Perturbation analysis of trapped-particle dynamics in axisymmetric dipole geometry

The bounce-action-angle coordinates (J,zeta) for charged particles trapped in an axisymmetric dipole magnetic field are constructed by perturbation analysis. First, the lowest-order bounce-action-angle coordinates (J(0), zeta(0)) are derived for deeply trapped particles in the harmonic-oscillator approximation. Next, the Lie-transform perturbation method is used to derive higher-order anharmonic action-angle corrections (J=J(0)+epsilon(t)J(1), zeta = zeta(0) + epsilon(t)zeta(1)), where the dimensionless parameter epsilon(t) (s(b)/r(e))(2) << 1 is defined as the ratio of the turning-point distance vertical bar s(b)vertical bar (measured from the equator) along a magnetic field line labeled by the equatorial distance r(e). Explicit expressions ( with anharmonic corrections) for the canonical parallel coordinates s(J,zeta) and p(parallel to)(J, zeta) are presented, which satisfy the canonical identity {s, p(parallel to)} 1. Lastly, analytical expressions for the bounce and drift frequencies (which include anharmonic corrections) yield excellent agreement with exact numerical results. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3486554]