Icarus 218 (2012) Contents lists available at SciVerse ScienceDirect. Icarus. journal homepage:

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1 Icarus 8 () 4 4 Contents lists available at SciVerse ScienceDirect Icarus journal homeage: The fate of sub-micron circumlanetary dust grains I: Aligned diolar magnetic fields Daniel Jontof-Hutter, Douglas P. Hamilton Astronomy Deartment, University of Maryland, College Park, MD 74-4, United States article info abstract Article history: Received 4 Aril Revised 7 August Acceted Setember Available online 7 October Keywords: Planetary rings Juiter, Rings Magnetic fields Saturn, Rings We study the stability of charged dust grains orbiting a lanet and subject to gravity and the electromagnetic force. Our numerical models cover a broad range of launch distances from the lanetary surface to beyond synchronous orbit, and the full range of charge-to-mass ratios from ions to rocks. Treating the sinning lanetary magnetic field as an aligned diole, we ma regions of radial and vertical instability where dust grains are driven to escae or crash into the lanet. We derive the boundaries between stable and unstable trajectories analytically, and aly our models to Juiter, Saturn and the Earth, whose magnetic fields are reasonably well reresented by aligned dioles. Ó Elsevier Inc. All rights reserved.. Introduction The discoveries of the faint dusty ring systems of the giant lanets beginning in the late 97s greatly changed our understanding of lanetary rings. Unlike Saturn s classical rings, which are most likely ancient (Canu, ), dusty rings are young and are continually relenished from source satellites. Individual ring articles have short lifetimes against drag forces and other loss mechanisms, and because dusty rings are so diffuse, they are essentially collisionless. Furthermore, dusty rings are affected by a host of non-gravitational forces including solar radiation ressure and electromagnetism, which can scult them in interesting ways. Since the giant lanets are far from the Sun and dusty rings are normally near their rimary, radiation ressure is usually a weak erturbation to the lanet s gravity. The electromagnetic force arising from the motion of charged dust grains relative to the lanetary magnetic field, however, can be quite strong. In articular, with nominal electric charges, dust grains smaller than a fraction of a micron in radius are more strongly affected by electromagnetism than gravity. Dust in sace acquires electric charges in several ways. Moving through the lasma environment roduces a negative charge on a grain, since the lasma electrons are much lighter and swifter than ions and hence are catured more frequently by orbiting dust grains (Goertz, 989). On the other hand, sunlight ejects hotoelectrons from the surface of a grain, and can cause ositive charges (Horányi et al., 988). Electron or ion imacts will also roduce secondary electron emission, which also favors a net ositive Corresonding author. address: danielj@astro.umd.edu (D. Jontof-Hutter). equilibrium charge on the grain (While, 98). These currents interact in comlicated ways; the charging of a dust grain deends on the hysical roerties of the grain itself and also on its charge history (Meyer-Vernet, 98). Gras et al. (8) rovide an excellent review of these rocesses. Many authors have investigated detailed asects of the motion of charged grains in lanetary magnetic fields, but no study has yet determined the orbital stability of grains for all charge-to-mass ratios launched at all distances in a systematic way. In this aer we exlore the local and global stability of both ositive and negative dust grains launched from ring article arent bodies which themselves orbit at the local Keler seed... Motion in the Keler and Lorentz limits As grains with radii greater than several microns have small charge-to-mass ratios, electromagnetic effects are weak, and the grains orbit the lanet along nearly Kelerian ellises. In the frame rotating with the mean motion of the dust article, the orbits aear as retrograde ellitical eicycles with a : asect ratio (Mendis et al., 98). When gravity acts alone, the vertical, radial and azimuthal motions all have recisely the same frequency. Equations governing the slow changes to the ellise s orbital elements due to weak electromagnetic erturbations from a rotating aligned diole magnetic field are given by Hamilton (99a). These equations show that the three frequencies diverge slightly and are functions of the sign and magnitude of the charge as well as the distance from the lanet and from synchronous orbit. Conversely, the very smallest dust grains aroach the Lorentz limit, where the electromagnetic force dominates over gravity. In this regime, the frequencies of radial, vertical and azimuthal 9-/$ - see front matter Ó Elsevier Inc. All rights reserved. doi:.6/j.icarus..9.

2 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () motions differ significantly. The radial oscillation is fastest and, as the electromagnetic force is erendicular to the rotating magnetic field, articles gyrate about local field lines on tyical timescales of seconds for dust, and microseconds for ions. Dust grains tyically oscillate vertically on a timescale of hours to days. Since this timescale is far slower than gyration, an adiabatic invariant exists and can easily be found. In the absence of forces other than electromagnetism, and in the absence of lanetary rotation, a dust grain s seed v remains constant: v ¼ v? þ v k, where v \ and v k are the seeds erendicular and arallel to the magnetic field lines, resectively. The v \ comonent determines the radius of the gyrocycle, while the v k comonent moves the center of gyration to regions of differing magnetic field strength. If changes to a non-rotating magnetic field! B are small over the size and time scales of gyromotion, the ratio v? =B, where B is the local strength of the field, is an adiabatic invariant (de Pater and Lissauer, ) and hence is nearly constant. These two conditions rovide an imortant constraint on the grain s motion arallel to the field lines. As a grain with a vertical velocity comonent climbs u a magnetic field line away from the equatorial lane, the field strength B increases, v \ also increases, and hence v k must decrease. There is thus a restoring force towards the equatorial lane where the magnetic field strength is a local minimum, and the motion arallel to the field lines takes the form of bounce oscillations between mirror oints north and south of the equator (Störmer, 9). Thomsen and van Allen (98) studied the bounce motion of articles in the Lorentz limit at Saturn. Their results neglected the effects of lanetary rotation, and hence are most alicable to slow rotators like Mercury and otentially some lanetary satellites. Finally, on the longest timescales (days), articles drift longitudinally with resect to the rotating magnetic field (de Pater and Lissauer, ), forced by a number of effects including gravity, the curvature of the magnetic field, and rb. Because these motions are usually slow comared to the gyration and bounce frequencies, it is often useful to assume that in the Lorentz limit, grains are tied to the local field lines... Dust affected by both gravity and electromagnetism For a broad range of grain sizes from nanometers to microns, both gravity and the Lorentz force are significant, and their combined effect causes a number of dynamical henomena that are distinct from either limiting case. As dust in this size range redominates in many lanetary rings (Burns et al., 999; de Pater et al., 999; Showalter et al., 8; Krüger et al., 9), their dynamics have attracted much attention. Schaffer and Burns (994) rovide a general framework for the motion of dust started on initially Kelerian orbits. Since the radial forces on a dust grain at launch are not balanced as they are for a large arent body on a circular orbit, these dust grains necessarily have non-zero amlitude eicyclic motions. For the magnetic field configurations of the giant lanets, a negatively-charged dust grain gyrates towards synchronous orbit while ositively-charged dust initially moves away from this location. In fact, some ositivelycharged grains are radially unstable and either crash into the lanet if launched inside synchronous orbit, or are exelled outwards if launched from beyond this distance. The latter have been detected as high-seed dust streams near Juiter (Grün et al., 99, 998) and Saturn (Kemf et al., ). Theoretical exlanations for the electromagnetic acceleration rocess have been given by Horányi et al. (99a,b), Hamilton and Burns (99b) and Gras et al. (). Mendis et al. (98, 98a) and Northro and Connerney (987) exlored the shae and frequency of eicycles for negativelycharged grains in the transitional regime, where both EM effects and gravity are comarable. The eicycles make a smooth transition from erfectly circular clockwise (retrograde) gyromotion in the Lorentz limit, where EM dominates, to : retrograde ellitical eicycles in the Keler limit. Mitchell et al. () studied the shaes of eicyclic motion for ositive grains and found that there is not a similarly smooth transition from rograde gyromotion to retrograde Keler eicycles, and that the eicyclic motions of intermediately-sized grains cannot be reresented as ellises. The effects of gravity and electromagnetism comete for intermediate charge-to-mass values and motion can be rimarily radial, leading to escae or collision (Horányi et al., 99a; Hamilton and Burns, 99b). Northro and Hill (98, 98a) and Northro and Connerney (987) studied the vertical motion of negatively-charged dust grains on circular uninclined orbits in a centered and aligned diole field, a configuration most closely realized by Saturn. They found that some small grains on initially centrifugally-balanced circular trajectories inside the synchronous orbital distance are locally unstable to vertical erturbations, climbing magnetic field lines to crash into the lanet at high latitudes. Some motions at high latitude, however, are stable: Howard et al. (999, ) identified non-equatorial equilibrium oints for charged dust grains, and showed that dust grains can orbit them stably. They characterized these halo orbits for ositive and negative charged grains on both rograde and retrograde trajectories. Howard and Horányi () used these analytical results to argue for a stable oulation of ositively-charged grains in retrograde orbits and develoed numerical models of such halo dust oulations at Saturn. Grains that may oulate these halos, however, are unlikely to result from the equatorial launches considered here. If one of the dust grain s natural frequencies matches a characteristic satial frequency of the rotating multiolar magnetic field, the article exeriences a Lorentz resonance (Burns et al., 98; Schaffer and Burns, 987, 99; Hamilton and Burns, 99a; Hamilton, 994). Lorentz resonances behave similarly to their gravitational counterarts and can have a dramatic effect on a dust grain s orbit, exciting large radial and/or vertical motions. These resonances have been rimarily studied in the Keler limit aroriate for the micron-sized articles seen in the dusty rings of Juiter. In our idealized roblem, with an axisymmetric magnetic diole, Lorentz resonances cannot occur. Variations in a dust grain s charge can also alter its trajectory over surrisingly raid timescales. Gradients in the lasma roerties, including density, temerature and even comosition affect the equilibrium otential of a grain by altering the direct electron and ion currents. This can result in resonant charge variation with gyrohase, causing radial drift. Working in the Lorentz limit, Northro and Hill (98b) noted that with large radial excursions, the grain s seed through the lasma can vary significantly with gyrohase, leading to enhanced charging at one extremity. A similar effect occurs in the Keler limit where resonant charge variation can cause a dramatic evolution in the orbital elements of a dust grain (Burns and Schaffer, 989). Northro et al. (989) found that the varying charge has a time lag that deends on the lasma density and grain caacitance. These time lags can cause grains to drift towards or away from synchronous orbit deending on the grain seed, and on any radial temerature or density gradients in the lasma. Schaffer and Burns (99) exlored the effects of stochastic charging on extremely small grains, where the discrete nature of charge cannot be ignored. They found that Lorentz resonances are robust enough to survive even for small dust grains with only a few electric charges. The dynamics of time-variable charging may lay an imortant role in determining the structure of Saturn s E ring (Juhász and Horányi, 4) and Juiter s main ring and halo (Horányi and Juhász, ). Another examle of charge variation occurs when the insolation of a dust grain is interruted during transit through

3 4 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () 4 4 the lanetary shadow. This induces a variation in charge that resonates with the grain s orbital frequency (Horányi and Burns, 99). Hamilton and Krüger (8) found that this shadow resonance excites radial motions while normally leaving vertical structure unaltered. This effect can exlain the aearance of the faint outward extension of Juiter s Thebe ring, and the roerties of its dust oulation samled by the Galileo dust detector (Krüger et al., 9)... Research goals In this study, we consider the orbits of charged grains launched in lanetary ring systems. Our aim is to exlore the boundaries between stable and unstable orbits in aligned and centered diolar magnetic fields. Diolar fields have the advantage of being analytically tractable while still caturing most of the imortant hysics. Under what conditions are grains unstable to vertical erturbations? Which grains escae the lanet as high seed dust streams? And which grains will strike the lanet after launch? All of these instabilities deend on the launch distance of the grain and its charge-to-mass ratio. We first exlore grain trajectories numerically and then derive analytical solutions for the stability boundaries that we find. There are several standard choices for exressing the ratio of the Lorentz and gravitational forces. The charge-to-mass ratio q/ m in C/kg (Northro and Hill, 98) or in statcoulomb/g (Mitchell et al., ) may be the most straightforward, but it is cumbersome. For this reason, converting to the grain otential measured in Volts, which is constant for different-sized dust grains, is a common choice (Mendis et al., 98; Schaffer and Burns, 994; Howard et al., ; Mitchell et al., ). Yet another otion is to exress the charge-to-mass ratio in terms of frequencies associated with the rimary motions of the grain, such as the gyrofrequency, orbital frequency and the sin frequency of the lanet (e.g. Mendis et al., 98; Mitchell et al., ). We choose a related ath, namely to fold q/m and key lanetary arameters into a single dimensionless arameter L following Hamilton (99a). Consider the Lorentz force in a rotating magnetic field: ƒ! q F B ¼ ð~v c X! ~rþ! B ; where c is the seed of light, ~r and ~v are the grain s osition and velocity in the inertial frame, ~ X is the sin vector of the lanet, and B! is the magnetic field. We use CGS units here and throughout to simlify the aearance of the electromagnetic equations. The second comonent of Eq. () is qe!, where E! ¼ c ðx! ~rþ B! is the so-called co-rotational electric field which acts to accelerate charged grains across magnetic field lines. Since a diolar magnetic field obeys B! ¼ g R =r^z in the midlane (with g the magnetic field strength at the lanet s equator), E!, like gravity, is roortional to /r there. Thus the ratio of the electric force to gravity is both indeendent of distance and dimensionless: L ¼ qg R X GM mc : Here, R and M are the lanetary radius and mass, m is the dust grain mass, and G is the gravitational constant. Note that the sign of L deends on the roduct of two signed quantities, q and g. For all of the giant lanets, the magnetic north ole is in the northern hemishere, and g >. However, for the Earth at the current eoch, g < and the magnetic and geograhic oles are in oosite hemisheres. We have made a slight notational change L? L from Hamilton (99a,b) to avoid confusion with the L-shell of magnetosheric hysics. Choosing L as an indeendent variable takes the lace ðþ ðþ of assuming a articular electric otential, grain size and grain density. We focus our study rimarily on Juiter, the lanet with by far the strongest magnetic field, but also aly our results to Saturn and to the Earth.. Numerical simulations Aroximating Juiter s magnetic field as an aligned diole by including just g = 4.8 Gauss (Dessler, 98), we tested the stability of dust grain orbits over a range of grain sizes and launch distances both inside and outside synchronous orbit. We used a Runge Kutta fourth-order integrator and launched grains at the local Keler seed with a small initial latitude of k =.. This tiny nominal value ensures a launch close to the midlane, whilst avoiding otential numerical roblems of launching a grain recisely at k =. Non-zero launch seeds from the arent article do have a small effect on the stability boundaries, one that we will exlore in more deth in a future study. Our models treat the grain charge as constant and neglect J, other higher-order comonents of the gravitational field, and radiation ressure. For both negative and ositive grains, we ran simulations for a grid of 8 values of L and launch distances (r L ). The charge-to-mass ratio sans four decades from the Lorentz regime where EM dominates (jl j), to the Keler regime where gravity reigns (jl j). The range of launch distances extends from the lanetary surface to well beyond the synchronous orbital distance ( ), and trajectories were followed for u to. years. With some exerimentation, we determined that all relevant dynamical timescales are <. years and that for longer integration times, the aearance of our stability lots does not change significantly. In Fig. we lot the fate of 8 negative and 8 ositive dust grains and find comlex regions of instability. The negatively-charged dust grains in Fig. a dislay only vertical instability at moderate to high L and inside. Some are bound by high latitude restoring forces (locally unstable, light grey) whilst others crash into the lanet at high latitude (both locally and globally unstable, darker grey). To searate these globally stable grains from locally stable ones, we choose a latitude threshold at k m =. Although is a small latitude, it is far greater than the launch latitude of. ; any grains excited beyond k m are clearly locally unstable, and we determined that our results were fairly insensitive to actual value of k m. Northro and Hill (98) derived a boundary for the threshold between locally stable and unstable trajectories for negativelycharged dust and found that grains launched within a certain distance should leave the equatorial lane (NH8 curve in Fig. a). In the Lorentz limit, the vertical instability allows grains to climb u local magnetic field lines into regions of stronger magnetic field, while for smaller L the ath taken by these grains follows the lines of a seudo-magnetic field which includes the effects of lanetary rotation (Northro and Hill, 98). The Northro curve however, is not a good match to our data which reveal additional stable orbits (white areas) immediately inside this boundary and also close to the lanetary surface. These differences arise from the fact that Northro and Hill (98) assumed that grains are launched at their equilibrium circular seeds, which differ from the circular seeds of arent bodies when L. Conversely, we launch our grains at v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi GM =r, the circular seed of the arent body, which is aroriate for debris roduced by cratering imacts into these objects. In Section, we develo a vertical stability criterion aroriate for our launch conditions. The situation for ositive grains is quite different. Fig. b shows a less extensive region of vertical instability than Fig. a, and one that is not active close to Juiter. More dramatic, however, are two regions of radial instability (darkest grey areas), searated by the synchronous orbital distance. Grains inside are driven

4 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () (a) a d (μm).. Lorentz regime Keler regime r/r r L /R.. q < NH8 λ ( o ) T (hours) (b). H9 HB9 q > Fig.. The trajectory of a ositively-charged grain orbiting Juiter after launch at r L =.74R, with L =. (a d =. lm). We lot the scaled distance and latitude of the dust grain against time. The small, raid radial gyration is just visible in the uer lot. The dust grain is vertically unstable on a much longer timescale and ultimately crashes into the lanet. This trajectory is the left-most filled square in Fig. b. r L /R.7. r/r to strike Juiter, while those outside escae the lanet. If grains move beyond r esc =R, the inner magnetoshere, we consider them to have escaed. As with k m, our numerical results are fairly insensitive to the exact value chosen for r esc, so long as it is large. To characterize the individual trajectories that make u Fig., we exlore a few examles in detail, focusing on the ositivelycharged dust grains and roceeding from smaller to larger grains. Fig. shows the trajectory of a dust grain that becomes vertically unstable and crashes into the lanet at high latitude. These smallest grains siral u magnetic field lines, which for a diole are given by r/cos k = r L (de Pater and Lissauer, ); collision with the lanet or reflection from a high latitude mirror oint tyically occurs within a few tens of hours. By contrast, Fig. shows an electromagnetically-dominated grain that remains stable at low latitude. L * Fig.. Stability of Keler-launched (a) (negative) and (b) (ositive) dust grains at Juiter. We model the lanet with a sherically-symmetric gravitational field, and a centered and aligned diolar magnetic field. All grains were launched with an initial latitude of k =. and followed for. years. The horizontal dashed line in both anels denotes the synchronous orbital distance at =.4R. The grain radii (a d ) in microns along the uer axis are calculated assuming a density of g/cm and an electric otential of ±V so that j L j¼ :84=a d. Dust grains in the white regions and lightest grey areas survive the full. years, with the latter reaching latitudes k in excess of. Grains in the moderately-grey areas are vertically unstable and strike the lanet, also at high latitudes (k > ). The darkest regions, seen only in anel (b), are radially unstable grains that crash into the lanet (those with r L < ), or escae to beyond r esc =R (from r L > ) at latitudes less than. We overlot three analytically-derived stability boundaries, obtained by Northro and Hill (98) for negative grains, by Horányi et al. (99a) for small ositive grains, and by Hamilton and Burns (99b) for large ositive grains. Each oint on the lot is a trajectory, some of which (marked by filled squares), are illustrated in detail in Figs... λ ( o ) r/r λ ( o ) T (hours) Fig.. The trajectory of a stable ositively-charged grain orbiting Juiter after launch at r L =.74R, with L =.4 (a d =.97 lm). The grain undergoes radial oscillations much larger than in Fig. but its latitude remains low. Here the bounce eriod is 7 times longer than the gyroeriod T (hours) Fig. 4. The trajectory of a ositive grain inside (r L =.R,L =.98,a d =. lm). Here, unlike Fig., large radial motions ultimately excite vertical motions, forcing the trajectory to end with a collision at the lanetary surface after just a few orbits.

5 44 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () 4 4 r/r λ ( o ) T (days) Fig.. The trajectory of a ositive grain outside (r L =.7R,L =.49,a d =.6 lm). As in Fig. 4, large radial oscillations eventually excite large vertical oscillations. Since the dust grain has L <, it is energetically required to remain bound (Hamilton and Burns, 99b). Here T is measured in Earth days. A more subtle interlay between radial and vertical motions is illustrated in Fig. 4. This grain is outside the radial instability region in which grains collide with the lanet at low latitude (darkest grey). Instead, large radial motions lead to instability in the vertical direction, and ultimately, the grain strikes the lanet at high latitude. Notice the two white dots near (L =.4, r L /R =.) in Fig. b, signifying grains that survive the full. years integration. These trajectories are indeed stable (for at least years) and, as the effect is much more rominent for the Earth, we discuss it in more detail in Section 6. Finally, Fig. shows a dust grain just inside the Hamilton and Burns (99b) L ¼ stability limit. Although the dust grain does not escae, the non-linearity of its radial oscillation is large enough to excite substantial vertical motions. A glance at Fig. shows that most stability boundaries are unexlained. The Northro and Hill (98) vertical stability boundary does not match the numerical data esecially well, and only alies to negative grains. For ositive grains, Horányi et al. (99b) rovided an aroximate criterion for radial escae, which they alied far from synchronous orbit near Io. Their criterion is based on a comarison between the radius of gyromotion r g, and the length scale over which the magnetic field changes substantially, namely where jb/(r g rb)j, with the gyroradius calculated in the Lorentz limit. Although not intended for use near synchronous orbit where r g?, we nevertheless lot it on the left side of Fig. b. Finally, the Hamilton and Burns (99b) L ¼ limit, derived from an energy argument, is a good match to the largest escaing grains. There is however, no analytical model for the broad class of grains that strike the lanet. Accordingly, we seek to develo a unified theory that can cleanly determine all of these boundaries. We take u this task first for radial and then for vertical motions.. Local radial stability analysis Consider a centered magnetic diole field that rotates with frequency X around a vertical axis aligned in the z-direction. Northro and Hill (98) derived the Hamiltonian for a charged dust grain in the rotating frame in cylindrical coordinates: H ¼ Uðq; zþþ _q þ _z ; ðþ where _q and _z are the radial and vertical velocity comonents. The otential is given by Uðq; zþ ¼ / q m GM q L þ GM Xr r L q ; ð4þ r where the sherical radius r satisfies r = q + z (Northro and Hill, 98; Schaffer and Burns, 994; Howard et al., ; Mitchell et al., ). Eq. (4) is the sum of two energetic comonents: first the azimuthal secific kinetic energy, which can be exressed as a function of r using the conservation of angular momentum, and then the otential associated with both the corotational electric field and gravity. Note that we have chosen the zero of our otential to be aroached as q?. Because U(q,z) is indeendent of /, the azimuthal coordinate, the canonical conjugate momentum / is a constant of the motion. For our launch condition from a large arent body on a circular orbit at r = r L : / m ¼ r L ðn L þ X gl Þ (Schaffer and Burns, 994), where n L and X gl are the Keler frequency and gyrofrequency evaluated at the launch distance r L : sffiffiffiffiffiffiffiffi GM n L ¼ ; ð6þ and r L X gl ¼ qb mc ¼ n L L X : Notice that in the gravity limit (L? ), Eq. () reduces to r L n L, the secific angular momentum about the lanet, while in the Lorentz limit (L? ), it is r X L gl, the secific angular momentum about the center of gyromotion that moves with the magnetic field. If the motion of the article is radially stable, it exhibits eicyclic motion about an equilibrium oint determined from Eq. (4). The existence of equilibrium oints ¼, in the equatorial lane (Northro and Hill, 98) and at high latitudes (Howard et al., 999, ). The local stability of the equilibrium oints, defined as whether oscillations about these oints remain small, is then determined by considering the second derivatives of the otential. Given our launch condition, we focus on the equatorial equilibrium oints which are of greatest interest. these, U q¼q c ;z¼ ¼, r? q, and radial and vertical motions are initially decouled and may be considered searately (Northro and Hill, 98; Mitchell et al., ). The equilibrium oint is the guiding center of eicyclic motion. Grains launched at the guiding center have canonical conjugate momenta that are different from our Keler-launched grains: namely, / ¼ m qðx c c þ X gc Þ, where x c is the orbital frequency of a grain at the guiding center, X gc is the gyrofrequency at the guiding center, and q c is the guiding center distance in the equatorial lane. A local radial stability analysis is most relevant for our Kelerlaunched grains if an equilibrium oint is not too distant. Accordingly, it is imortant to distinguish between quantities evaluated at the Keler launch osition and those determined at the guiding center. Here and throughout, we use the subscrit c for the guiding center and the subscrit L for the launch osition. At the ¼, which evaluates to: q¼qc ;z¼ x c q c þ GM L q c x c GM ¼ : X q c Physically, Eq. (8) just imlies a balance of forces in the rotating frame, whereby the centrifugal force, the Lorentz force and gravity sum to zero. We solve Eq. (8) for the angular seed of the guiding center x c, and find two real roots for L <, which includes all negative charges. For L > conversely, two equilibrium oints exist only if ðþ ð7þ ð8þ

6 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () q c R syn L 6 4ðL Þ : Two equilibria always exist inside and everywhere for L and L. There are no equilibrium oints in a region starting at (L =, q c = ) in Fig. b, and oening uward to include an increasing range of L values for increasing distance q c. In this region, no equilibrium oint exists, and grains are guaranteed to be locally unstable. Not surrisingly, this region is fully contained within the unstable ortion of Fig. b (darkest grey region outside ). The existence of an equilibrium oint, therefore, is a necessary rerequisite for stability. Additional instability in Fig. b comes from two sources: (i) the intrinsic instability of the equilibrium oint, if it exists, and (ii) large amlitude motions about a locally stable equilibrium oint. Large oscillations are beyond the scoe of a local stability analysis and so we focus on small amlitude radial motion near an equilibrium oint, which takes the form ð9þ A ¼ n L L X ; n L L X þ ; B ¼ n LL X C ¼ n L L X þ ; D ¼ L : To determine the radius of the eicycles (r g ) induced by a Keler launch, we follow the rocedure of Schaffer and Burns (994), and solve for the distance to the otential q¼qc ¼. Note that this is only valid to first order in small ;z¼ quantities, since we are effectively assuming that the otential is symmetric about the equilibrium oint. Evaluating the derivative, multilying by r, setting r = r L + r g, and assuming r g r L, we obtain the eicycle radius for a grain launched at r L in terms of arameters known at launch: q q ¼ : ðþ r L ðx n L ÞX gl r g ¼ X gl X : ðþ glðx þ n L Þþn L Small radial motions are stable U ¼ j c >, which, ;z¼ from Eq. (4) can be written as: j c ¼ x c 4x cx gc þ X gc ðþ (Mendis et al., 98; Northro and Hill, 98; Mitchell et al., ). Note here that the gyrofrequency X gc is evaluated at the guiding center, and is given by Eq. (7) with the subscrit change: L? c. The eicyclic frequency j c reduces to the Keler orbital frequency n c at the guiding center r c in the gravity limit, and to the gyrofrequency X gc in the Lorentz limit. Radial excursions in both of these cases are small and, since j c >, are guaranteed to be stable. Radial motions are also initially small near synchronous orbit where electromagnetic forces are very weak (Eq. ()), and so a local stability analysis is also alicable. Atsynchronous orbit, x c = n c = X and Eq. () reduces to j c ¼ X 4L þ L, which is ositive for small or large L. For ffiffiffi < L < þ ffiffiffi, however, Eq. () shows that radial motions near synchronous orbit are locally unstable. Comaring this analysis with Fig. b, we see that all orbits with r L that are locally stable are, not surrisingly, also globally stable. The converse, however, does not hold: although most of the locally unstable orbits are also globally unstable, some are in fact globally stable (e.g. L < just outside R syn in Fig. b). In conclusion, the local analysis is consistent with our numerical exeriments but cannot fully account for our stability boundaries. Accordingly, we turn to a global analysis, ausing first to ut the otential of Eq. (4) into a more useful form and to derive the radius of gyration, r g... Radius of gyration With our launch condition, grains are often far enough from an equilibrium oint that the small oscillation aroximation of Eq. () is invalid. This is articularly true far from and for L. Returning to the effective otential of Eq. (4) with the canonical conjugate momentum determined by launching the grain at the Keler seed (Eq. ()), and limiting our attention to lanar orbits for which z = and r = q, we exress the otential as a quartic olynomial function of distance and a quadratic function of L : Uðr; L Þ¼ GM A r4 L r L r þ B r L 4 r þ C r L r þ D r L ; ðþ r with dimensionless coefficients In this limit, the radial range of motion of a dust grain is simly jr g j, and the grain reaches a turning oint at r t = r L +r g. Note the sign conventions used here; r g and X gl may be either ositive or negative; thus negative grains (with X gl < ) always gyrate towards. Eq. () corrects a sign error in Schaffer and Burns (994) which led to an artificial disagreement between the numerical and analytical (a) r/r (b) r/r a d (μm).. L *. q < q >. Fig. 6. The radial range of (a) negative and (b) ositive grains launched azimuthally with the Keler seed v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi GM=r L at.r. Both numerical data (oints) and the analytical results (curves from Eq. ()) are included. The total radial excursion is twice the eicyclic radius r g.

7 46 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () 4 4 model in their Fig. 6. Eq. (), by contrast, shows excellent agreement with our numerical data for negative grains (Fig. 6a). The eak in Fig. 6a, for oscillations towards synchronous orbit, occurs at r g ¼ r L n L X n L þ X ; L ¼ X : n L ð4þ Eq. (4) redicts that grains with L ¼ X n L launched near reach about halfway to the synchronous orbital distance, in agreement with Fig. 6a. For the ositive grains, Eq. () gives the roer radial range about stable local minima in both the Lorentz limit and in the Keler limit (Fig. 6b). At critical values of L, however, jr g j? and the assumtions under which Eq. () was derived are violated. This is readily aarent in the decreasing quality of the match between the theory and the data for intermediate-sized grains in Fig. 6b. Note that this is the same region where Mitchell et al. () find large non-ellitical gyrations. Nevertheless, the relatively close agreement between theory and numerical data in Fig. 6 confirms that the eicyclic model is usually a good assumtion in lanetary magnetosheres. 4. Global radial stability analysis Our local radial stability analysis makes a number of successful redictions, but cannot fully account for the boundaries in Fig. b, rimarily because of the large radial excursions exerienced by the ositive grains. The quartic otential within the equatorial lane given by Eq. () contains all the information necessary to determine which grains strike the lanet and which escae into interlanetary sace. 4.. Escaing grains Close to the lanet, the A/r 4 term of Eq. () dominates, and U(r?,L )? +, while for the distant articles we have U(r?,L )?. Accordingly, the quartic otential can have at most three stationary oints (one local maximum and two local minima). Setting r = r L gives a simle form for the launch otential Uðr L ; L Þ¼ GM L : ðþ r L Energetically, a article is able to escae if U(r L,L )>U(r?,L ) = and we immediately recover the L < stability criterion of Hamilton and Burns (99b). Note that only ositive grains can escae from Juiter s diolar magnetic field and that, in rincile, grains with L > at all launch distances, both inside and outside are energetically able to escae. Whether or not they do so deends on the form of U(r,L ), in articular, on the ossible existence of an exterior otential maximum with U(r eak,l )>U(r L,L ). Analysis of Eq. () shows that the otential revents all grains launched with Keler initial conditions from crossing. Positive grains gyrate away from, while negative grains cannot reach (Eq. (), Fig. 6). Outside, U(r,L ) monotonically decreases for L J. Thus L ¼ is a global stability boundary and it matches Fig. b very well. For larger L (smaller grains), the toograhy is illustrated in Fig. 7. Stability is determined by the height of the distant eak in the otential. For L no such eak exists. For larger L, however, the radial otential decreases with distance from r L, then increases to the distant eak, and finally declines to zero as r?. Consider the quartic equation U(r,L ) U(r L,L ) =, which by construction, has one root at r = r L, and one root at a more distant turning oint r = r t. The critical quartic, where the turning oint is also a local maximum (as in Fig. 7) has a double root at r = r t.by U(r,L*)/ U(r L,L*) factoring out (r r L ), and then differentiating with resect to r, we find a quadratic equation for the location of the turning oint; r t varies smoothly from r t = r L at synchronous orbit to r t ¼ r L for r L. The stability boundary, r L (L ) starts at r ¼ ; L ¼ þ ffiffiffi and asymtotes to r L ¼ L 7 ð6þ for r L. Eq. (6) for r is a useful aroximation for the boundary far from, which nicely comliments the exact value we have found at the synchronous orbital distance. The full solution for the boundary r L (L ) is given by a rather messy cubic equation and so we resort to numerical methods for its solution, which we lot on Fig. 9b. 4.. Grains that strike the lanet Inside, the surface of the lanet resents a hysical boundary to radial motion. Particles that strike the atmoshere are slowed and removed from orbit. The otential at the lanet s surface U(q, z) varies with latitude, and so for simlicity, we restrict our attention to lanar motions where Eq. () alies. Since, for ositive grains in the equatorial lane, the otential declines as the grain moves inwards from its launch distance r L, it can have at most one local maximum within r L. There are thus two ways in which a grain can be revented from striking the lanet: (i) the otential U at the surface is greater than the launch otential, or (ii) a otential eak exists between the surface and the launch osition and its value is greater than or equal to the launch otential. These two scenarios are illustrated in Fig. 8. For case (i), the stability criterion is where U(R,L )=U(r L,L ). Using Eq. (), we find a quadratic exression in L that imlies two boundaries: n L r L X R r/r r L L R þ n Lr L XR! L þ r L R L * =8. bound L * =7.7 escaes Fig. 7. Potential wells for ositive grains launched from the solid oint just outside Juiter s synchronous orbit, at r L =.4R. For L = 8. (a d =.96 lm), a distant local maximum bounds the motions. If L = 7.7 (a d =.67 lm) the distant eak in the otential is at the radial turning oint, and the otential is equal to the launch otential; this is the stability threshold. For smaller L, the eak is lower and escae occurs. ¼ : ð7þ The two quadratic roots of Eq. (7), L and L, may be obtained analytically and are lotted on Fig. 9b. The roots obey the simle exression

8 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () a d (μm) U(r,L*)/ U(r L,L*) (i) (ii) r L /R (a)... Lorentz regime Keler regime NH8 q < κ c = Ω b r/r. Fig. 8. Potential wells for two lanar trajectories launched from the solid oints which are inside Juiter s synchronous orbit. Distances are in lanetary radii, and the otential is scaled to the launch value. Curve (i) (r L =.6R, L =.99,a d =. lm) has a otential eak higher than the launch otential inside the lanetary surface. Equating U(r L )=U(R ) gives an analytic solution (Eq. (7)) for the stability boundary in Fig. 9b. Curve (ii) (r L =.8R,L =.77, a d =.48 lm) has a otential eak outside the lanetary surface. In this case, the stability boundary is best obtained numerically. Both grains deicted here are oised on the stability threshold. r L /R (b). q > L L ¼ r LR < : ð8þ R syn. Eq. (8) conveniently highlights several features of the lower curves in Fig. 9b: The two curves marking the grains on the threshold of collision with the lanet are centered on L <, as required by Eq. (8). In addition, for smaller r L, the center of the instability shifts to smaller L, hence the left-most curve is steeer than the rightmost. Finally, a lanet with a larger (e.g. the Earth) will have roots that shift to very low L near the lanet. The curves determined by Eq. (7) match our numerical data cleanly with two imortant excetions. Firstly, because our method is only valid for grains that collide with the lanet in the equatorial lane (recall our assumtion z = ), it misses the high latitude collisions near (r L =R,L =) in Fig. 9b. All collisions exterior to the boundaries given by Eq. (7) necessarily involve substantial vertical motions, and the greyscale shading of Fig. 9b shows that they do. Secondly, our criterion redicts instability for a small region near (r L =,L =.) that our numerical data show in fact are stable. These grains encounter a high eak, similar to curve (ii) in Fig. 8, that revents them from reaching the lanetary surface. Thus U(r L,L )>U(R,L ) is a necessary condition for radial instability in the equator lane, but it is not sufficient. The additional requirement for instability is that U(r L,L )>U(r eak,l ), where r eak is the location of an intervening maximum. Just as for the escaing grains exterior to synchronous orbit, evaluation of this condition necessarily involves a cubic and a semi-analytic method. We find that no corrections to Eq. (8) are needed for the high L radial boundary and for all grains near the lanet. Only for the right-most curve near is there a discreancy. Our new curve is lotted in Fig. 9b and it erfectly matches the numerical instability boundary. Although the stability curve in this region can only be obtained semi-analytically, the oint at which it becomes necessary occurs when the otential maximum is located at the r¼r ¼ and U(r L,L )=U(R, L ). Evaluating these conditions, we find L ¼ r L R : þ r L R R = syn r= L R ð9þ For Juiter, the critical oint that satisfies both Eqs. (7) and (9) is at L =., r L =.694R (solid oint in Fig. 9b). The stability curve meets r L = at L ¼ ffiffiffi, a result suggested by our local stability analysis of Section. Note that our energy arguments yield analytic exressions both inside and outside. Arguments involving the location of otential maxima, conversely, require semi-analytic methods.. Local Vertical Stability Analysis The stability of grains against vertical erturbations was first exlored by Northro and Hill (98). In their model, a grain is launched on a circular orbit at the equilibrium orbital frequency x c in the otential of Eq. (4) so that there is no gyromotion around magnetic field lines. If the grain orbit at the equilibrium oint is stable to vertical erturbations, the square of the bounce frequency X b,givenby X b q¼qc ;z¼ ¼ GM q c L _ / c X þ L *.! ¼ x c n c. Fig. 9. Our new analytic results (heavy solid lines) are lotted over the numerical data from Fig.. (a) Northro s solution (dotted line) is suerseded by our two semianalytic boundaries where X b ¼ from Eq. (9). The new boundaries are a significantly better fit to the data and indicate an inner stability zone. The jj c j =X b curve indicates the : resonance between the eicyclic and the vertical bounce frequencies; it matches the data oints well. (b) We extend our vertical stability boundary to ositive grains. The radial stability boundaries for grains that escae or crash into the lanet are discussed in the text (Section 4). Between the oen circles at r L ¼ and L ¼ ffiffiffi, orbits are locally, radially unstable. The solid circle is the critical oint defined by Eqs. (7) and (9). ðþ

9 48 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () 4 4 (Northro and Hill, 98) is ositive. Here / _ c ¼ x c X is the dust grain s azimuthal frequency in the frame rotating with the magnetic field. When multilied by z, Eq. () gives the centrifugal (first term), and gravitational (last term) accelerations along a nearly vertical magnetic field line. For X b <, the vertical motion is unstable; note that the gravitational acceleration is negative and thus destabilizing. This follows from the fact that the diolar magnetic field curves toward the lanet and so a grain leaving the equatorial lane along a field line moves downhill in the gravitational otential. The (Northro and Hill, 98) solution for the boundary where X b ¼ is lotted in Figs. a and 9a. At distances closer to the lanet than a critical distance q crit, gravity forces grains to leave the equatorial lane... Vertical Instability in the Lorentz Limit In the limit of high charge-to-mass ratio, Eq. () can be solved exactly: q crit ¼ð=Þ :87: ðþ The effect of our initial condition, launching grains at the Keler seed, however, necessarily causes eicyclic gyromotion as the grain orbits the lanet. This leads to a stabilizing magnetic mirror force, in which the grain resists moving out of the equatorial lane to regions of higher magnetic field strength as discussed in Section.. Following the rocedure of (Lew (96)) and (Thomsen and van Allen (98)), the magnetic mirror force for equatorial itch angles near 9 adds a comonent of strength 9r g X gc =q c to Eq. (). In the Lorentz limit, Eq. () simlifies to r g X gc ¼ q c ðx n c Þ, and the bounce frequency can be found from X b ¼ X n c þ 9 ðn c XÞ : ðþ As above, the first two terms are due to the centrifugal and gravitational forces on a grain tied to a nearly-vertical magnetic field line. The third term of Eq. () is the magnetic mirror term, generalized to account for a rotating magnetic field. The three vertical accelerations add linearly, and are valid in the limit that L! and r g!. Fig. comares the (Northro and Hill (98)) bounce T b (hours).. r L /R Fig.. The bounce eriod for L ¼ 4 grains at Juiter over a range of launch distances. Northro s solution (Eq. (), dotted line) and our solution (Eq. (), solid lines) with T b ¼ =X b, are lotted alongside numerical data (oints). Note that our solution for T b is smaller than Northro s everywhere, excet at ¼ :4R. In the qffiffi limit r c, Eq. () shows that grains satisfy X b! n c, while for r c, qffiffiffiffi X b! X and T b! :6 h. eriod Eq. ()) with our Eq. ()) that accounts for eicyclic motion for small dust grains at Juiter. The Northro formalism erroneously redicts bounce eriods that are too long both inside and outside synchronous orbit and, more seriously, misses the second solution near the lanet. The third term in Eq. ()) is ositive everywhere inside the Northro boundary and thus leads to enhanced vertical stability. The stability boundaries in the high jl j limit are determined by Eq. ()); setting X b ¼, we find: r L ¼ = 9 ffiffiffi 6 :8; :84: ðþ These limits are valid for both ositive and negative grains with jl j!. Between these limits, X b < and grain orbits are locally unstable; the enhanced stability from the mirroring force moves the vertical stability boundary inwards from Northro s.87 to.84. A more imortant change, regained stability inside.8, is due to the higher launch seeds relative to the field lines, larger gyroradii, and a stronger magnetic mirror force. For Juiter these distances are at :9R and :87R resectively (see Fig. 9). Hints of this inner stability zone were seen numerically by (Northro and Hill (98a)) and (Northro and Connerney (987)); here we have derived analytical solutions for vertical stability in the Lorentz limit... Vertical instability for all charge-to-mass ratios To extend our model for bounce motion to all charge-to-mass ratios we must, in rincile, account for the variation in the strengths of the vertical gravitational, centrifugal and electromagnetic accelerations over one gyrocycle. Extending the electromagnetic mirror acceleration requires breaking the assumtion of erfectly circular gyrocycles, and is beyond the scoe of this work. The remaining two accelerations, however, can be extended to second order in r g =q c while retaining circular gyrations. We begin by writing the vertical acceleration as a function of the eicyclic hase z ¼ GM zðhþ q ðhþ! L /ðhþ _ þ : ð4þ X To first order in r g, the eicycles are circles in the guiding center frame. Setting h ¼ at the closest oint to the lanet, we find qðhþ ¼q c jr g j cos h; and ðþ _/ðhþ ¼ _ / c j c jr g j q c cos h: ð6þ Due to the geometry of a diole near its equator, an eicycle is tilted by an angle k (where k is the latitude). Hence the vertical offset is given by: zðhþ ¼z c jr gj q c cos h: ð7þ To calculate the bounce frequency, we average the restoring acceleration over an eicycle, a rocedure that is valid as long as j c X b : X b ¼ D hzi E ¼ U z zðhþdh: ð8þ Using Eqs. () and (6) to eliminate qðhþ and /ðhþ _ in Eq. (4), we exand to Oðr gþ, integrate Eq. (8)), and add in the magnetic mirroring term from Eq. () to obtain:

10 D. Jontof-Hutter, D.P. Hamilton / Icarus 8 () X b ¼ x c n c þ 9 ðn c XÞ r g 9 q c X _ gc / c þ n c : ð9þ The frequencies in Eq. (9): x c (Eq. (8)), n c (Eq. (6)), X gc (Eq. (7)), and _ / c ¼ x c X, are all evaluated at the guiding center of motion q c ¼ r L þ r g, which is determined by Eq. (). Our calculation adds two additional destabilizing terms that are strongest for intermediate values of L where gyroradii are largest (Fig. 6). How does our solution comare to numerical data? In Fig. 9, we lot our theoretical curves against the numerical data for both negative and ositive grains launched at the Keler rate in an aligned diole field for Juiter. We find the curves tracing the unstable zone semi-analytically by setting X b ¼ in Eq. (9). Within the regions bordered by the curves, trajectories are locally unstable but may remain globally bound due to high-latitude restoring forces. Our model closely matches the outer stability boundary for negative grains but is less successful for the inner boundary, esecially for moderate L. This is recisely where our derivation is weakest; recall that we have not accounted for higher-order corrections to the magnetic mirror force which are strongest close to the lanet and for jl j. Near jl j¼ eicycles become large and distorted for negative grains and even more so for ositive grains (Mendis et al., 98, ). Figure 6b shows that the eicyclic model matches the radial range of ositively-charged grains well for values L >. This is exactly where the numerical data deart from the theory in Fig. 9b. Aarently, large gyroradii and interference from the roximate radial instability stri lead to unmodeled effects and excess vertical instability. The curvature of the outer boundary in Fig. 9a is similar to that for the Northro instability, albeit dislaced to locations closer to the lanet. Notice that, with decreasing jl j, the instability region curves towards the lanet for negative grains, and away from it for ositive grains (Fig. 9). This is rimarily due to the x c n c term that determines the Northro boundary. For negative grains inside synchronous orbit, n c > x c, and x c increases with decreasing jl j due to a weakening outwardly-directed electromagnetic force. It thus takes a greater value of n c to make x c n c change sign, which destabilizes the vertical motion. Hence the boundary curves to lower launch distances in Fig. 9a. For the ositive grains in the Lorentz limit, by contrast, x c decreases as L decreases, and a smaller n c will destabilize the grain. Thus with decreasing L, the boundary in Fig. 9b curves u to higher launch radii. Finally, notice the band of locally unstable but globally stable oints that stretches from jl j. at the surface of the lanet to jl j at large distances in Figs. a and 9a. These grains are affected by a jj c j =X b resonance that coules their radial and vertical motions. Energy is transferred from the radial oscillation to a vertical oscillation and back again. Near the synchronous orbit, gyroradii are initially small and therefore there is not as much radial motion to transform into vertical motion; these grains do not reach our k m = threshold and aear as white sace in Fig. 9a. The existence of stable trajectories within the Northro boundary is an imortant result, articularly for small slowly-rotating lanets with distant synchronous orbits like Earth. Small dust grains generated by the collisional grinding of arent bodies on Kelerian orbits can remain in orbits near the lanetary surface. High energy lasma, like that found in Earth s van Allen radiation belts, is more stable than we have calculated here by virtue of exceedingly raid gyrations and a greatly enhanced mirroring force. Our analysis to this oint is comletely general and, although we have focused on Juiter, can be easily alied to other lanets. Saturn and Earth are logical choices, as their magnetic fields are also dominated by the g aligned diolar comonent. The aearance of the stability ma for any lanet deends on only the arameters and R, and not on the substantially different magnetic field strengths which, due to our use of L, only affect the conversion to grain radius a d. The synchronous orbital distance is somewhat closer to the lanetary surface at Saturn ( =.86R ) than at Juiter ( =.4R ), while at Earth ( = 6.6R )itis much further away. This leads to interesting differences between the lanets, as we shall see below. 6. Saturn and Earth A centered and aligned diole is an excellent aroximation for Saturn s magnetic field. We take g =.4 Gauss from Connerney et al. (984) and lot both our numerical data and analytical stability boundaries in Fig.. The Cassini measurement of g does not vary significantly from the older value that we use (Burton et al., 9). A lower synchronous orbit at Saturn ushes the local vertical instability inward, as exected from Eq. (). Comaring Fig. to Fig. 9, we see that the roximity of the surface at Saturn causes all the locally vertically unstable grains to hysically collide with the lanet. This is true for both negative and ositive grains. Outside synchronous orbit in Fig. b, the solutions derived for ositive escaing grains in Section 4 aly at Saturn to very high accuracy, for both the low L and high L boundaries. As in r L /R r L /R (a) (b).... Lorentz regime a d (μm).. L * κ c = Ω b Keler regime. q < q >. Fig.. Stability of charged grains at Saturn modeled with a centered and aligned diole field. All initial conditions and theoretical curves are as in Fig. 9. Also as in Fig. 9, the darkest shade of grey signifies low latitude collision or escae, the middle shade indicates high latitude collisions, and the lightest grey signifies large vertical excursions. For negative charges (anel a) only a tiny stable region exists near (r L = R,L = ) due to Saturn s smaller. Furthermore, due to the roximity to Saturn, nearly all grains that are locally vertically unstable do in fact hit the lanet, unlike their counterarts at Juiter. (b) Positive charges. As with the negative grains, nearly all the vertically unstable grains hit Saturn. Saturn s radial instability region (darkest grey) looks much like Juiter s.