An Asteroid in a Penning Trap
Year: 2026
Type: preprint
Abstract: The triangular Lagrange point of the restricted three-body problem and the radial motion of a charged particle in a Penning trap are both equilibria held at a maximum of an effective potential by a gyroscopic force. Each possesses one mode of negative energy, and each loses linear stability when its two frequencies collide—one at Routh’s mass ratio, the other at the trapping limit. Both systems belong to a single family of planar gyroscopic Hamiltonians. On every level of the conserved angular-momentum invariant, the degree-four normal form of any member at such a collision coincides with the reduced Hamiltonian of an axisymmetric Penning trap whose detuning is shifted by a term proportional to the invariant and whose quartic coefficient is fixed by a single normal-form invariant. On the level containing orbits that start at rest, this reduction yields closed-form laws for the largest excursion, the escape threshold, and the escape time. These laws depend only on the sign of the quartic coefficient and on scaled variables. At the triangular point the coefficient is positive, placing the collision in the soft class: orbits remain confined after stability is lost, with an excursion that grows like the square root of the distance past the critical value. Direct integration confirms the soft law to a fraction of a percent at the smallest detunings tested. Two traps whose quartic coefficient matches that of the triangular point reproduce the same law. Traps with the opposite sign escape at the predicted threshold and in the predicted times. A randomized ensemble of members confirms the classification by the sign of the coefficient. Two further results hold for every member of the family. Before the collision the linear transient gain is exactly the inverse square root of the frequency splitting. With isotropic velocity damping the first-order growth rates depend only on the undamped frequencies, and at the collision they scale with the square root of the damping rate. The first of these explains earlier reports of apparent escape just below Routh’s value as ordinary linear transients that reach a fixed radius. The second recovers the classical magnetron-instability formula as a special case. The results are local, planar, and classical. What transfers between the asteroid and the trap is the normal-form dynamics on each level of the invariant, the transient gain, and the damped linear rates. What does not transfer without further work is the behaviour beyond the normal-form neighbourhood, where the gravitational terms of every order cause the triangular point to deviate from any purely quartic trap.
Author: Doucette, Doug
Institution: SAIT Polytechnic
Language: English
Subfield: Aerospace Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225594337