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Generalizations of the St¨o rmer Problem for Dust Grain Orbits H.R.Dullin Department of Mathematical Sciences,Loughborough University Leicestershire,LE113TU,United Kingdom 1M.Hor´a nyi and J.E.Howard Laboratory for Atmospheric and Space Plasmas University of Colorado,Boulder,CO 80309-0392Abstract:We consider the generalized St¨o rmer Problem that includes the electromagnetic and gravitational forces on a charged dust grain near a planet.For dust grains a typical charge to mass ratio is such that neither force can be neglected.Including the gravitational force gives ri to stable circular orbits that encircle that plane entirely above/below the equatorial plane.The effects of the different forces are discusd in detail.A modified 3rd Kepler’s law is found and analyzed for dust grains.PACS:96.30.Wr,45.50.Jf,96.35.Kx Keywords:Stormer Problem,Dust Grains,Halo Orbits,Stability
1Introduction
尽多音字组词One of the early milestones of space physics was St¨o rmer’s theoretical analysis of charged particle motion in a purely magnetic dipolefield[1,2].This minal study provided the basic physical framework that led to the understanding of the radiation belts surrounding the Earth and other magnetized planets.The radiation belts are now known to be compod of individual ions and electrons who motion is often well described by magnetic forces alone.The classical results are also relevant to the dynamics of charged dust grains in planetary magnetospheres.However,the much smaller charge-to-mass ratios produce a more complex dynamics,as planetary gravity and the corotational electric field must also be taken into account[3-6].
In a ries of recent papers[7-10]equilibrium and stability conditions were derived for charged dust grains orbiting about Saturn.The orbits can be highly non-Keplerian and include both positively and negatively charged grains,in prograde or retrograde orbits.Thefirst article was restricted to equa-torial orbits,while the cond treated nonequatorial“halo”bits which do not cross the equatorial plane.Both assumed Keplerian gravity,an ideal aligned and centered magnetic dipole rotating with the planet,and con-comitant corotational electricfield.The third paper dealt with the effects of planetary oblateness(J2),magnetic quadrupolefield,and radiation pressure. While thefirst two forces were found to have a negligible effect on particle confinement,the effects of radiation pres
sure could be large for distant orbits. Interestingly,J2and radiation pressure can act synergistically to lect out one-micron grains in the E-Ring[11].Thefinal paper in this ries allowed the surface potential of a grain,and hence its charge,to adjust to local photoelec-tric and magnetospheric charging currents.It was concluded that stable halo orbits were mostly likely to be compod of rather small(≈100nm)positively charged grains in retrograde orbits.A dust grain“road map”was drawn for the Cassini spacecraft now en route to Saturn,showing where to expect dust grains of a given composition and radius.
This paper prents a more comprehensive treatment of dust grain dynam-ics,but under the simplified assumptions of Keplerian gravity,pure dipole magneticfield,and no radiation pressure.Some of the results were already prented in the letters[7,8],here wefill in the details of the necessary calcula-tion and also prent new results.Our goals are a mathematically rigorous yet simplified derivation of equilibrium and stability conditions which highlights the relative importance of the veral different forces acting on an individual grain.
As is well known,there are no stable equilibrium circular orbits for the pure St¨o rmer problem of charged particle motion in a pure dipole magneticfield.It
1
is the addition of planetary gravity and spin that gives ri to stable families of equatorial and nonequatorial orbits.We begin with a general discussion of charged particle motion in axisymmetric geometry,which is then specialized to the motion of charged grains in a planetary magnetosphere.Equilibrium conditions are derivedfirst for equatorial orbits,then for halo orbits.Next we take up the issue of stability for each family of equilibrium orbits.Results are prented for four distinct problems:the Classical St¨o rmer Problem(CSP) in which a charged particle moves in a pure dipole magneticfield,the Rota-tional St¨o rmer Problem(RSP),with the electricfield due to planetary rotation included,the Gravitational St¨o rmer Problem(GSP),with Keplerian gravity included but not the corotational electricfield,and the full system(RGSP) including bothfields.For each ca one must also consider each charge sign in prograde or retrograde orbits.Our results may be summarized as follows: CSP:As is well known,no stable circular orbits,equatorial or nonequatorial exist.However,under adiabatic conditions important families of guiding center orbits confined to a potential trough called the Thalweg exist.
Such trajectories lie outside the scope of the prent paper.
RSP:Stable equatorial equilibria exist for both charge signs.There are no halo orbits.
GSP:Stable equatorial equilibria exist for both charge signs.Positive halos are retrograde and negative halos are prograde.Both types are stable wherever they exist.
RGSP:Stable equatorial equilibria exist for both charge signs.There is a range of positive charge-to-mass ratios without stable equatorial equi-libria.Negative halos are prograde,while positive halos can be pro-or retrograde.For stability the frequency must be sufficiently different from twice the rotation rate of the planet.
Therefore halo orbits do exist with and without the corotational electricfield. However,the corotational electricfield is required in order to sustain stable positive prograde halos.
2Charged particle dynamics in axisymmetric geometry
The equations of motion of a particle of mass m and charge q in R3,r= (x,y,z)t,are
q
m¨r=
where the potential U(r)generates the forces of gravity and perhaps corota-tional electricfield.Denote
by R a rotation around the z axis and assume that the magneticfield B and the potential U are symmetric with respect to this rotation:
B(R r)=R B(r),∇U(R r)=R∇U(r).
In particular this is true for thefield B=∇×A of a centered magnetic dipole of strength M and dipole axis the z axis,for which the vector potential is,in the Coulomb gauge,
A(r)=M(y,−x,0)t/r3,r2=x2+y2+z2.
The equations of motion can be transformed to a rotating coordinate sys-tem using a rotation matrix R corresponding to the angular velocityΩ= (0,0,Ω)t which rotates around the z-axis with with angular speedΩ.For given angular velocityΩthe z axis is chon in the same direction,so thatΩis positive.Note that the magnetic moment M can be positive or negative. Direct differentiation then gives
r=R q,˙r=R(˙q+Ω×q),¨r=R(¨q+2Ω×˙q+Ω×(Ω×q)).
so that
m¨q=(q
c
B×(Ω×q)−∇U(q),
where−2mΩ×˙q is the Coriolis force and−mΩ×(Ω×q)is the centrifugal force.Following the standard argument the term q
c B×(Ω×r)=γΩ∇Ψ,Ψ=
x2+y2
In an inertial frame the potential now reads
U(r)=−σg
µm
2m p−q
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x2+y2,φ=arctan(y,x),z)H becomes
H=1
ρ2
(pφ−γΨ)2 −σgµm
•Parameters describing the dust particle’s mass m and charge,measured byγ=q M/c.
•The angular momentum pφand total energy h=H are constants of the motion determined by the initial conditions.Fixing both h and pφdefines a region of possible motion in configuration space.
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We now introduce a convenient scaling to reduce the number of parame-ters.Time is measured by the inver frequency of the planetary spin rateΩ. Distances are measured in terms of the radius of the Keplerian synchronous orbit
R=(µ/Ω2)1/3
while mass is measured in units of the particle mass m.The scaled Hamiltonian is then
ˆH=1
ˆρ−δˆρ
ˆr
+σrδ
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γandδ=
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m
M
GM
=
ωcΩ
GM/R3s is the Kepler frequency,with R s the
planetary radius,and the parameter p is just the angular momentum pφmea-sured in the new units.The single parameter for the dust grain isδ,which is esntially the charge-to-mass ratio.Recall that the z axis is oriented so that Ω>0.In the following we will looly talk about positive/negative charge when we mean positive/negativeδ.This correspondence is correct if the mag-netic dipole moment M is he spin and thefield are aligned.This is true for Saturn,the main application that we have in mind.Our results are valid in both cas.
3Equilibria
3.1Equatorial Orbits
Here we shallfind it advantageous to work in spherical coordinates(r=
