One should have in mind that this
can only somehow approximate the reality for earth-injected H-pick-up
ions of geocoronal origin or for the case of He where (see Eq. (28)) and
is the Dirac delta function. This means that
here we assume an injection of pick-up ions at
with vanishing velocity in the solar frame (i.e the original peculiar
velocity of the parent neutral atom is taken to be negligibly small in
comparison with the solar wind velocity The system of SDE's given by Eqs. (10-12) has been solved numerically by use of the Cauchy-Euler finite-difference method for a statistically relevant set of particles (in the present calculations about particles have been used). A set of calculations has been carried out for various values of the governing parameters to find out their role in determining the phase space evolution of pick-up protons in the solar wind. The essential information on all runs is presented in Table 1. Column 1 gives the run number. Columns 2, 3, and 4 give the level of Alfvénic turbulence at the Earth's orbit , the ratio of abundances of backward and forward moving waves , and the dissipative scale , respectively. The last column gives the colatitude (n.b.: corresponds to the ecliptic plane). Figs. 1-5 show spatial distributions of pick-up protons at
subsequent times = 1, 2, 3, 4, 5, where the
dimensionless time (i.e the solar wind passage
time over 1 AU) is introduced. Consequently a solar wind parcel,
started at = 0 from where the constant is defined with the normalization
In Figs. 1-5 one can study the spatial broadening of pick-up protons which were initially injected according to a spatial -function representing runs 1 through 5 given in Table 1. This broadening of the spatial distribution can be explained as due to an individual parallel (along the magnetic field lines) motion and thus spatial diffusion of protons with an anisotropic pitch-angle distribution. The pitch-angle anisotropy is also reflected by the fact that the average velocity of the pick-up proton bulk is smaller than the solar wind velocity. This is clearly seen in Figs. 1a,c-e. The maxima of spatial distributions in these figures are systematically shifted towards the Sun with respect to the isochronical solar wind bulk. Only in the case labeled as run 2 (see Table 1 and Fig. 2) the bulk of the pick-up protons comoves with the isochronical solar wind parcel. Run 2 differs from runs 1 and 3 by the adopted much higher level of turbulence, and from runs 4 and 5 by the fact that an identical level of turbulence, however here, together with an absence of a dissipative range in the turbulence spectrum is adopted (see Table 1). As we show in the following figures this results from the fact that the velocity distribution in run 2 is isotropic. As it was expected the maxima in Fig. 5 are closer to the Sun than in Fig. 4 since the magnetic field in the first case () is closer to a radial field than in the second case (, ecliptic plane!). To reveal the degree of the velocity anisotropy we present Figs. 6 and 7. These figures show pitch-angle averaged velocity distributions of pick-up protons corresponding to run 1 at times t = 1 and 5. The velocity distribution functions , , and shown in Figs. 6 and 7 are given by the following formulae where the following normalization is used: Fig. 6 shows velocity distributions at the maxima of the
spatial distributions at and 5 (see
Fig. 1), while Fig. 7 shows velocity distributions at
Figs. 8-12 show pitch-angle distributions of pick-up protons at the maxima of the spatial distributions (see Figs. 1-5 for different run numbers. The numbers in the figures indicate various times. The pitch-angle distribution function shown in Figs. 8-9 is given by the formula
Effects of velocity diffusion can be seen in Figs. 13-17. These figures show velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. As usually the numbers in the figures indicate various times. The velocity distribution function is given by Eqs. (38),(41). Considerable stochastic acceleration of pick-up protons takes place only for a sufficiently high level of turbulence (Fig. 14). The presence of the dissipation range in the spectrum of magnetic field fluctuations reduces the rate of acceleration (Fig. 16). As it could be expected, stochastic acceleration does not operate in the case when Alfvénic waves propagate only in one direction (i.e. = 0).
Fig. 9 shows that the velocity distribution corresponding to run 2 is already isotropic at = 1 (see the shapes of the spatial distributions in Fig. 1b), while in the case with the lower level of turbulence (Fig. 8) isotropy is reached at = 3 only. Fig. 10 shows the case when Alfvénic waves propagate only in one direction ( = 0). In this case the velocity distribution of the bulk of pick-up protons still is anisotropic even at = 5 because of the well-known resonance gap in the pitch-angle scattering (Schlickeiser, (1989)). Comparison between runs 2 and 4 (Figs. 9 and 11, respectively) which differ only by the presence or absence of the dissipation range in the last case shows that a deficit of high frequency waves in the solar wind can essentially modify the process of scattering towards isotropy. More isotropic velocity distribution of pick-up protons at (Fig. 12) than at Fig. 11 at times can be explained by the fact that adiabatic focusing transports particles from negative pitch angles to positive ones more efficiently under radial magnetic field conditions (it can be seen from Eqs. (9), (12)). © European Southern Observatory (ESO) 1998 Online publication: June 18, 1998 |