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| Titel |
Planar charged-particle trajectories in multipole magnetic fields |
| VerfasserIn |
D. M. Willis, A. R. Gardiner, V. N. Davda, V. J. Bone |
| Medientyp |
Artikel
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| Sprache |
Englisch
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| ISSN |
0992-7689
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| Digitales Dokument |
URL |
| Erschienen |
In: Annales Geophysicae ; 15, no. 2 ; Nr. 15, no. 2, S.197-210 |
| Datensatznummer |
250012627
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| Publikation (Nr.) |
 copernicus.org/angeo-15-197-1997.pdf |
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| Zusammenfassung |
| This paper provides a complete generalization
of the classic result that the radius of curvature (ρ) of a
charged-particle trajectory confined to the equatorial plane of a magnetic
dipole is directly proportional to the cube of the particle's equatorial
distance (ϖ) from the dipole (i.e. ρ ∝ ϖ3).
Comparable results are derived for the radii of curvature of all possible planar
charged-particle trajectories in an individual static magnetic multipole of
arbitrary order m and degree n. Such trajectories arise wherever
there exists a plane (or planes) such that the multipole magnetic field is
locally perpendicular to this plane (or planes), everywhere apart from possibly
at a set of magnetic neutral lines. Therefore planar trajectories exist in the
equatorial plane of an axisymmetric (m = 0), or zonal,
magnetic multipole, provided n is odd: the radius of curvature varies
directly as ϖn+2. This
result reduces to the classic one in the case of a zonal magnetic dipole (n =1).
Planar trajectories exist in 2m meridional planes in the case of the
general tesseral (0 < m < n)
magnetic multipole. These meridional planes are defined by the 2m roots
of the equation cos[m(Φ – Φnm)]
= 0, where Φnm = (1/m)
arctan (hnm/gnm); gnm and hnm denote the spherical harmonic coefficients. Equatorial planar trajectories also
exist if (n – m) is odd. The polar axis
(θ = 0,π) of a tesseral magnetic
multipole is a magnetic neutral line if m > 1. A
further 2m(n – m) neutral lines exist at
the intersections of the 2m meridional planes with the (n – m) cones defined by the (n – m) roots of the
equation Pnm(cos θ) = 0 in the range 0 < θ < π, where Pnm(cos θ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the
equator and the magnetic field is then perpendicular to the equator everywhere
apart from the 2m equatorial neutral lines. The radius of curvature of an
equatorial trajectory is directly proportional to ϖn+2
and inversely proportional to cos[m(Φ – Φnm)]. Since
this last expression vanishes at the 2m equatorial neutral lines, the
radius of curvature becomes infinitely large as the particle approaches any one
of these neutral lines. The radius of curvature of a meridional trajectory is
directly proportional to rn+2,
where r denotes radial distance from the multipole, and inversely
proportional to Pnm(cos θ)/sin θ;.
Hence the radius of curvature becomes infinitely large if the particle
approaches the polar magnetic neutral line (m > 1)
or any one of the 2m(n – m) neutral lines
located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived
by stepwise numerical integration of the exact equations of particle motion, are
presented for low-degree (n ≤ 3) magnetic multipoles.
These computed particle trajectories clearly demonstrate the
"non-adiabatic'' scattering of charged particles at magnetic neutral lines.
Brief comments are made on the different regions of phase space defined by
regular and irregular trajectories. |
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