Ring Self-gravity Research Articles

Axisymmetric viscous overstability in fully self-gravitating systems. Conditions for the onset of overstable oscillations

ABSTRACT We examine the onset of viscous overstability in dense self-gravitating particle rings. This oscillatory instability offers a plausible explanation for the periodic radial density variations seen at several locations of Saturn’s B and inner-A ring. So far, the theoretical understanding of overstable ring systems relies mainly on analytical results based on approximate treatments of ring self-gravity which omit the emergence of self-gravity wakes (small-scale gravitational instabilities). Therefore, the interplay between the two mechanisms, self-gravity and overstability, is still not well understood. Here, we address numerically the factors that determine the onset of overstability in self-gravitating rings. We confirm that weak self-gravity promotes overstability whereas strong self-gravity, with prominent wake structure, weakens the overstable pattern. This strong gravity regime corresponds to optical depths beyond a threshold value. In systems where overstable oscillations and wakes co-exist, we detect a strong anticorrelation between the strength of wakes and overstable oscillations. It is this interaction that eventually leads to the suppression of overstability, with strong transient wake structure preventing the phase synchronization between density and velocity oscillations required to maintain a coherent overstable pattern. Finally, we derive a composite criterion for the onset of overstability valid even in the regime of strong gravity wakes, framed in terms of collective properties of the system: the central-plane filling factor and the ratio of gravitational and collisional viscosities. In the limit where gravity wakes are omitted, our new criterion agrees with that found by the kinetic theory analysis of linear stability.

ANN-BODY INTEGRATOR FOR GRAVITATING PLANETARY RINGS, AND THE OUTER EDGE OF SATURN'S B RING

A new symplectic N-body integrator is introduced, one designed to calculate the global 360 degree evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. This freely-available code is called epi_int, and it is distinct from other such codes in its use of streamlines to calculate the effects of ring self-gravity. The great advantage of this approach is that the perturbing forces arise from smooth wires of ring matter rather than discreet particles, so there is very little gravitational scattering and so only a modest number of particles are needed to simulate, say, the scalloped edge of a resonantly confined ring or the propagation of spiral density waves. The code is applied to the outer edge of Saturn's B ring, and a comparison of Cassini measurements of the ring's forced response to simulations of Mimas' resonant perturbations reveals that the B ring's surface density at its outer edge is 195+-60 gm/cm^2 which, if the same everywhere across the ring would mean that the B ring's mass is about 90% of Mimas' mass. Cassini observations show that the B ring-edge has several free normal modes, which are long-lived disturbances of the ring-edge that are not driven by any known satellite resonances. Although the mechanism that excites or sustains these normal modes is unknown, we can plant such a disturbance at a simulated ring's edge, and find that these modes persist without any damping for more than ~10^5 orbits or ~100 yrs despite the simulated ring's viscosity of 100 cm^2/sec. These simulations also indicate that impulsive disturbances at a ring can excite long-lived normal modes, which suggests that an impact in the recent past by perhaps a cloud of cometary debris might have excited these disturbances which are quite common to many of Saturn's sharp-edged rings.

DYNAMICS OF THE SHARP EDGES OF BROAD PLANETARY RINGS

(Abridged) The following describes a model of a broad planetary ring whose sharp edge is confined by a satellite's m^th Lindblad resonance (LR). This model uses a streamline formalism to calculate the ring's internal forces, namely, ring gravity, pressure, viscosity, as well as a hypothetical drag force. The model calculates the streamlines' forced orbit elements and surface density throughout the perturbed ring. The model is then applied to the outer edge of Saturn's B ring, which is maintained by an m=2 inner LR with the satellite Mimas. Ring models are used to illustrate how a ring's perturbed state depends on the ring's physical properties: surface density, viscosity, dispersion velocity, and the hypothetical drag force. A comparison of models to the observed outer B ring suggests that the ring's surface density there is between 10 and 280 gm/cm^2. The ring's edge also indicates where the viscous torque counterbalances the perturbing satellite's gravitational torque on the ring. But an examination of seemingly conventional viscous B ring models shows that they all fail to balance these torques at the ring's edge. This is due ring self-gravity and the fact that a viscous ring tends to be nearly peri-aligned with the satellite, which reduces the satellite's torque on the ring and makes the ring's edge more difficult to maintain. Nonetheless, the following shows that a torque balance can still be achieved in a viscous B ring, but only in an extreme case where the ratio of the ring's bulk/shear viscosities satisfy ~10^4. However, if the dissipation of the ring's forced motions is instead dominated by a weak drag force, then the satellite can exert a much stronger torque that can counterbalance the ring's viscous torque.

The Secular Evolution of a Close Ring‐Satellite System: The Excitation of Spiral Density Waves at a Nearby Gap Edge

The secular perturbations exerted by an inclined satellite orbiting in a gap in a broad planetary ring tends to excite the inclinations of the nearby ring particles, and the ring's self-gravity can allow that disturbance to propagate away in the form of a spiral bending wave. The amplitude of this spiral bending wave is determined, as well as the wavelength, which shrinks as the waves propagate outwards due to the effects of the central planet's oblateness. The excitation of these bending waves also damps the satellite's inclination I. This secular I damping is also compared to the inclination excitation that is due to the satellite's many other vertical resonances in the ring, and the condition for inclination damping is determined. The secular I damping is likely responsible for confining the orbits of Saturn's two known gap-embedded moons, Pan and Daphnis, to the ring plane.

The Circumbinary Ring of KH 15D

The light curves of the pre-main-sequence star KH 15D from the years 1913-2003 can be understood if the star is a member of an eccentric binary that is encircled by a vertically thin, inclined ring of dusty gas. Eclipses occur whenever the reflex motion of a star carries it behind the circumbinary ring; the eclipses occur with a period equal to the binary orbital period of 48.4 days. Features of the light curve, including the amplitude of central reversals during mid-eclipse, the phase of the eclipse with respect to the binary orbit phase, the level of brightness out of eclipse, the depth of the eclipse, and the eclipse duty cycle, are all modulated on the timescale of nodal regression of the obscuring ring, in accord with the historical data. The ring has a mean radius near 3 AU and a radial width that is likely less than this value. While the inner boundary could be shepherded by the central binary, the outer boundary may require an exterior planet to confine it against viscous spreading. The ring must be vertically warped to maintain a nonzero inclination. Thermal pressure gradients and/or ring self-gravity can readily enforce rigid precession. In coming years, as the node of the ring regresses out of our line of sight toward the binary, the light curve from the system should cycle approximately back through its previous behavior. Near-term observations should seek to detect a mid-infrared excess from this system; we estimate the flux densities from the ring to be 3 mJy at wavelengths of 10-100 μm.

Three‐dimensional Dynamics of Narrow Planetary Rings

Narrow planetary rings are eccentric and inclined. Particles within a given ring must therefore share the same pericenter and node. We solve for the three-dimensional geometries and mass distributions that enable the Uranian α- and β-rings and the Saturnian Maxwell and Colombo (Titan) rings to maintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticle collisions, and the quadrupole field of the host planet balance each other to achieve this equilibrium. We prove that such an equilibrium is linearly stable. Predictions for the Saturnian ringlets to be tested by the Cassini spacecraft include (1) ringlet masses are of order a few × 1019 g, (2) surface mass densities should increase from ring midline to ring edges, and (3) rings are vertically warped such that the fractional variation of inclination across the ring is of order 10%. Analogous predictions are made for the Uranian rings. Simultaneous apsidal and nodal locking forces the narrowest portion of the ring—its "pinch," where self-gravitational and collisional forces are strongest—to circulate relative to the node and introduces previously unrecognized time-varying forces perpendicular to the planet's equator plane. We speculate that such periodic stressing might drive kilometer-scale bending waves at a frequency twice that of apsidal precession; such flexing might be observed over a few weeks by Cassini.

Apse Alignment of the Uranian Rings

An explanation of the dynamical mechanism for apse alignment of the eccentric uranian rings is necessary before observations can be used to determine properties such as ring masses, particle sizes, and elasticities. The leading model (P. Goldreich and S. Tremaine 1979, Astron J. 84, 1638–1641) relies on the ring self-gravity to accomplish this task, yet it yields equilibrium masses which are not in accord with Voyager radio measurements. We explore possible solutions such that the self-gravity and the collisional terms are both involved in the process of apse alignment. We consider limits that correspond to a hot and a cold ring, and we show that pressure terms may play a significant role in the equilibrium conditions for the narrow uranian rings. In the cold ring case, where the scale height of the ring near periapse is comparable to the ring particle size, we introduce a new pressure correction pertaining to a region of the ring where the particles are locked in their relative positions and jammed against their neighbors and the velocity dispersion is so low that the collisions are nearly elastic. In this case, we find a solution such that the ring self-gravity maintains apse alignment against both differential precession ( m=1 mode) and the fluid pressure. We apply this model to the uranian α ring and show that, compared to the previous self-gravity model, the mass estimate for this ring increases by an order of magnitude. In the case of a hot ring, where the scale height can reach a value as much as 50 times the particle size, we find velocity dispersion profiles that result in pressure forces which act in such a way as to alter the ring equilibrium conditions, again leading to a ring mass increase of an order of magnitude. We find that such a velocity dispersion profile would require a different mechanism than is currently envisioned for establishing a heating/cooling balance in a finite-sized, inelastic particle ring. Finally, we introduce an important correction to the model of E. I. Chiang and P. Goldreich (2000, Astrophys. J. 540, 1084–1090.). These authors relied on collisional forces in the last ∼100 m of an ∼10 km wide ring to increase ring equilibrium masses by up to a factor of ∼100. However, their treatment of ring edges as one-sided surface density drops leads to a strong dependence of the ring mass on the adjustable parameter λ (the length scale over which the ring's optical depth drops from order unity to zero at the edge). A treatment of the ring edges that takes into account their ridgelike structure retains the increase of ring mass of the order of ∼100 for a 10 km wide ring, while exhibiting weak dependence on λ. We conclude that a modified Chiang–Goldreich model can likely account for the masses of narrow, eccentric planetary rings; however, the role of shepherd satellites both in forming ring edges and in altering the streamline precession conditions near them needs to be explored further. It is also unclear whether a fully self-consistent ring model allows for the possibility of rings with negative eccentricity gradients.

Viscous Overstability in Saturn's B-Ring: II. Hydrodynamic Theory and Comparison to Simulations

We investigate the viscous oscillatory instability (overstability) of an unperturbed dense planetary ring, an instability that might play a role in the formation of radial structure in Saturn's B-ring. We generalize existing hydrodynamic models by including the heat flow equation in the analysis and compare our results to the development of overstable modes in local particle simulations. With the heat flow, in addition to the balance equations for mass and momentum, we take into account the balance law for the energy of the random motion; i.e., we allow for a thermal mode in a stability analysis of the stationary Keplerian flow. We also incorporate the effects of nonlocal transport of momentum and energy on the stability of the ring. In a companion paper (Salo, H., J. Schmidt, and F. Spahn 2001. Icarus, doi:10.1006/icar.2001.6680) we describe the determination of the local and nonlocal parts of the viscosity, the heat conductivity, the pressure, as well as the collisional cooling, together with their dependences on temperature and density, in local event-driven simulations of a planetary ring. The ring's self-gravity is taken into account in these simulations by an enhancement of the frequency of vertical oscillations Ω z >Ω. We use these values as parameters in our hydrodynamic model for the comparison to overstability in simulated rings of meter-sized inelastic particles of large optical depth with Ω z /Ω=3.6. We find that the inclusion of the energy-balance equation has a stabilizing influence on the overstable modes, shifting the stability boundary to higher optical depths, and moderating the growth rates of the instability, as compared to a purely isothermal treatment. The non-isothermal model predicts correctly the growth rates and oscillation frequencies of overstable modes in the simulations, as well as the phase shifts and relative amplitudes of the perturbations in density and radial and tangential velocity.

Apse Alignment of Narrow Eccentric Planetary Rings

The boundaries of the Uranian ∈, α, and β rings can be fitted by Keplerian ellipses. The pair of ellipses that outline a given ring share a common line of apsides. Apse alignment is surprising because the quadrupole moment of Uranus induces differential precession. We propose that rigid precession is maintained by a balance of forces due to ring self-gravity, planetary oblateness, and interparticle collisions. Collisional impulses play an especially dramatic role near ring edges. Pressure-induced accelerations are maximal near edges because there (1) velocity dispersions are enhanced by resonant satellite perturbations and (2) the surface density declines steeply. Remarkably, collisional forces felt by material in the last ~100 m of a ~10 km wide ring can increase equilibrium masses up to a factor of ~100. New ring surface densities are derived that accord with Voyager radio measurements. In contrast to previous models, collisionally modified self-gravity appears to allow for both negative and positive eccentricity gradients; why all narrow planetary rings exhibit positive eccentricity gradients remains an open question.

Hydrodynamical Simulations of Narrow Planetary Rings. I. Scaling

We investigate the global dynamics of the uranian rings using a modified 2-D smoothed particle hydrodynamic code combined with a 2-D tree code used to compute the particle-to-particle gravitational interactions. This code includes epicyclic fluid motion, nonaxisymmetric flow, local and nonlocal shear viscosity, self-consistent scale height evolution, ring–satellite gravitational interaction and coevolution, and ring self-gravity. To follow the scale height of each SPH particle we solve the vertical momentum equation for the flow using a Runge-Kutta scheme with a second order polynomial fit to the vertical behavior of the fluid pressure (N. Borderies et al., 1985, Icarus63, 406–420). The behavior of the fluid viscosity is obtained from I. Mosqueira (1996, Ph.D. thesis, Cornell University, Ithaca, NY), who found good agreement between a low optical depth extension of the high optical depth nonlocal viscosity model of Borderies et al. (1985) with the results of a local patch-code ring simulation. Our present viscosity model includes a further correction which accounts for the epicyclic limit to the mean free path (P. Goldreich and S. Tremaine, 1978, Icarus34, 227–239). This treatment covers both the high and the low ring density regimes so long as the fluid treatment remains valid. All energy source and sink terms in the presence of a nonzero fluid velocity divergence are self-consistently considered, including terms not considered in prior studies of rings. Furthermore, a new method has been tested and used to remove the noise contribution to the viscous momentum flux which occurs in traditional treatments of the viscosity within the SPH framework. In the present context this correction is needed to assure a physical behavior for the ring. Even within a 2-D framework the uranian rings are so narrow compared to their semimajor axes that radial resolution requires too many particles given our present computer resources. To address this issue we have developed a physical scaling that reduces the semimajor axis of the ring while preserving its width and, we believe, retains the relevant global satellite–ring dynamics. In the case of ring resolution on the order of kilometers, with a value of the scaling parameter that reduces the ring's semimajor axis by a factor of 20, our scaling allows for savings between a factor of 500 in the case of synodic time scales to a factor of 10,000 for viscous time scales. We expect this scaling to figure significantly in numerical studies of the global dynamics of planetary rings in the future. In the present study we use this scaling to follow the evolution of an epsilon-like ring in the presence of an m=2 mode (but no satellites) with and without self-gravity. In these runs, chosen as an initial demonstration of the usefulness and applicability of our scaling, the ring can be seen to evolve in the presence of shear.

Collisional Simulations of Satellite Lindblad Resonances: II. Formation of Narrow Ringlets

The effect of a perturbing satellite on a planetary ring at an isolated 2:1 inner Lindblad resonance is examined with direct N-body simulations. Particle-particle impacts with the velocity dependent coefficient of restitution are accurately calculated in the course of three-dimensional simulation, but the ring self-gravity is neglected. An isolated ringlet is observed to form in the resonance zone, resembling the narrow ringlets characteristic of the C-ring and Cassini Division. The carefully chosen simulation parameters lead to realistic dissipation that makes the ringlet formation possible; in our previous simulations (paper I) this effect was suppressed by the rapid inward drift of particles due to the unrealistically strong dissipation. Extra care has also been taken in scaling the simulation system parameters to the Prometheus 2: 1 inner Lindblad resonance, so that the lower limit for the mass of Prometheus can be estimated. Even if the mechanism of the torque in the collisional system is different from the self-gravitating systems, the theoretical linear standard torque (derived for systems with self-gravitating density waves) is found to be of the correct order of magnitude. The formation of the ringlet and the gap decreases the strength of the torque. With the simulation parameter values the observed nonlinear torque is about three times larger than the theoretical nonlinear estimate.

Precession of the epsilon ring of Uranus

It is noted that the outer and inner boundaries of the epsilon ring of Uranus can be fitted by aligned Keplerian ellipses. Four possible mechanisms for maintaining uniform precession in the epsilon ring are considered: the ring's self-gravity, precession due to a satellite, smooth pressure gradients, and shocklike phenomena. It is proposed that apse alignment is maintained by the self-gravity of the ring. In this case, a ring mass of approximately 5 x 10 to the 18th g and a mean surface density at quadrature of about 25 g/sq cm are estimated.