Fundamental Technologies Galileo Spacecraft Pages

2.3 Adiabatic Invariants

The guiding center equation is an approximation to the Lorentz force equation. It simplifies matters enough to deduce the three basic periodic motions that charged particles perform in magnetospheres: gyration, bounce, and drift. However, the drift [2.16] and mirror [2.13] equations must still be integrated over several degrees of drift and many bounce periods to predict the future state of the particles. This is still a problem even with today’s computational power for most cases.

Fortunately, each of the three typical, nearly periodic, motions have associated with them an adiabatic invariant. These adiabatic invariants are conditional on the fact that the forces providing the motion must vary infinitesimally slowly when compared to the time scales of the particle motion. The quantity called action remains invariant for slow changes in the system:

where p is the conjugate momentum and q is the generalized coordinate. Each of the three invariants is derived from Equation [2.17].

• 2.3.1 The First Adiabatic Invariant
• 2.3.2 The Second Adiabatic Invariant
• 2.3.3 The Third Adiabatic Invariant

Next: 2.4 Pitch Angle Diffusion

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Updated 10/18/02, T. Hunt-Ward
tizby@ftecs.com