Recent Progress on Planetary
Dust Grain Dynamics
J. E. Howard
Laboratory for Atmospheric and Space Physics
and
Center for Integrated Plasma Studies
University of Colorado, Boulder CO 80309 USA
Abstract. We review recent research at Colorado on the dynamics of charged dust grains in
planetary magnetospheres, with emphasis on the tenuous E-Ring of Saturn. After summarizing
earUer research on the eqiulibrium and stability of single particle orbits, we describe current work on
the crucial effects of plasma sputtering on particle confinement and its impUcations for the Cassini
mission. Finally, we suggest a few challenging problems for mathematicians.
Keywords: Celestial mechanics, planetary rings, dust dynamics, plasma sputtering.
PACS: 45.20 Jj, 45.50 Jf, 96.15 Uv
INTRODUCTION
Source: NASA/JPL-Caltech
FIGURE 1.
Recent view of Saturn by the Cassini spacecraft
The complex motion of tiny charged dust grains in planetary magnetospheres continues to present exciting challenges to celestial mechanicians [1]. Such particles are to
be found orbiting all the gaseous giants, and are thought to be produced primarily by
collisions of micrometeroids with moons and moonlets. In the case of Saturn most of
the visible dust is located in the tenuous outer E-Ring [4], which extends from about
3Rs to 20Rs, according to recent measurements. Our knowledge of Saturn's rings comes
from ground-based observations as well as the two Voyager encounters and the Cassini
CP946, Applications ofMathematics in Engineering and Economics '33, 33^ International Conference
edited by M. D. Todorov
© 2007 American Institute of Physics 978-0-7354-0460-l/07/$23.00
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Spacecraft now orbiting Saturn [5]. Cassini was launched in 1997 and went into orbit
about Saturn in June 2004. It is the largest interplanetary spacecraft ever built and contains many instruments, among them the Cosmic Dust Analyzer (CDA), designed and
built at the Max-Planck Institute in Heidelberg [6]. Figure 1 is a mosaic of recent Cassini
images of Saturn, revealing two previously unknown rings.
Single-particle models have proven especially effective in studies of the E-Ring [712], whose low density makes particle motion essentially coUisionless. The complexity
of the orbital motion is due to the simultaneous action of a large number of forces,
primarily planetary gravity and dipolar magnetic field. In addition, for a conducting
magnetosphere, planetary rotation produces a corotational electric field in an inertial
frame of reference [9]. Solar radiation pressure [13] and planetary oblateness also
significantly perturb the orbits of small dust grains, while plasma sputtering by hot
ions and electrons [15], plasma drag, and Poynting-Robertson drag act over longer
time scales. For larger grains (r^ > 1 jUm) the motion is gravitationally dominated and
therefore regular (non-chaotic). Smaller grains (r^ < 100 nm) on the other hand, are
dominated by the electromagnetic forces, producing primarily adiabatic guiding center
motion. In between these limits a grain feels several different forces with differing
characteristic frequencies, often resulting in highly chaotic motion. Figure 2 depicts
typical regular and chaotic orbits within the E-Ring of Saturn.
We review here research carried out over the past decade at the University of Colorado
on single particle dynamics of charged dust grains near Saturn and Jupiter. The primary
aim of this work is to suggest observations for the Cassini dust measurement group
at Heidelberg. The physical models used in this work are described, along with the
Hamiltonian mathematical model employed in numerical and analytic calculations. Our
earlier results on the equilibrium and stabihty of circular orbits are briefly summarized,
then the role of symmetry breaking due to radiation pressure and the magnetic tilt
(of Jupiter), followed by brief accounts of dynamic charging and ergodicity induced
by chaotic dynamics. We then discuss our more recent work concerning the crucial
effects of plasma sputtering on particle confinement in the E-ring of Saturn, which can
significantly reduce the size of a dust grain in a matter of decades, altering its electronic
charge in the process. Finally we suggest a few new research directions of interest to
apphed mathematicians.
PHYSICAL MODEL
Saturn is a very axisymmetric and highly oblate planet, whose gravitational field is
known to high accuracy. Unlike Earth and Jupiter, its magnetic field is also apparently
axisymmetric and may be well characterized as a centered dipole aligned with the planetary spin axis, conveniently described by a vector potential A = ^ ( p , z ) ^ / p , where
W is the magnetic stream function, expressed in cylindrical coordinates {p,(j>,z). Thus,
B = V X A = V ^ X ^ / p . In addition, in an inertial frame of reference charged particles
experience a "corotational" electric field Ec normal to the local magnetic surface and
given by Ec = (p X Q) X B = —qQ.yW, where Q. is the planetary spin rate. Solar radiation pressure can be calculated using Mie theory [13] and is only important for a small
range of grain radius, rg « 100 — 200 nm. In some cases solar radiation pressure acts
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(dsecx.f):po=4,P| = 6,p = 0, Eg =-9.0731 e-9, O = 700
4
' /^^!^
^^k \
/^^m
2
0
2
4
)
(
^
(dsecx.f): pg =4, p, =7.65, p = 0, Eg =-9.0731 e-9, 0 = 700
-
2
-
0
- 'w ^
"
,
,
FIGURE 2.
V--.-'''''
'
/^^' u" = E
/mfM\
4
u" = E
-
2
"
4
)
^^S
-
( ^ ^ ^
-
^^^p
-
^^^^^ i
"
,
,
"
' ' ' ' • • ^ ' ' ' ' '
Typical regular and chaotic dust grain orbits near Saturn
synergistically with oblateness terms in the gravitational field to pump up orbital eccentricity and deliver particles in the planetary surface [16]. However, with the exception of
solar radiation pressure, the orbital dynamics near Saturn is entirely axisymmetric.
Since dissipative forces (due, e.g., to interparticle collisions) are negligible, a Hamiltonian model has been adopted in most theoretical studies. Neglecting radiation pressure,
the resultant Hamiltonian may be written [10]
H
i^^P
U + qQ.'V
2mp^
(1)
where p,p = mp^(j> + qW is the conserved angular momentum and U{p,z) is the gravitational potential. Radiation pressure can be included by adding the term Fp cos (j> to
(1), where F is a constant derived from Mie theory. Single particle orbits may then be
calculated from Hamilton's equations
Pz =
Pp =
dH
(2)
dH
(3)
(^Pz
The solutions of these highly nonlinear equations are typically chaotic, interspersed
with regular regions filled with quasiperiodic orbits. In general both charge signs must
be considered, in both prograde and retrograde orbits (i.e., moving with the planetary
rotation or counter to it). Of these four possibilities, the most important is negatively
charged grains in prograde orbits, for reasons to be explained below.
'' = d]rr'''
Sources of Dust. The rings of Saturn are in fact comprised of several ring systems,
each with its own composition of particles which range from submicron size, to fist-size
rocks, to large boulders, to moonlets, which are responsible for creating some of the
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famous gaps in the rings [1]. Despite their great diversity in size, most ring particles are
composed of water ice, with a small admixture of sihcates, iron, and organic compounds
in the inner rings. It is not our purpose to describe the rings in detail here, but rather
to focus on the E-ring, which reaches from about 3Rs to lORg and contains the bulk
of dust grains orbiting Saturn. The larger bodies may have resulted from an ancient
moon which wandered too close to the planet and was torn asunder by the gravitational
gradient. Some dust is generated by colhsions among these bodies, but most is believed
to be produced by collisions of micrometeroids with moonlets and rocks and especially
with the moon Enceledus. Recently, however, a surprising new major source of dust has
been discovered by the Cassini spacecraft [5]: icy volcanos on Enceledus! [17]. This
unexpected phenomenon was first observed when the Cassini spacecraft passed close
to Enceledus, a small moon of 250 km radius located within the E-Ring at r = 3.95Rs.
Even more recently it has been discovered that the moon Dione, of radius 560 km and
located at r = 6.26Rs, also experiences cryovolcanism. While the data are still new,
it now appears that most of the existing dust population in the E-Ring is produced by
cryovolcanism on Enceledus, with only a small contribution from Dione [18]. Whichever
of these mechanisms is dominant, most dust grain orbits are prograde, in the same
sense as the moon or moonlet that produced them. After ejection, a grain finds itself
in a nearly circular, nearly equatorial orbit with velocity close to that of the parent
body. It is then exposed to several populations of ion and electron fluxes in the hot
plasma environment of the magnetosphere, in addition to solar radiation pressure. After
a characteristic charging time the grain, generally assumed spherical, comes to charge
equilibrium, with (more or less) constant surface potential 4>j, with charge-to-mass ratio
q
W<P,
m A%pmd^M
esu/g
(4)
where 4>j is the surface potential in volts, pm is the mass density in gm/cm^, and a^ is
the grain radius in microns.
As time goes on these particles experience various perturbations which change their
orbital elements. For example, planetary oblateness can act together with solar radiation pressure to pump up the orbital eccentricity. Radiation pressure also possesses a
nonequatorial component which can cause the orbital inclination to increase appreciably. Local equatorial instabilities can also produce nonequatorial orbits.
Several caveats are in order:
• Dust grains are not spherical, but highly irregular. Thus, the surface potential must
be interpreted as a spatial average.
• The radius of a dust grain decreases slowly in time owing to plasma sputtering.
It can be shown that a variable charge-to-mass ratio cannot be described by a
Hamiltonian model.
• The surface potential 4>j varies with position along an orbit due to the changing
plasma environment. This effect has been successfully modelled [12].
• Dust grains are often not in electrical equilibrium with their environment. This is a
serious drawback, which in some cases can only be handled numerically.
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• Very small grains have small capacitance and can therefore only hold a few electronic charges. This means that any change in surface potential can produce a relatively large orbital perturbation.
As we shall see, submicron dust grains do not live forever, perhaps only 50—100
Earth years. One of the major revelations of the Cassini observations is that the entire
system of rings is much younger and much more dynamic than thought only a few years
ago. But now we must turn to the main subject of this review, which is to describe the
role of mathematical analysis in understanding the dynamics (and therefore the ultimate
fate) of planetary dust grains.
SUMMARY OF PREVIOUS RESULTS
Equilibrium and Stability. We have seen that, neglecting radiation pressure and
plasma sputtering, the motion of a charged dust grain is governed by the two-degreeof-freedom autonomous Hamiltonian (1). The dynamics is therefore expected to be typically chaotic, with regular regions organized around periodic orbits, the most important
being circular equilibrium orbits. Thus, the first issue in a single particle analysis is
the existence and stability of circular orbits [9]. In this way one can for example glean
information about the radial extent of the Main Ring [7]. This problem was completely
solved by the author and his collaborators, first for equatorial orbits [9] and subsequently
for more general nonequatorial "halo" orbits [10]. Figure 3 shows a projection of a halo
orbit floating above the E ring. Similar orbits have been observed, both in numerical simulations, and possibly by the Cosmic Dust Analyzer, on board the Cassini spacecraft.
3
Source: Howard and Hordnyi, 2002
FIGURE 3.
Projection of a halo orbit floating above the E-Ring
Circular equilbrium orbits are then found by setting Vf/^ = 0, where
U'{p,z) = U{p,z) + -
2mp^
q£lW
(5)
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is the effective potential. In our studies we usually assume Keplerian gravity, U =
—GMs/r, but occasionally include an oblateness term. For a dipole magnetic field
W = {BoR^y )p^/'^ • For equatorial equilibria (zo = 0), this gives a quadratic for the orbital
frequency co:
PQ(O^ - (OcCO- {(ol - ©cQ) = 0
(6)
where po is a reference radius, (O^ = \/GMS/RI is the Kepler frequency, and ©c = qBo/m
is the cyclotron frequency, with Bo referenced to a point on the planetary equator. The
solutions to the above quadratic then give the prograde and retrograde orbits for each
choice of sgn(ig). The stability of each mode is given by the Hessian D^V, i.e., an
equilibrium orbit is Lyapunov stable if both trD^f/^ > 0 and detD^f/^ > 0. Due to the
Z2 symmetry of U^ these conditions reduce to Upp > 0 for radial stability and U^>0 for
transverse stability. Eliminating co between these conditions and the equilibrium yields
explicit stabihty bounds. When Upp changes sign a saddle-node bifurcation occurs,
when U^ changes sign a pitchfork bifurcation. For nonequatorial equilibria, a more
complicated calculation yields explicit stability bounds for halo orbits. In this case Up^
does not vanish and only saddle-node bifurcations are allowed. An improved method is
reported in Ref. [19]. Figure 4 illustrates these two bifurcations for equatorial orbits.
Symmetry Breaking via Radiation Pressure. Since radiation pressure is unidirectional in a satumian coordinate system, it is nonaxisymmetric and constitutes a small
perturbation to the axisymmetric Hamiltonian (1). It is also nonequatorial, due to the
obliquity of Saturn's spin axis relative to the ecliptic. As this perturbation term depends
on the azimuth (j), the total angular momentum p^ is no longer conserved so that the motion is now three- rather than two-dimensional. Nevertheless, for most conditions this
perturbation strength is so small that p,p oscillates about a constant average, making the
dynamics quasi-two-dimensional. The conditions for the validity of this picture and its
consequences are explored elsewhere [14].
Symmetry Breaking Due to Magnetic Tilt. Jupiter also has a system of rings [1].
While Saturn's magnetic axis is well aligned with its spin axis, the situation is quite
different for Jupiter, whose magnetic axis is tilted by about 9.5° with its spin axis [1].
Interestingly, this also causes the total angular momentum to oscillate about a constant
value, but only for a range of grain radii [21]. Preliminary orbital simulations show that
smaller grains are chaotic and rapidly lost to the planet.
Dynamical Charging. As a grain visits different regions of Saturn's magnetosphere
its surface potential changes in response to local plasma conditions. Extensive studies
of equilibrium and stability of circular orbits including variable 4>(r) have been carried
out and used to make quantitative predictions of possible Cassini observations [12].
Chaos and Ergodicity. Most dust grain orbits are initialized near the equatorial plane
of Saturn and turn out to be very regular. Over time, however, nonequatorial forces such
as radiation pressure and local instabihties can produce highly inclined orbits. Such
orbits have in fact been detected by the Cassini Orbiter. These orbits are of considerable
interest to a dynamicist, as they are typically chaotic and therefore ergodic. Lyapunov
220
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1.5
-
-
E
0.5
0
; y
^).5
^
-1
^
^
^
(b)
-
-1.5
0
0
06
1
1
2
15
2
25
3
4
3
36
5
r
5oMrce.- Howard and Hordnyi, 1999
FIGURE 4.
(a-b) a saddle node bifurcation (po = 2), (c-d) a pitchfork bifurcation (po = 4)
exponents and Poincare sections have been computed for these orbits (see Figure 2),
in order to determine how their chaoticity depends on orbital parameters. In reahty the
confinement time of regular orbits is finite, owing to the effects of plasma sputtering, as
described in the following section.
NEW RESULTS: PLASMA SPUTTERING
A dust grain orbiting in a planetary magnetosphere is constantly bombarded with ions
and electrons which for spherical grains may be assumed to remove material isotropically. Also assuming that all material is removed at sufficiently high velocity that it
does not re-impact the particle, the motion of the grain is governed by Gylden's equa-
221
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3DDipoleq/m (0-100) Vg = 0 1, xg = 30°, a= 0005
3D Dipole q/m (400-500) Vg = 0 1, xg = 30°, a= 0005
3D Dipole q/m (900-1000) Vg = 0 1, xg = 30°, a= 0005
04
^^..^^
03
" ^
02
%
01
07
08
00
2
07
08
00
12
07
08
00
m
Source: Howard, 2007
FIGURE 5.
A guiding center orbit evolving under mass and charge loss
tion [22, 23]
OT(?)f = F ( r , v )
(7)
where F represents all the various forces described in the previous sections. Except in
special cases, eq. (7) is not Hamiltonian, and must be directly integrated, using some
physical model for m{t). In particular, the angular momentum pij, is not conserved, even
under axisymmetric conditions. Jurac [15] has shown that the radius of a dust grain in a
hot plasma will decrease linearly in time, so that m{t) =OTO(1— kt)^, where k is chosen
such that a 1 micron grain disappears in 50 years. For relatively short times, say t = ly,
the grain is confined by a slowly varying effective potential U^{p,z;t), which depends
on time through m{t) and pij,. It can be shown that the charge q also decreases linearly
in time, so that the charge-to-mass ratio increases in time. Thus, if one can predict
the time evolution of pij, and E, then setting U^ = E(t) will describe the quasistatic
evolution of the confining potential well, with implications for Cassini measurements.
This calculation has been successfully carried out for the guiding center motion of very
small grains [23]. Figure 5 shows that the evolution of a dust grain in a magnetic dipole
field closely follows its confining potential V. We are currently extending our analysis
to encompass the corotational electric field and the gravitational field [24] of Saturn.
DISCUSSION
The problems outlined in this brief review present many challenges to mathematicians
and celestial mechanicians alike. What is the role of chaotic diffusion in dust dynamics?
Do global invariants exist for nonhamiltonian systems with symmetries? Can one construct a useful effective potential in the presence of the three major forces? Is there an
adiabatic invariant associated with slow variations of the charge-to-mass ratio? How can
one predict the transition to global chaos with a slowly increasing symmetry-breaking
force such as radiation pressure? What is the mechanism for the observed high velocity
streaming particles from Saturn [25]?
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ACKNOWLEDGMENTS
Many thanks to Larry Esposito, Mihaly Horanyi, Antal Juhasz, Glenn Stewart, and
Holger Dullin for many helpful discussions and to Brandon Gonzales for help with the
calculations. This work was supported by the Cassini Project.
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