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Astron. Astrophys. 353, 473-478 (2000)

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2. Analysis of forces in a rotating reference frame

Usually the motion of a particle along rotating magnetic field lines is treated in the bead-on-the-wire approximation where a bead is assumed to follow the rotating field line and experiences centrifugal acceleration (or deceleration) while moving in the outward direction (e.g. Machabeli & Rogava 1994; Chedia et al. 1996; Cao 1997). This simple approach yields interesting results, though, as we will show further below, such an approximation breaks down in the region near the light cylinder.

Let us consider the forces acting on a particle in a rotating frame of reference (Gangadhara 1996; Gangadhara & Lesch 1997). A particle with rest mass m and charge q, which is injected at time [FORMULA] and position [FORMULA] with initial velocity [FORMULA] parallel to the magnetic field line [FORMULA] experiences a centrifugal force in the radial direction given by

[EQUATION]

where [FORMULA] is the Lorentz factor of the particle and [FORMULA] is the angular velocity of the field. Additionally, there is also a relativistic Coriolis force in the noninertial frame governed by the equation

[EQUATION]

which acts as a deviation-force in the azimuthal direction. In the inertial rest frame the particle sees the field line bending off from its initial injection position. Hence, it experiences a Lorentz force, which may be written as

[EQUATION]

where [FORMULA] is the relative velocity between the particle and the magnetic field line and where the convention [FORMULA] is used. Due to the Lorentz force a charged particle tries to gyrate around the magnetic field line. Initially, the direction of the Lorentz force is perpendicular to the direction of the Coriolis force, but as a particle gyrates, it changes direction and eventually becomes antiparallel to the Coriolis force. Hence one expects that the bead-on-the-wire approximation holds, if the Lorentz force is not balanced by the Coriolis force. In this case, the accelerated motion of the particle's guiding center due to the centrifugal force is given by

[EQUATION]

where r denotes the radial coordinate and [FORMULA]. The bead-on-the-wire motion for the guiding center breaks down, if the Coriolis force exceeds the Lorentz force, i.e. if the following inequality, given by the azimuthal components of the forces, holds:

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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