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Heikki Salo - One of the best experts on this subject based on the ideXlab platform.
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a weakly nonlinear model for the damping of resonantly forced density waves in dense Planetary Rings
arXiv: Earth and Planetary Astrophysics, 2018Co-Authors: Marius Lehmann, Jürgen Schmidt, Heikki SaloAbstract:In this paper we address the stability of resonantly forced density waves in dense Planetary Rings. Already by Goldreich & Tremaine (1978) it has been argued that density waves might be unstable, depending on the relationship between the ring's viscosity and the surface mass density. In the recent paper Schmidt et al. (2016) we have pointed out that when - within a fluid description of the ring dynamics - the criterion for viscous overstability is satisfied, forced spiral density waves become unstable as well. In this case, linear theory fails to describe the damping, but nonlinearity of the underlying equations guarantees a finite amplitude and eventually a damping of the wave. We apply the multiple scale formalism to derive a weakly nonlinear damping relation from a hydrodynamical model. This relation describes the resonant excitation and nonlinear viscous damping of spiral density waves in a vertically integrated fluid disk with density dependent transport coefficients. The model consistently predicts density waves to be (linearly) unstable in a ring region where the conditions for viscous overstability are met. Sufficiently far away from the Lindblad resonance, the surface mass density perturbation is predicted to saturate to a constant value due to nonlinear viscous damping. The wave's damping lengths of the model depend on certain input parameters, such as the distance to the threshold for viscous overstability in parameter space and the ground state surface mass density.
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DYNAMICS OF SELF-GRAVITY WAKES IN DENSE Planetary Rings. I. PITCH ANGLE
The Astrophysical Journal, 2015Co-Authors: Shugo Michikoshi, Akihiko Fujii, Eiichiro Kokubo, Heikki SaloAbstract:We investigate the dynamics of self-gravity wakes in dense Planetary Rings. In particular, we examine how the pitch angles of self-gravity wakes depend on ring parameters using N-body simulations. We calculate the pitch angles using the two-dimensional autocorrelation function of the ring surface density. We obtain the pitch angles for the inner and outer parts of the autocorrelation function separately. We confirm that the pitch angles are 15°–30° for reasonable ring parameters, which are consistent with previous studies. We find that the inner pitch angle increases with the Saturnicentric distance, while it barely depends on the optical depth and the restitution coefficient of ring particles. The increase of the inner pitch angle with the Saturnicentric distance is consistent with the observations of the A ring. The outer pitch angle does not have a clear dependence on any ring parameters and is about 10°–15°. This value is consistent with the pitch angle of spiral arms in collisionless systems.
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dynamics of self gravity wakes in dense Planetary Rings i pitch angle
arXiv: Earth and Planetary Astrophysics, 2015Co-Authors: Shugo Michikoshi, Akihiko Fujii, Eiichiro Kokubo, Heikki SaloAbstract:We investigate the dynamics of self-gravity wakes in dense Planetary Rings. In particular, we examine how the pitch angle of self-gravity wakes depend on ring parameters using N-body simulations. We calculate the pitch angles using the two-dimensional autocorrelation function of the ring surface density. We obtain the pitch angles for the inner and outer parts of the autocorrelation function separately. We confirm that the pitch angles are 15 to 30 degrees for reasonable ring parameters, which are consistent with previous studies. We find that the inner pitch angle increases with the Saturnicentric distance, while it barely depends on the optical depth and the restitution coefficient of ring particles. The increase of the inner pitch angle with the Saturnicentric distance is consistent with the observations of the A ring. The outer pitch angle does not have the clear dependence on any ring parameters and is about 10 - 15 degrees. This value is consistent with the pitch angle of spiral arms in collisionless systems.
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N-body simulations of viscous instability of Planetary Rings
Icarus, 2010Co-Authors: Heikki Salo, Jürgen SchmidtAbstract:We study viscous instability of Planetary Rings in terms of N-body simulations. We show that for Rings composed of fairly elastic particles (e.g. as in Hatzes et al. [Hatzes, A., Bridges, F.G., Lin, D.N.C., 1988. Collisional properties of ice spheres at low impact velocities. Mon. Not. R. Astron. Soc. 231, 1091–1115]) the instability may lead to the spontaneous formation of dense ringlets in a background of lower density. In most parts of Saturn’s Rings the particle collisions are probably much more dissipative, as suggested by the presence of self-gravity wakes, and classic viscous instability should be suppressed. However, our results demonstrate that the mechanism of viscous instability itself is valid. The dynamical effects of size-dependent elasticity in a system with a size distribution have never been studied before. We show that this may in principle lead to a size-selective viscous instability, small particles concentrating on ringlets against the more uniform background of large particles.
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simulations of dense Planetary Rings iv spinning self gravitating particles with size distribution
Icarus, 2006Co-Authors: Ryuji Morishima, Heikki SaloAbstract:Abstract Previous self-gravitating simulations of dense Planetary Rings are extended to include particle spins. Both identical particles as well as systems with a modest range of particle sizes are examined. For a ring of identical particles, we find that mutual impact velocity is always close to the escape velocity of the particles, even if the total rms velocity dispersion of the system is much larger, due to collective motions associated to wakes induced by near-gravitational instability or by viscous overstability. As a result, the spin velocity (i.e., the product of the particle radius and the spin frequency) maintained by mutual impacts is also of the order of the escape velocity, provided that friction is significant. For the size distribution case, smaller particles have larger impact velocities and thus larger spin velocities, particularly in optically thick Rings, since small particles move rather freely between wakes. Nevertheless, the maximum ratio of spin velocities between the smallest and largest particles, as well as the ratio for translational velocities, stays below about 5 regardless of the width of the size distribution. Particle spin state is one of the important factors affecting the temperature difference between the lit and unlit face of Saturn's Rings. Our results suggest that, to good accuracy, the spin frequency is inversely proportional to the particle size. Therefore, the mixing ratio of fast rotators to slow rotators on the scale of the thermal relaxation time increases with the width of the particle size distribution. This will offer means to constrain the particle size distribution with the systematic thermal infrared observations carried by the Cassini probe.
Keiji Ohtsuki - One of the best experts on this subject based on the ideXlab platform.
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Collisions and Gravitational Interactions between Particles in Planetary Rings
Progress of Theoretical Physics Supplement, 2012Co-Authors: Keiji OhtsukiAbstract:Particles in Planetary Rings orbit the central planet, and undergo collisions and gravitational interactions with other particles. As a result, their orbits become inclined and non-circular. Also, ring particles likely have rough surfaces, and an oblique impact between them leads to rotation. In the case of dilute Rings where collision frequency is sufficiently smaller than the orbital frequency, particles’ orbits evolve through successive two-body collisions and/or gravitational encounters, and the evolution can be described by the formulation based on the three-body problem. We describe basic equations for such cases, and derive evolution equations for particle velocity dispersion and spin rates. We also discuss effects of Rings’ self-gravity, which become dominant in dense Rings. Collisions and gravitational interactions between particles result in angular momentum transfer in Planetary Rings. We discuss ring viscosity, which determines the rate of angular momentum transfer in Rings. Viscosity in dilute Rings can be expressed by particles’ velocity dispersion and is proportional to the ring surface density for a given particle size, while the dependence of the viscosity on ring surface density is stronger in dense self-gravitating Rings, where angular momentum is transferred by interactions between gravitational wakes.
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Viscosity in Planetary Rings with Spinning Self-gravitating Particles
The Astronomical Journal, 2012Co-Authors: Yuki Yasui, Keiji Ohtsuki, Hiroshi DaisakaAbstract:Using local N-body simulation, we examine viscosity in self-gravitating Planetary Rings. We investigate the dependence of viscosity on various parameters in detail, including the effects of particle surface friction. In the case of self-gravitating Rings with low optical depth, viscosity is determined by particle random velocity. Inclusion of surface friction slightly reduces both random velocity and viscosity when particle random velocity is determined by inelastic collisions, while surface friction slightly increases viscosity when gravitational encounters play a major role in particle velocity evolution, so that viscous heating balances with increased energy dissipation at collisions due to surface friction. We find that including surface friction changes viscosity in dilute Rings up to a factor of about two. In the case of self-gravitating dense Rings, viscosity is significantly increased due to the effects of gravitational wakes, and we find that varying restitution coefficients also change viscosity in such dense Rings by a factor of about two. We confirm that our numerical results for viscosity in dense Rings with gravitational wakes can be well approximated by a semianalytic expression that is consistent with a previously obtained formula. However, we find that this formula seems to overestimate viscosity in dense Rings far from the central planet, where temporary gravitational aggregates form. We derive semianalytic expressions that reproduce our numerical results well for the entire range of examined parameters.
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Local N-Body Simulations for the Rotation Rates of Particles in Planetary Rings
The Astronomical Journal, 2005Co-Authors: Keiji Ohtsuki, Daisuke ToyamaAbstract:Using N-body simulations, we examine rotation rates caused by mutual collisions of particles in Planetary Rings. In the case of particles with a power-law size distribution, we find that the size-dependence of the rms rotation rates can be well approximated by a power law. The overall rotation rates increase for a smaller tangential coefficient of restitution, an increased size of the largest particles, and/or a shallower size distribution of particles. However, the slope of the size dependence of the rotation rates is rather insensitive to these parameter values. This size dependence of the particles' rotation rates needs to be taken into account when developing energy balance models that are used to interpret observations of thermal emission from Saturn's Rings.
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On the rotation of a moonlet embedded in Planetary Rings
Icarus, 2004Co-Authors: Keiji OhtsukiAbstract:Abstract We examine the rotation of a small moonlet embedded in Planetary Rings caused by impacts of ring particles, using analytic calculation and numerical orbital integration for the three-body problem. Taking into account the Rayleigh distribution of particles' orbital eccentricities and inclinations, we evaluate both systematic and random components of rotation, where the former arises from an average of a large number of small impacts and the latter is contribution from large impacts. Calculations for parameter values corresponding to inner parts of Saturn's Rings show that a moonlet would spin slowly in the prograde direction if most impactors are small particles whose velocity dispersion is comparable to or smaller than the moonlet's escape velocity. However, we also find that the effect of the random component can be significant, if the velocity dispersion of particles is larger and/or impacts of large particles comparable to the moonlet's size are common: in this case, both prograde and retrograde rotations can be expected. In the case of a small moonlet embedded in Planetary Rings of equal-sized particles, we find that the systematic component dominates the moonlet rotation when m / M ≪ 0.1 (m and M are the mass of a particle and a moonlet, respectively), while the random component is dominant when m / M ≳ 0.3 . We derive the condition for the random component to dominate moonlet rotation on the basis of our results of three-body orbital integration, and confirm agreement with N-body simulation.
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A new formulation of the viscosity in Planetary Rings
Icarus, 2002Co-Authors: Hidekazu Tanaka, Keiji Ohtsuki, Hiroshi DaisakaAbstract:Abstract We present a new formulation of the viscosity in Planetary Rings, where particles interact through their gravitational forces and direct collisions. In the previous studies on the viscosity in self-gravitating Rings, the viscosity consists of three components, which are defined separately in different ways. The complex definitions make it difficult to evaluate the viscosity in N -body simulation of Rings. In our new formulation, the viscosity is expressed in terms of changes in orbital elements of particles due to particle interactions. This makes the expression of the viscosity simple. The new formulation gives a simple way to evaluate the viscosity in N -body simulation. We find that for practical evaluation of the viscosity of Planetary Rings, only energy dissipation at direct inelastic collisions is needed. For tenuous particle disks (i.e., optically thin disks), we further derive a formula of the viscosity. The formula requires only a numerical coefficient that can be obtained from three-body calculation. Since planetesimal disks are also tenuous, the viscosity in planetesimal disks can be also obtained from this formula. In a subsequent paper, we will evaluate this coefficient through three-body calculation and obtain the viscosity for a wide range of parameters such as the restitution coefficient and the radial location in Rings.
Ove Havnes - One of the best experts on this subject based on the ideXlab platform.
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The Charging of Planetary Rings
Space Science Reviews, 2008Co-Authors: A. L. Graps, Geraint H. Jones, A. Juhasz, Mihaly Horanyi, Ove HavnesAbstract:This chapter will review what is known about the charging of Planetary Rings, in particular the sum of the individual currents from the time-varying charge dQ/dt, of the Planetary ring particle. For the smallest ring particles, in addition to checking the plasma conditions for the charging currents, one must consider if collective effects in the ring environment are relevant. Two Planetary ring environments that have held a strong interest for ring scientists in the last two decades are Saturn’s spokes in the B Ring and the environment of Saturn’s E ring. Two sections of this chapter will describe these Planetary ring charging environments in detail. Finally, we describe two charging effects that demonstrate areas of future studies while providing fresh examples of the intriguing effects from Planetary ring charging processes.
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Mach cones in dusty plasmas in Planetary Rings and in laboratory experiments
Planetary and Space Science, 2001Co-Authors: Ove Havnes, T Aslaksen, T W Hartquist, F. Li, A BrattliAbstract:Abstract We have discussed in more detail the possibilities of extracting information on the dusty plasma conditions in Planetary Rings and in laboratories by observing the V-shaped Mach cone pattern around a charged body moving through or close to a layer of dusty plasma. Based on the existing theories for dust acoustic waves and accelerations of dust orbits at the front of the body we find that, if the normal plasma parameters are known, we should be able to extract information on the dust average sizes and size distribution, the dust number density and material density. With more refined theories for the dust acoustic wave and dust bow shocks, including a dust size distribution it should be possible to find additional and more accurate information on the total plasma conditions.
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Low frequency dust wave modes in Planetary Rings
Planetary and Space Science, 2000Co-Authors: Ove HavnesAbstract:Abstract The effects of gravity are taken into consideration for low frequency wave modes in a Planetary ring. The electrons and ions are considered to be magnetized and corotate with the planet, while the dust grains are unmagnetized. Under the action of gravity the dust particles oscillate normal to the ring plane and move around the planet on Keplerian orbits. Two wave modes of low frequency in such dusty plasma, propagating along the ring in the azimuthal direction, have been analyzed based on the susceptibilities derived from the kinetic theory. The first is a gravity-drift wave which is found as an instability in Planetary Rings. The instability exists in a limited space region around the synchronous orbit. Grains of small sizes are most likely to produce the instability, and the unstable wavelength increases with the distance from the synchronous orbit. This instability could be of some importance for the evolution of spokes in Saturn’s Rings. The second is a dust-magnetosonic wave in the gravity-influenced dust plasma of the ring. This mode has different dispersion and polarization character than a magnetosonic wave in an electron–ion plasma. A two-steam instability in the ring which can generate a dust-magnetosonic wave has also been analyzed.
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probing the properties of Planetary ring dust by the observation of mach cones
Journal of Geophysical Research, 1995Co-Authors: Ove Havnes, Frank Melandso, T Aslaksen, T W Hartquist, Gregor Eugen Morfill, Tore NitterAbstract:Compressive dust acoustic waves can be excited in dusty plasmas. Big boulders in Planetary Rings move at the Keplerian velocity, while smaller dust particles move at a slightly different velocity due to the action of the Lorentz force. If the difference in velocity Δυ is larger than the dust acoustic wave velocity, αd, a wake will be formed with an opening angle of 2θ where sin θ = |αd/Δυ|. The discovery of wakes and the measurement of their opening angles by the space experiment Cassini to Saturn will yield added information on the dusty plasma conditions in regions through which Cassini will not pass. We find that in some regions the waves that are excited by the boulders may be weak because a large fraction of the interacting dust is absorbed rather than deflected by the boulder. For a given dust size the most favourable conditions for the observations of wakes exist in two fairly narrow regions, one inside and one outside the corotation radius. The favorable regions are closest to the corotation radius for the smallest dust particles and progressively further away for larger dust particles.
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A kinetic model for dust acoustic waves applied to Planetary Rings
Journal of Geophysical Research, 1993Co-Authors: Frank Melandso, T Aslaksen, Ove HavnesAbstract:We have derived a kinetic model for the propagation of low-frequency waves in a dusty plasma containing dust particles and drifting plasma particles. The model includes Landau damping or growth and damping from charge variation on the dust particles, and is applied to dust-acoustic waves in Planetary Rings. Analytic expressions for the dispersion function are used to examine the stability of this wave mode. The dispersion properties are also found numerically for dense dust clouds or large drift velocities, where the analytical expressions are not applicable. We show how the stability condition depends on the density of dust particles and the wavelength, for plasma and dust parameters which may apply to Saturn's F ring, G ring, and E ring, and to Jupiter's ring.
Glen R Stewart - One of the best experts on this subject based on the ideXlab platform.
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negative diffusion in Planetary Rings with a nearby moon
Icarus, 2011Co-Authors: Mark C Lewis, Glen R Stewart, Jason Leezer, Amy WestAbstract:Abstract This paper analyzes a process that has been observed in simulations of numerous systems where ring material is strongly perturbed by a nearby moon. If the ring particles can be imparted with a forced eccentricity on the order of 10 −5 in a single pass by the moon, particle orbits are observed to move towards regions of higher density as a result of the organized collisions that occur in the dense peaks of the satellite wake. The width of the ring can decrease by as much as 90% if the forced eccentricity is greater than 3 × 10 −5 and the unperturbed geometric optical depth is greater than 0.03. The fractional change in ring width is relatively insensitive to the particle size so long as the particle radius is much less than the product of the semimajor axis and the forced eccentricity. Including a power law particle size distribution with slope of −2.8 spanning a decade in particle radius reduces the fractional width change by about 10% compared to the uniform particle-size case. Adding gravitational interactions between ring particles only has a significant effect on ring confinement if the unperturbed geometric optical depth exceeds .03, but a 40% reduction in ring width is still achieved in a self-gravitating ring of geometric optical depth 0.3 if the forced eccentricity exceeds 3 × 10 −5 . This process does not require the material to be in resonance with the moon, nor does it have any minimum mass constraints because particle self-gravity is not required. The collisional damping of satellite wakes therefore provides a simple mechanism by which a single moon can reduce the radial extent of any ringlet that is close to it and has sufficient optical depth for collisions to be significant.
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collisional dynamics of perturbed Planetary Rings i
The Astronomical Journal, 2000Co-Authors: Mark C Lewis, Glen R StewartAbstract:Local simulations of the outer edge of the Encke gap are presented that use particles of approximately the proper size and optical depth but ignore self-gravity. These simulations clearly show the formation and damping of wakes caused by the moonlet Pan. In this paper we focus primarily on shear reversal and the values of the pressure tensor in this system. We observe angular momentum luminosity reversal in the simulations, lending support to the prediction that this process could be responsible for the sharp edges seen at the Encke gap. We also find evidence for vertical splashing of particles out of the ring plane at the wake peaks. This vertical splashing violates assumptions made in many analytic treatments of this region but does not appear to invalidate the main conclusions drawn from those models. In addition, we find that the maximum amplitude of the wakes is limited by localized displacements of the particles' semimajor axes near the wake peaks.
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nonlinear satellite wakes in Planetary Rings i phase space kinematics
Icarus, 1991Co-Authors: Glen R StewartAbstract:Abstract An explicit expression is derived for the phase-space density of a Planetary ring perturbed by a nearly satellite. The derivation is facilitated by working in guiding center variables instead of local position and velocity variables and by neglecting collisions between ring particles. The usual equations for perturbed streamlines are recovered by taking first-order moments of the phase-space density. The local surface density and the local mean velocity in a nonlinear satellite wake are obtained in the form of convergent infinite series. Unlike previous estimates based on streamline crowding, this surface density is positive definite because the finite velocity dispersion of ring particles is taken into account. In other words, the phase-space density describes streamlines of finite width. The finite width of streamlines limits the maximum value of the local surface density that can result from streamline crowding. This result is consistent with numerical phase-space fluid simulations of perturbed Rings reported by Brophy, Esposito, and Stewart (1991) . The local mean velocity components in the satellite wake are found to deviate from the sinusoidal form of the streamline equations, and this deviation grows as the wake moves downstream from the shepherding satellite. These results suggest that collisional stresses between neighboring streamlines should depend on the second-order derivatives of the streamline parameters because the local mean velocity is itself a strong function of the first-order derivatives of the streamline parameters.
Henrik N. Latter - One of the best experts on this subject based on the ideXlab platform.
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Planetary Rings and Other Astrophysical Disks
arXiv: Earth and Planetary Astrophysics, 2018Co-Authors: Henrik N. Latter, Gordon I. Ogilvie, Hanno ReinAbstract:This chapter explores the physics shared by Planetary Rings and the various disks that populate the Universe. It begins with an observational overview, ranging from protoPlanetary disks to spiral galaxies, and then compares and contrasts these astrophysical disks with the Rings of the Solar System. Emphasis is placed on fundamental physics and dynamics, and how research into the two classes of object connects. Topics covered include disk formation, accretion, collisional processes, waves, instabilities, and satellite-disk interactions.
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Planetary Rings and other astrophysical disks
prs, 2018Co-Authors: Henrik N. Latter, Gordon I. Ogilvie, Hanno ReinAbstract:This chapter explores the physics shared by Planetary Rings and the various disks that populate the Universe. It begins with an observational overview, ranging from protoPlanetary disks to spiral galaxies, and then compares and contrasts these astrophysical disks with the Rings of the solar system. Emphasis is placed on fundamental physics and dynamics, and how research into the two classes of object connects. Topics covered include disk formation, accretion, collisional processes, waves, instabilities, and satellite–disk interactions. INTRODUCTION Disks are ubiquitous in astrophysics and participate in some of its most important processes. Most, but not all, feed a central mass: by facilitating the transfer of angular momentum, they permit the accretion of material that would otherwise remain in orbit (Lynden-Bell and Pringle, 1974). As a consequence, disks are essential to star, planet, and satellite formation (McKee and Ostriker, 2007; Williams and Cieza, 2011; Papaloizou and Terquem, 2006; Peale, 1999). They also regulate the growth of supermassive black holes and thus indirectly influence galactic structure and the intracluster medium (Volonteri, 2010; Fabian, 2012). Although astrophysical disks can vary by ten orders of magnitude in size and differ hugely in composition, all share the same basic dynamics and many physical phenomena. This review explores these areas of overlap. The prevalence of flattened astrophysical systems is a result of dissipation and rotation (Goldreich and Tremaine, 1982). A cloud of gas or debris in orbit around a central mass conserves its total angular momentum but not its energy, as there are numerous processes that may cool the cloud (inelastic physical collisions, Bremsstrahlung, molecular line emission, etc.). As a result, particles’ random velocities are steadily depleted – where “random velocity” is understood to be the component surplus to the circular orbit fixed by the angular momentum. The system contracts into a flat circular disk, the lowest energy state accessible. The contraction ends, and an equilibrium balance is achieved, once the cooling is met by heating (supplied by external irradiation or an internal viscous stress). Let us define a cylindrical coordinate system with its origin at the central mass and the vertical pointing in the direction of the total angular momentum vector. We describe systems as cold when the pressure gradient is weak and the final equilibrium very thin: radially the dominant force balance is between the centrifugal force and gravity, while vertically it is between pressure and gravity.
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the viscous overstability nonlinear wavetrains and finescale structure in dense Planetary Rings
Icarus, 2009Co-Authors: Henrik N. Latter, Gordon I. OgilvieAbstract:Abstract This paper addresses the fine-scale axisymmetric structure exhibited in Saturn's A and B-Rings. We aim to explain both the periodic microstructure on 150–220 m, revealed by the Cassini UVIS and RSS instruments, and the irregular variations in brightness on 1–10 km, reported by the Cassini ISS. We propose that the former structures correspond to the peaks and troughs of the nonlinear wavetrains that form naturally in a viscously overstable disk. The latter variations on longer scales may correspond to modulations and defects in the wavetrains' amplitudes and wavelength. We explore these ideas using a simple hydrodynamical model which captures the correct qualitative behaviour of a disk of inelastically colliding particles, while also permitting us to make progress with analytic and semi-analytic techniques. Specifically, we calculate a family of travelling nonlinear density waves and determine their stability properties. Detailed numerical simulations that confirm our basic results will appear in a following paper.
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Dense Planetary Rings and the viscous overstability
Icarus, 2008Co-Authors: Henrik N. Latter, Gordon I. OgilvieAbstract:Abstract This paper examines the onset of the viscous overstability in dense particulate Rings. First, we formulate a dense gas kinetic theory that is applicable to the saturnian system. Our model is essentially that of Araki and Tremaine [Araki, S., Tremaine, S., 1986. Icarus 65, 83–109], which we show can be both simplified and generalised. Second, we put this model to work computing the equilibrium properties of dense Planetary Rings, which we subsequently compare with the results of N-body simulations, namely those of Salo [Salo, H., 1991. Icarus 90, 254–270]. Finally, we present the linear stability analyses of these equilibrium states, and derive criteria for the onset of viscous overstability in the self-gravitating and non-self-gravitating cases. These are framed in terms of particle size, orbital frequency, optical depth, and the parameters of the collision law. Our results compare favourably with the simulations of Salo et al. [Salo, H., Schmidt, J., Spahn, F., 2001. Icarus 153, 295–315]. The accuracy and practicality of the continuum model we develop encourages its general use in future investigations of nonlinear phenomena.