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

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5. Numerical results

In 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 [FORMULA] on a spherical shell with radius [FORMULA]. 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+-pick-up ions mainly originating at regions close to the sun, while heliospheric pick-up protons in order to be described realistically would need the adequate consideration of a continuous injection all over the heliosphere according to the actual, local injection rates. However, for the above mentioned injection the source term in Eq. (5) attains the following form

[EQUATION]

where [FORMULA] (see Eq. (28)) and [FORMULA] is the Dirac delta function. This means that here we assume an injection of pick-up ions at [FORMULA] 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 U). Thus here we intend to treat a clean test case to be able to study a clearcut response to all diffusion effects which are operating. In a forthcoming paper we shall then also consider realistic, spatially distributed sources and continuous production rates.

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 [FORMULA] 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 [FORMULA], the ratio of abundances of backward and forward moving waves [FORMULA], and the dissipative scale [FORMULA], respectively. The last column gives the colatitude (n.b.: [FORMULA] corresponds to the ecliptic plane).


[TABLE]

Table 1. Parameters of runs


Figs. 1-5 show spatial distributions of pick-up protons at subsequent times [FORMULA] = 1, 2, 3, 4, 5, where the dimensionless time [FORMULA] (i.e the solar wind passage time over 1 AU) is introduced. Consequently a solar wind parcel, started at [FORMULA] = 0 from r = 1 AU, reaches at times [FORMULA] = 1, 2,... solar distances of r = 2, 3,...AU. The number density F of particles in Figs. 1-5 is given by the formula

[EQUATION]

where the constant [FORMULA] is defined with the normalization

[EQUATION]

[FIGURE] Fig. 1. Spatial distributions of pick-up protons injected at [FORMULA] = 0 and [FORMULA]. The numbers denote the subsequent moments: [FORMULA] = 1, 2, 3, 4, 5. a) run 1.

[FIGURE] Fig. 2. Spatial distributions of pick-up protons injected at [FORMULA] = 0 and [FORMULA]. The numbers denote the subsequent moments: [FORMULA] = 1, 2, 3, 4, 5. b) run 2.

[FIGURE] Fig. 3. Spatial distributions of pick-up protons injected at [FORMULA] = 0 and [FORMULA]. The numbers denote the subsequent moments: [FORMULA] = 1, 2, 3, 4, 5. c) run 3.

[FIGURE] Fig. 4. Spatial distributions of pick-up protons injected at [FORMULA] = 0 and [FORMULA]. The numbers denote the subsequent moments: [FORMULA] = 1, 2, 3, 4, 5. d) run 4.

[FIGURE] Fig. 5. Spatial distributions of pick-up protons injected at [FORMULA] = 0 and [FORMULA]. The numbers denote the subsequent moments: [FORMULA] = 1, 2, 3, 4, 5. e) run 5.

In Figs. 1-5 one can study the spatial broadening of pick-up protons which were initially injected according to a spatial [FORMULA]-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 ([FORMULA]) is closer to a radial field than in the second case ([FORMULA], 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 [FORMULA], [FORMULA], and [FORMULA] shown in Figs. 6 and 7 are given by the following formulae

[EQUATION]

[EQUATION]

[EQUATION]

where the following normalization is used:

[EQUATION]

[EQUATION]

Fig. 6 shows velocity distributions at the maxima of the spatial distributions at [FORMULA] and 5 (see Fig. 1), while Fig. 7 shows velocity distributions at r = 2AU at time [FORMULA] = 1 and r = 6AU at time [FORMULA] = 5, that is in a parcel comoving with the solar wind. The dashed curves labeled by (-) represent particles with [FORMULA] and the solid ones labeled by (+) represent particles with [FORMULA]. First of all one can identify considerable adiabatic cooling which is in accordance with most recent findings in observational data obtained with the SWICS instrument (Fisk et al., (1997)). Fig. 6 shows that at [FORMULA] = 1 the bulk of pick-up protons have negative pitch angles, that is, they move inward as judged by the solar wind reference frame (see Gloeckler et al., (1995); Möbius et al., (1997)). At [FORMULA] = 5 the distribution is almost isotropic. Another interesting feature in Fig. 6 is a shift between maxima of the curves (+) and (-) at [FORMULA] = 1 denoting that particles with negative pitch angles suffer more efficient cooling than particles with positive ones. An explanation of this effect has recently been given by Fisk et al. (1997): particles propagating inward in the frame of the solar wind have a longer than average dwell time in the inner heliosphere and as result of it suffer substantial adiabatic cooling. Fig. 7 as distinct from Fig. 6 shows velocity distributions at a parcel comoving with the solar wind. One can see that in this case particles with positive pitch angles prevail and even at [FORMULA] = 5 the distribution is still anisotropic.

[FIGURE] Fig. 6. Velocity distributions of pick-up protons corresponding to run 1 at times [FORMULA] = 1, 5. The dashed curves labeled by (-) represent particles with [FORMULA] and the solid ones labeled by (+) represent particles with [FORMULA] (see text). a) velocity distributions at the maxima of the spatial distributions at [FORMULA] = 1 and 5.

[FIGURE] Fig. 7. Velocity distributions of pick-up protons corresponding to run 1 at times [FORMULA] = 1, 5. The dashed curves labeled by (-) represent particles with [FORMULA] and the solid ones labeled by (+) represent particles with [FORMULA] (see text). b) velocity distributions at r = 2AU (at [FORMULA] = 1) and r = 6AU (at [FORMULA] = 5).

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 [FORMULA] shown in Figs. 8-9 is given by the formula

[EQUATION]

with the normalization

[EQUATION]

[FIGURE] Fig. 8. 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. a) run 1.

[FIGURE] Fig. 9. 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. b) run 2.

[FIGURE] Fig. 10. 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. c) run 3.

[FIGURE] Fig. 11. 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. d) run 4.

[FIGURE] Fig. 12. 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. e) run 5.

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 [FORMULA] 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. [FORMULA] = 0).

[FIGURE] Fig. 13. Velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. The numbers in the figures indicate various times. a) run 1.

[FIGURE] Fig. 14. Velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. The numbers in the figures indicate various times. b) run 2.

[FIGURE] Fig. 15. Velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. The numbers in the figures indicate various times. c) run 3.

[FIGURE] Fig. 16. Velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. The numbers in the figures indicate various times. d) run 4.

[FIGURE] Fig. 17. Velocity distributions of pick-up protons at the maxima of the spatial distributions for different run numbers. The numbers in the figures indicate various times. e) run 5.

Fig. 9 shows that the velocity distribution corresponding to run 2 is already isotropic at [FORMULA] = 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 [FORMULA] = 3 only. Fig. 10 shows the case when Alfvénic waves propagate only in one direction ([FORMULA] = 0). In this case the velocity distribution of the bulk of pick-up protons still is anisotropic even at [FORMULA] = 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 [FORMULA] (Fig. 12) than at [FORMULA] Fig. 11 at times [FORMULA] 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)).

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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