3. Invariants and acceleration
The motion of a particle in a slowly time- and space-varying magnetic field in the absence of collisions can be characterized by a drift motion of the so-called center of gyration around which the particle performs its gyration orbit. For a particle of charge q, rest mass m and Lorentz factor , the perpendicular drift of the gyrocentre under the influence of a force (Alfvén & Fälthammar 1963) is given by
Up to first order in and in , where is the angular frequency, and the wave vector of the field perturbations, the total gyrocentre speed (in the case of a particle in the WD magnetosphere) is (Roederer 1970, Goldston & Rutherford 1995)
and we have assumed the drift motion to be sub-relativistic. The perpendicular drifts in Eq. (9) correspond to, respectively, the electric field drift, the gradient drift, the curvature drift, and the inertial drift. We have left out the drift due to gravity as it is relatively unimportant for particles which have enough energy to remain in the magnetic fields surrounding the white dwarf.
In the same approximation - as long as the cyclotron frequency of the particle is much larger than the oscillation frequency of the field, and as long as the spatial scale of the field is much larger than the particle cyclotron radius -the following quantity is conserved
which is the so-called first adiabatic invariant. Here is the particle momentum perpendicular to the magnetic field. It is important to realize that the quantity (11) relates to individual particles following their motion as they drift across the field. Therefore the perpendicular energy changes as
Finally, the evolution of the parallel energy is governed by (Goldston & Rutherford 1995)
Note that Eqs. (12) and (13) demonstrate that the total particle energy only changes in the presence of electric fields.
the longitudinal invariant, is conserved when the timescale of the field is much larger than the bounce time of a particle on a trapped orbit, , where (Lyons & Williams 1984)
and the last two equalities are for a dipole field with and is the particle pitch angle (the angle of the velocity vector to the local magnetic field direction) when it crosses the magnetic equator at a radial distance (the quantity is known as the magnetic shell parameter).
A third adiabatic invariant, the flux invariant, exists when the particle executes a bounce averaged periodic drift motion and the timescale of field changes is large compared to the bounce average drift period, . The flux invariant is defined as the magnetic flux linked in the particle's drift averaged over the bounce motion
where the last result is for a proton and .
Clearly as in our case the dominant perturbations in the fields are at periods below 33 s the third invariant does not exist. However, both the first (11) and the second invariant (14) exist as the inequalities (see Eq. (7)) and (see Eq. (15)) are easily satisfied for fast particles as long as they can be considered collisionless.
The above arguments suggest a number of single-step (non-stochastic) acceleration processes may occur in the magnetosphere of AE Aqr. For instance the magnetic field will vary in time by a factor of a few, and cause a similar increase in particle energy on account of (11). A more interesting possibility is when a particle somehow drifts from the companion to the surface of the white dwarf. Two magnetic shells of the rotating (dipolar) magnetosphere have an electric potential difference of magnitude
where is the magnetic flux trapped between both shells. This may lead to the production of TeV particles which have possibly been detected (Meintjes et al. 1994).
If collisions or fluctuations on a small spatio-temporal scale (either or ) occur the quantities (11) and (14) are not invariants during the short-lasting interaction. The combination of the existence of invariants during part of the time combined with pitch-angle scattering causes a rich variety of stochastic particle acceleration mechanisms to appear under periodic (instead of single-step) field changes, depending on the relative ordering of the scattering timescale and the particle orbital periods (e.g. Alfvén & Fälthammar 1963, and for planetary magnetospheres, Möbius 1994). Here we will concentrate on one such process of particular importance for AE Aqr; that of magnetic pumping (sometimes termed betatron acceleration).
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998