# Planar charged-particle trajectories in multipole magnetic fields

**Abstract.** 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* ∝ ϖ^{3}). 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 ϖ^{n}^{+2}. This result reduces to the classic one in the case of a zonal magnetic dipole (*n *=1). Planar trajectories exist in 2*m* meridional planes in the case of the general tesseral (0 < *m* < *n*) magnetic multipole. These meridional planes are defined by the 2*m* roots of the equation cos[*m*(*Φ* – *Φ _{n}^{m}*)] = 0, where

*Φ*= (1/

_{n}^{m}*m*) arctan (

*h*/

_{n}^{m}*g*);

_{n}^{m}*g*and

_{n}^{m}*h*denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (

_{n}^{m}*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 2

*m*(

*n*–

*m*) neutral lines exist at the intersections of the 2

*m*meridional planes with the (

*n*–

*m*) cones defined by the (

*n*–

*m*) roots of the equation

*P*(cos

_{n}^{m}*θ*) = 0 in the range 0 <

*θ*<

*π*, where

*P*(cos

_{n}^{m}*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 2

*m*equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ϖ

^{n}^{+2}and inversely proportional to cos[

*m*(

*Φ*–

*Φ*)]. Since this last expression vanishes at the 2

_{n}^{m}*m*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

*r*

^{n}^{+2}, where

*r*denotes radial distance from the multipole, and inversely proportional to

*P*(cos

_{n}^{m}*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 2

*m*(

*n*–

*m*) neutral lines located at the intersections of the 2

*m*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.