In an Earth-centered inertial frame, the motion of a charged dust grain of mass m, is governed by the equation:
;
Equation 18
where 
is the Earth's gravitational
force, 
is the light pressure force, 
is the Lorentz force, and 
is the solar gravitational force. [25,
34]. In GEO, we may neglect
the neutral gas and plasma (Coulomb) drag forces on the dust particle [16,
35].
The gravitational acceleration due to the Earth is:
;
Equation 19
where G is the gravitational constant, ME is the Earth's mass,
RE is the Earth's radius, r is the particle's position vector, 
is the grain's latitude as measured from the equatorial plane, 
and 
are the unit vectors, and J2 denotes the 2nd zonal harmonic coefficient,
which refers to the Earth's oblateness.
The acceleration due to light-pressure force is:
;
Equation 20
where Jo is the solar radiation energy flux at 1 A.U. (=1.36 106
ergs-cm-2 s-1, pointing outward from the Sun), QPR is the radiation pressure coefficient
described in the optical properties portion of this report,
is the particle's density, d is the planet's distance in A.U. from the Sun, c is the
speed of light, and a is the radius of the charged spherical particle.
The Lorentz acceleration is given by:
;
Equation 21
Here 
is
the local magnetic field, and, assuming a rigidly corotating magnetosphere, 
is the corotational electric field, with the rotation rate 
, and 
is the particle's velocity vector.
The gravitational acceleration due to the Sun is:
;
Equation 22
where MS is the mass of the Sun.
The force due to light radiation pressure is usually the second strongest after Earth's gravitational force on 1 micron-sized particles. (below, and [11]). However, if the 1 micron-sized particle is highly-charged, for example, with a potential > ~1000V, then the second strongest force will be the Lorentz force.
In the next four figures, we illustrate relative strengths of forces
on, first, a 1 micron-sized conducting particle in Quiet and Active plasma conditions in GEO, then, secondly, on
a dielectric particle in Quiet and Active plasma conditions in GEO. In these calculations, the particle has reached
equilibrium potential.
Conducting Particle, Quiet Plasma
Figure 33: The relative strengths in dynes of the forces versus 360 degree
local time in units of 0 to 24 hours for a canonical case of a conducting 1 micron-sized spherical particle to
compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
266]. The material properties are
= 1.5, Em = 250 eV, 
= 1.0, and the plasma conditions are Quiet.
Conducting Particle, Disturbed Plasma
Figure 34: The relative strengths in dynes of the forces versus 360 degree
local time in units of 0 to 24 hours for a canonical case of a conducting 1 micron-sized spherical particle to
compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
26]. The material properties are
= 1.5, Em = 250 eV, =1.0, and the plasma conditions are Active (Disturbed).
Dielectric Particle, Quiet Plasma
Figure 35: The relative strengths in dynes versus 360 degree local time
in units of 0 to 24 hours of the forces for a canonical case of a dielectric spherical 1 micron-sized particle
to compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
26]. The material properties
are
= 2.4, Em = 400 eV, =0.1, and the plasma conditions are Quiet.
Dielectric Particle, DisturbedPlasma
Figure 36: The relative strengths in dynes versus 360 degree local time
in units of 0 to 24 hours of the forces for a canonical case of a dielectric spherical 1 micron-sized particle
to compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
26]. The material properties
are
= 2.4, Em = 400 eV, =0.1, and the plasma conditions are Active.
In the next four figures, we illustrate relative strengths of forces on, first, a 1 micron-sized conducting particle at 16RE, and then, secondly, on a dielectric particle at 16RE. In these calculations, the particle has reached equilibrium potential.
Conducting Particle
Figure 37: The relative strengths in dynes versus 360 degree local time
in units of 0 to 24 hours of the forces for a canonical case of a conducting spherical 1 micron-sized particle
to compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
26]. The material properties
are
= 1.5, Em = 250 eV, = 1.0.
Dielectric Particle
Figure 38: The relative strengths in dynes versus 360 degree local time
in units of 0 to 24 hours of the forces for a canonical case of a dielectric spherical 1 micron-sized particle
to compare with Horányi, Juhász and their co-workers [14,
23,
24,
25,
26]. The material properties
are
= 2.4, Em = 400 eV, =0.1.
The possible effect of the dynamic, and some very energetic Earth environment is a large charge carried by the dust/debris particle. Our "Case 3" [Table 6] is an extreme case, so we examine here the dynamics of a particle in GEO, through one orbit, carrying a large charge. We look at the dynamics of [1, 10, 100, 1000]-sized spherical particles.
The results show that, even though the charge carried by the particle is quite high in the thousands of Volts, which would enter into the regime of high potentials that induce electrostatic fragmentation, the only size range of the listed 4 sizes whose orbit is noticeably affected by Lorentz force is the 1 micron-sized particle.
In addition, the smaller sized particles: 1 and 10 micron never reach equilibrium potential during the time that they travel one orbit. (See the last panel of each set of plots for each particle size.) If we sum the currents, which we call "qp", then the "qp" value shown in the third panel of each figure oscillates through the zero point. The charging time of the 100 micron and 1000 micron-sized particles is progressively shorter, so those larger particles reach an equilibrium potential value often during the time that the particle travels one orbit.
1 micron at GEO, Case 3
Figure 39: The dynamics of a spherical 1 micron particle in GEO, for one
orbit, under Active plasma conditions, starting at x=-6.6RE. The material properties are
= 1.5, Em = 250 eV, =0.1. The first panel shows a top-down view onto
the x-y plane, with the two axes in units of RE. The next two panels show the potential charge and the
sum of the currents "qp" carried by the particle during the time that the particle orbits one revolution.
10 micron at GEO, Case 3
Figure 40: The dynamics of a spherical 10 micron particle in GEO, for
one orbit, under Active plasma conditions, starting at x=-6.6RE. The material properties are
= 1.5, Em = 250 eV, =0.1. The first panel shows a top-down view onto
the x-y plane, with the two axes in units of RE. The next two panels show the potential charge and the
sum of the currents "qp" carried by the particle during the time that the particle orbits one revolution.
100 micron at GEO, Case 3
Figure 41: The dynamics of a spherical 100 micron particle in GEO, for
one orbit, under Active plasma conditions, starting at x=-6.6RE. The material properties are
= 1.5, Em = 250 eV, =0.1. The first panel shows a top-down view onto
the x-y plane, with the two axes in units of RE. The next two panels show the potential charge and the
sum of the currents "qp" carried by the particle during the time that the particle orbits one revolution.
1000 micron at GEO, Case 3
Figure 42: The dynamics of a spherical 1 micron particle in GEO, for one
orbit, under Active plasma conditions, starting at x=-6.6RE. The material properties are
= 1.5, Em = 250 eV, =0.1. The first panel shows a top-down view onto
the x-y plane, with the two axes in units of RE. The next two panels show the potential charge and the
sum of the currents "qp" carried by the particle during the time that the particle orbits one revolution.
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Dynamics
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