Bounce‐averaged Hamiltonian for charged particles in an axisymmetric but nondipolar model magnetosphere

Abstract

In order to facilitate bounce‐averaged guiding center simulations of geomagnetically trapped particles, we express the kinetic energy of a particle with magnetic coordinates (L,ϕ) as an analytic function of the first two adiabatic invariants (M, J) and the L value of the field line. The magnetic field model is axisymmetric, consisting of a dipolar B field plus a uniform southward magnetic field parallel to the dipole moment μE. This model magnetosphere is surrounded by a circular equatorial neutral line whose radius b is an adjustable parameter. The L value of a field line is (by definition) inversely proportional to the flux enclosed by the corresponding magnetic shell of equatorial radius r0, and the L value at the neutral line (r0 = b) is denoted L*. The azimuthal coordinate ϕ measures magnetic local time. The best functional representation found for the normalized difference (L³a³/μE)(Bm ‐ B0) between mirror‐point field Bm and equatorial field B0 along any field line is a 5‐term expansion in powers (2/3 through 6/3) of the quantity X ≡ (La/μE)1/2K, where K ≡ (J²/8m0M)1/2 is an adiabatically conserved quantity independent of particle energy, m0 is the rest mass of the particle, and a is the radius of the Earth. This functional form is motivated by results for limiting cases in which particles mirror very near and very far from the magnetic equator. Expansion coefficients corresponding to various powers of X are obtained from least squares fits to numerically computed results for X as a function of L and Bm. These are accurately expressible as fourth‐order polynomials in (r0/b)³, hence indirectly as functions of L/L* = 3La/2b. This representation, which leads (except for a manageably small region of parameter space) to better than 1% accuracy in the specification of Bm as a function of K and L, allows bounce‐averaged guiding center simulations to be performed without actually tracing the bounce motions of individual particles. Bounce‐averaged drifts L (meridional) and ϕ (azimuthal) are proportional to derivatives of the Hamiltonian H (sum of kinetic and potential energies) with respect to ϕ and L, respectively. Our formulation thus provides a computationally efficient method for tracing the bounce‐averaged adiabatic motion (conserving all three invariants) and nonadiabatic transport (violating the third invariant while conserving the first two invariants) of geomagnetically trapped particles in the model magnetosphere.