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Astron. Astrophys. 354, 321-327 (2000)

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3. Particle trajectories

In order to solve Eqs. (3), (4) and (8) we must specify the initial position and velocity of the particle. We will assume that the particles are initially at rest at some height [FORMULA] on the z-axis.

[EQUATION]

and

[EQUATION]

This assumption is justified if the particles' velocity and the size of the source region are much smaller than the macroscopic flow velocity and the ion gyroradius.

Integrating Eqs. (3) and (4) we obtain

[EQUATION]

[EQUATION]

so the Eq. (8) becomes

[EQUATION]

Using

[EQUATION]

and

[EQUATION]

we can write Eq. (13) as:

[EQUATION]

The solution to this equation with the initial conditions indicated above is:

[EQUATION]

Substituting the above expression into Eqs. (11) and (12) and integrating again we obtain

[EQUATION]

[EQUATION]

In contrast to the case when [FORMULA] is constant for which the particles necessarily have oscillatory trajectories we find that in the presence of a velocity shear there are 2 types of solutions depending on the value of the parameter [FORMULA]. As discussed by Perez-Tijerina et al. (1999), the particles describe oscillatory trajectories when

[EQUATION]

and hyperbolic trajectories when [FORMULA]. The motion of ions with hyperbolic trajectories is analyzed in Perez-Tijerina et al. (1999) where effects such as particle accelerations are also studied.

From Eq. (14) we see that [FORMULA] is real if [FORMULA] and the particles describe oscillatory trajectories according to:

[EQUATION]

[EQUATION]

[EQUATION]

In this case the particles path is a cycloid with period independent of the local velocity of the flow. However, the length of the cycloid, [FORMULA], depends on the local flow velocity as:

[EQUATION]

Fig. 2 shows the trajectories of particles at 3 different initial positions across the velocity shear. The dependence of the length of the cycloid with the coordinate z is immediately seen. For these cases we considered parameters of the solar wind and cometary ionosphere obtained with the spacecraft ICE near comet Giacobini-Zinner (Bame 1986); that is: [FORMULA], [FORMULA] and [FORMULA]([FORMULA] is the angle of magnetic field and y-axis). [FORMULA] correcting to [FORMULA] ions in a [FORMULA] magnetic field ([FORMULA]Gauss) with a boundary layer size of [FORMULA].

[FIGURE] Fig. 2. Trajectories of particles originating at 3 different initial positions for the case in which [FORMULA] and [FORMULA].

We can verify the agreement of our results with previous work if we consider the particular case, in which the magnetic field does not have a component in the direction of the flow, [FORMULA]. In this case, Eqs. (17)-(19) reduce to

[EQUATION]

[EQUATION]

[EQUATION]

which were presented by PD96.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000
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