http://www.lpi.usra.edu/meetings/lpsc1992/pdf/1135.pdf
The two body dynamics of an orbiting particle is usually governed by Newton's
laws of gravity and mechanics. Here the Kepler problem has been studied with the
addition of the Lorentz force of magnetism. For micrometeoroids and small man made
debris particles electromagnetic interactions may become important. The space
environment is filled with charged particles, electrons and ions, that may be deposited on
a 1 - 100 microns sized particle. The orbits of such charged particles have been
modeled in earth orbit, with the dipole magnetic field superposed on the gravity field.
An initial study of this problem has been conducted. Figures 1 illustrates a 600
second time frame of particles in a 300 kilometer altitude Earth orbit with a 5.7'
inclination. The solid line is the pure Kepler problem. The dashed lines are the paths
taken by various sized particles with a single positive charge of 1.6x10-'~ coulombs.
The additional energy required to deflect these particles from a Kepler orbit
comes from the deposition of charge at the initial stage in the calculation. From there
the energy of the particle is conserved since both the gravitational and magnetic forces
are conservative, i.e. $F d r = 0.
The orbit of 1 micron sized particles has been computed and the results a r e
shown in figures 2 through 5. The results of this numerical exercise in magneto-orbital
dynamics a r e rather unexpected, but appear physically correct when studied. The
particle was set in a 2000km orbit with a 5.7' inclination. Figure 2 displays how the
magnetic field torques the orbital plane. Figure 3 then shows how the orbit is flipped
over and shoved toward the north pole. This is analogous to the spin orbit interaction in
the quantum atom. He r e the magnetic moments of the Earth's magnetic field and the
orbit a r e parallel. The flip in the orbit is a result of the system's requirement to reach a
lower magnetic energy by antiorienting the magnetic moments. The energy lost by
reorienting the magnetic moment goes into gravitational potential energy by shoving the
orbit above the north pole.
Since this model is classical, and the electromagnetic radiation emitted by the
accelerating charge is negligible, events depart from the Bohr quantum analog above.
The charged particle finds itself in a magnetic bottle near the north pole, figure 4. The
spiralling charged particle is deflected upward by the converging magnetic field. The
spiral then climbs upward only to be drawn back down by gravity. The spiral bounces up
and down gradually reaching on average a lower altitude with each bounce. This
ephemeris run went for 84000 iterations, with three seconds per iteration, before the
particle crashed into the Earth. Figure 5 is a close up of the oscillating orbital spiral
shortly before it crashes into the earth.
This problem explores a hole in the current understanding of orbital mechanics of
particles. Ions and electrons in the Earth environment are treated as obeying the laws of
electromagnetism, and collectively according to plasma physics.
