PLANAR CHARGED-PARTICLE TRAJECTORIES IN MULTIPOLE MAGNETIC-FIELDS

This paper provides a complete generalization of the classic result th at the radius of curvature (rho) of a charged-particle trajectory conf ined to the equatorial plane of a magnetic dipole is directly proporti onal to the cube of the particle's equatorial distance (pi) from the d ipole (i.e. rho proportional to pi(3)). Comparable results are derived for the radii of curvature of all possible planar charged-particle tr ajectories in an individual static magnetic multipole of arbitrary ord er m and degree n. Such trajectories arise wherever there exists a pla ne (or planes) such that the multipole magnetic held is locally perpen dicular 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 m ultipole, provided n is odd: the radius of curvature varies directly a s pi(n+2). This result reduces to the classic one in the case of a zon al magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multi pole. These meridional planes are defined by the 2m roots of the equat ion cos[m(phi - phi(n)(m))] = 0, where phi(n)(m) = (1/m) arctan (h(n)( m)/g(n)(m)); g(n)(m) and h(n)(m) denote the spherical harmonic coeffic ients. Equatorial planar trajectories also exist if (n - m) is odd. Th e polar axis (theta = 0,pi) of a tesseral magnetic multipole is a magn etic 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 P-n(m)(cos theta) = 0 in the range 0 < theta < pi, where P-n(m)(cos theta) denotes the associat ed Legendre function. If(n - m) is odd, one of these cones coincides w ith the equator and the magnetic held is then perpendicular to the equ ator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to pi(n+2) and inversely proportional to cos[m(phi - phi(n)(m))]. Since t his last expression vanishes at the 2m equatorial neutral lines, the r adius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridion al trajectory is directly proportional to r(n+2), where r denotes radi al distance from the multipole, and inversely proportional to P-n(m)(c os theta)/sin theta. 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 intersecti ons of the 2m meridional planes with the (n - m) cones. Illustrative p article trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are presented for low-degree (n l ess than or equal to 3) magnetic multipoles. These computed particle t rajectories clearly demonstrate the ''non-adiabatic'' scattering of ch arged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.