Self-gravitating Ring Research Articles

Planetesimal Formation by the Gravitational Instability of Dust Ring Structures

We investigate the gravitational instability (GI) of dust ring structures and the formation of planetesimals by their gravitational collapse. The normalized dispersion relation of a self-gravitating ring structure includes two parameters that are related to its width and line mass (the mass per unit length). We survey these parameters and calculate the growth rate and wavenumber. Additionally, we investigate the formation of planetesimals by growth of the GI of the ring that is formed by the growth of the secular GI of the protoplanetary disk. We adopt a massive, dust-rich disk as a disk model. We find the range of radii for fragmentation by the ring GI as a function of the width of the ring. The innermost radius for the ring GI is smaller for a smaller ring width. We also determine the range of the initial planetesimal mass resulting from the fragmentation of the ring GI. Our results indicate that the planetesimal mass can be as large as 1028 g at its birth after the fragmentation. It can be as low as about 1025 g if the ring width is 0.1% of the ring radius, and the lower limit increases with the ring width. Furthermore, we obtain approximate formulae for the upper and lower limits of the planetesimal mass. We predict that the planetesimals formed by the ring GI have prograde rotations because of the Coriolis force acting on the contracting dust. This is consistent with the fact that many trans-Neptunian binaries exhibit prograde rotation.

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.

Swirling self-gravitating vortex as the imagination of the Hoag’s ring galaxy

It is shown that based on the concept of a toroidal self-gravitating ring vortex with a swirl (with an orbital motion along the torus generatrix), it is possible to explain all the main morphological features of the famous Hoag ring galaxy, including the still unexplained rotation of the central elliptical galaxy. The astronomical observations can be used, thus, to study complex vortex motions [The article based on the authors’ reports Bannikova et al., XIII International Conference to 100 years of Illia Mikhailovich Lifshitz, Kharkiv, KhNU (2017) and Kontorovich et al., XIV International Science Conference “Physical phenomena in solids”, Kharkiv, KhNU (2019) at the Conferences on condensed matter physics at Kharkiv National University, the first of which was dedicated to the 100th anniversary of the birth of Ilya Mikhailovich Lifshitz, a teacher and friend of Mark Azbel.].

An implementation of viscous pressure-force (‘soft-sphere’) model in REBOUND for local ring simulations

ABSTRACT Numerical simulations are an essential tool for studying dynamics of dense, self-gravitating planetary rings. Here, we present a new implementation of a soft-sphere collision model into the parallel gravitational N-body code REBOUND. Specifically, our module is designed for simulations done in the local shearing sheet approximation, and includes a simple viscoelastic model for the contact forces between colliding particles. The soft-sphere treatment is particularly important in regimes where collisions cannot be treated as independent binary impacts, as is done in the hard-sphere method already implemented in the code. The new module is fully parallelized with MPI and it can be used in conjunction with the available gravity octree method, also fully parallelized, to perform self-gravitating simulations. We check the validity of our soft-sphere model by successfully reproducing the main steady-state properties of non-gravitating and self-gravitating systems reported in earlier literature. Moreover, we show that it gives realistic results also in the high optical depth and strong self-gravity regimes, where the hard-sphere method fails due to particle overlaps. In addition, we test the CPU-scaling and efficiency of the MPI-parallelization. The algorithm proves to retain much of the original REBOUND speed without sacrificing the accuracy of the results. Our new tool can thus be applied with confidence to planetary ring studies, such as those aimed at understanding the onset and evolution of viscous overstability in dense self-gravitating rings.

Fragmentation of ring galaxies and transformation to clumpy galaxies

ABSTRACT We study the fragmentation of collisional ring galaxies (CRGs) using a linear perturbation analysis that computes the physical conditions of gravitational instability, as determined by the balance of self-gravity of the ring against pressure and Coriolis forces. We adopt our formalism to simulations of CRGs and show that the analysis can accurately characterize the stability and onset of fragmentation, although the linear theory appears to underpredict the number of fragments of an unstable CRG by a factor of 2. In addition, since the orthodox ‘density-wave’ model is inapplicable to such self-gravitating rings, we devise a simple approach that describes the rings propagating as material waves. We find that the toy model can predict whether the simulated CRGs fragment or not using information from their pre-collision states. We also apply our instability analysis to a CRG discovered at a high redshift, z = 2.19. We find that a quite high-velocity dispersion is required for the stability of the ring, and therefore the CRG should be unstable to ring fragmentation. CRGs are rarely observed at high redshifts, and this may be because CRGs are usually too faint. Since the fragmentation can induce active star formation and make the ring bright enough to observe, the instability could explain this rarity. An unstable CRG fragments into massive clumps retaining the initial disc rotation, and thus it would evolve into a clumpy galaxy with a low surface density in an interclump region.

Lyapunov exponent and criticality in the Hamiltonian mean field model

We investigate the dependence of the largest Lyapunov exponent (LLE) of an N-particle self-gravitating ring model at equilibrium with respect to the number of particles and its dependence on energy. This model has a continuous phase-transition from a ferromagnetic to homogeneous phase, and we numerically confirm with large scale simulations the existence of a critical exponent associated to the LLE, although at variance with the theoretical estimate. The existence of strong chaos in the magnetized state evidenced by a positive Lyapunov exponent is explained by the coupling of individual particle oscillations to the diffusive motion of the center of mass of the system and also results in a change of the scaling of the LLE with the number of particles. We also discuss thoroughly for the model the validity and limits of the approximations made by a geometrical model for their analytic estimate.

Ensemble inequivalence and Maxwell construction in the self-gravitating ring model

The statement that Gibbs equilibrium ensembles are equivalent is a base line in many approaches in the context of equilibrium statistical mechanics. However, as a known fact, for some physical systems this equivalence may not be true. In this paper we illustrate from first principles the inequivalence between the canonical and microcanonical ensembles for a system with long range interactions. We make use of molecular dynamics simulations and Monte Carlo simulations to explore the thermodynamics properties of the self-gravitating ring model and discuss on what conditions the Maxwell construction is applicable.

Thickness of Saturn’s B ring as derived from seasonal temperature variations measured by Cassini CIRS

Structural and thermal properties of Saturn’s B ring and its particles are derived from orbital and seasonal temperatures variations observed by the Cassini CIRS spectrometer between 2004 and 2009 equinox. Our multiscale thermal model (Ferrari, C., Reffet, E. [2013]. Icarus 223, 28–39), which assumes negligible heat transfer by vertical motion of particles in dense rings, is adjusted to the data. Most observations were focused on the center of the B ring, at 105,000km from Saturn. A very good fit is obtained for conductive particles embedded in a moderately conductive ring medium. Assuming a bulk composition of water ice, the thermal inertia of particles is found to be Γ1=160–200J/m2/K/s1/2 and to vary with seasons as part of the heat transfer is radiative, then temperature-dependent. For the same reason, the thermal inertia of the ring, Γ0, varies with seasons, between 30 and 35J/m2/K/s1/2. It is very comparable to the thermal inertia of icy satellite surfaces. The porosity of particles p1 found to fit this thermal inertia is very high (0.93) and may emphasize an inappropriate modeling of particles by an effective porous medium. The ring filling factor is fairly high, D=0.34±0.01, but stays typical of a compact medium and compatible with the output of numerical simulations of dense ring dynamics. The thickness of the B ring at 105,000km from Saturn is estimated at HS=2.2±0.2m. The observed correlation of its optical depth with the thermal gradient between lit and unlit sides is easily reproduced by the model if the radial variations in optical depth are due to varying thickness HS(a) with constant filling factor. This thickness varies between 1 and 3m across theB2,B3 and B4 regions. It is thinner than the neighboring C ring and Cassini Division. This can be understood as a consequence of self-gravity. The ring surface mass density Σ=(1-p1)ρ0DHS(a) derived from these structural parameters is too low for a self-gravitating ring, due to the high porosity found. But its radial variations, if deduced from coupling HS(a) with known aspect-ratios H/λ of self-gravity wakes, range between 300 in the inner B ring and most certainly exceed 1000kg/m2 in the B3 ring, values compatible with a self-gravitating ring. The inferred minimum total mass of Saturn's rings is MR=1.4±0.2·1019kg, close to the mass expected for a viscous disk evolving over the age of the Solar System and compatible with current higher estimates.

Self-gravitating ring of matter in orbit around a black hole: the innermost stable circular orbit

We study analytically a black-hole-ring system which is composed of a stationary axisymmetric ring of particles in orbit around a perturbed Kerr black hole of mass $M$. In particular, we calculate the shift in the orbital frequency of the innermost stable circular orbit (ISCO) due to the finite mass $m$ of the orbiting ring. It is shown that for thin rings of half-thickness $r\ll M$, the dominant finite-mass correction to the characteristic ISCO frequency stems from the self-gravitational potential energy of the ring (a term in the energy budget of the system which is quadratic in the mass $m$ of the ring). This dominant correction to the ISCO frequency is of order $O(\mu\ln(M/r))$, where $\mu\equiv m/M$ is the dimensionless mass of the ring. We show that the ISCO frequency increases (as compared to the ISCO frequency of an orbiting test-ring) due to the finite-mass effects of the self-gravitating ring.

Dynamics and physical interpretation of quasistationary states in systems with long-range interactions.

The time evolution of the one-particle distribution function of an N-particle classical Hamiltonian system with long-range interactions satisfies the Vlasov equation in the limit of infinite N. In this paper we present a new derivation of this result using a different approach allowing a discussion of the role of interparticle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed a quite comprehensive study of the quasistationary states (QSSs) though many aspects of the physical interpretations of these states still remain unclear. In this paper a proper definition of time scale for long time evolution is discussed, and several numerical results are presented for different values of N. Previous reports indicate that the lifetimes of the QSS scale as N1.7 or even the system properties scale with exp(N). However, preliminary results presented here indicates that time scale goes as N2 for a different type of initial condition. We also discuss how the form of the interparticle potential determines the convergence of the N-particle dynamics to the Vlasov equation. The results are obtained in the context of the following models: the Hamiltonian mean field, the Self-gravitating ring model, and one- and two-dimensional systems of gravitating particles. We have also provided information of the validity of the Vlasov equation for finite N.

Geometry of the energy landscape of the self-gravitating ring

We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapsed one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent λ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of λ and is also quantitatively correct at small energy densities and, in the limit N→∞, in the whole homogeneous phase.

VISCOSITY IN PLANETARY RINGS WITH SPINNING SELF-GRAVITATING PARTICLES

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.

Collisions and Gravitational Interactions between Particles in Planetary Rings

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.

FORMATION OF MOONLETS IN SATURN’S RINGS: THE ROLE OF THE CONSTRUCTIVE INTERFERENCE OF LIN-SHU-TYPE CIRCULAR AND SPIRAL DENSITY WAVES

Localized kilometer-sized propeller features, recently detected in CASSINI images of Saturn's A ring, may be interpreted as signatures of small hundred-meter-sized moonlets embedded within the broad ring. We propose that these moonlets are formed due to the constructive interference of the gravitationally unstable axisymmetric and nonaxisymmetric perturbations of the self-gravitating Saturnian ring disk. The hydrodynamical slab model of the disk is studied to determine its instability against low-frequency compression-type, or Lin-Shu-type, oscillations whose propagation vector is perpendicular to the axis of the disk's rotation. The linear instability analysis is performed in the following way. First, we derive the general dispersion relation for the gravity perturbations inside the slab within the standard local short-wavelength approach. Next, we match the solutions to the solutions of the Poisson equation outside the slab to arrive at the particular dispersion relation. The particular dispersion relation describes the development of symmetric Jeans' perturbations which cause density enhancements. From this dispersion relation, the growth rate of Jeans-unstable oscillations developing in the spatially homogeneous slab model is obtained. The dispersion relation shows that perturbations of both types, axisymmetric and nonaxisymmetric, are excited simultaneously due to the Jeans' gravitational instability. As a result, circular and spiral patterns develop. We suggest that clumping occurs due to the enhanced density increase at the intersections of the patterns. We also speculate that these clumps further develop into propeller moonlets.

Negative diffusion in planetary rings with a nearby moon

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.

Stochastic orbital migration of small bodies in Saturn’s rings

Many small moonlets, creating propeller structures, have been found in Saturn's rings by the Cassini spacecraft. We study the dynamical evolution of such 20-50m sized bodies which are embedded in Saturn's rings. We estimate the importance of various interaction processes with the ring particles on the moonlet's eccentricity and semi-major axis analytically. For low ring surface densities, the main effects on the evolution of the eccentricity and the semi-major axis are found to be due to collisions and the gravitational interaction with particles in the vicinity of the moonlet. For large surface densities, the gravitational interaction with self-gravitating wakes becomes important. We also perform realistic three dimensional, collisional N-body simulations with up to a quarter of a million particles. A new set of pseudo shear periodic boundary conditions is used which reduces the computational costs by an order of magnitude compared to previous studies. Our analytic estimates are confirmed to within a factor of two. On short timescales the evolution is always dominated by stochastic effects caused by collisions and gravitational interaction with self-gravitating ring particles. These result in a random walk of the moonlet's semi-major axis. The eccentricity of the moonlet quickly reaches an equilibrium value due to collisional damping. The average change in semi-major axis of the moonlet after 100 orbital periods is 10-100m. This translates to an offset in the azimuthal direction of several hundred kilometres. We expect that such a shift is easily observable.

Energy landscape and phase transitions in the self-gravitating ring model

We apply a recently proposed criterion for the existence of phase transitions, which is based on the properties of the saddles of the energy landscape, to a simplified model of a system with gravitational interactions referred to as the self-gravitating ring model. We show analytically that the criterion correctly singles out the phase transition between a homogeneous and a clustered phase and also suggests the presence of another phase transition not previously known. On the basis of the properties of the energy landscape we conjecture on the nature of the latter transition.

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n=1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy.

Regular motions of a self-gravitating ring in the field of a center of gravity

Information about the planetary rings of Saturn, Jupiter, Uranus, and Neptune obtained during the missions of Voyager, Galileo, and Cassini provided astronomers, physicists, and celestial mechanics with a large amount of data on striking and inexplicable phenomena, in particular, a small but noticeable eccentricity and thickness variability of some rings. Fridman and Gor’kavy œ [1] attribute the eccentricity to the interaction of the ring with the extended planetary atmosphere and attribute the variable thickness to the attraction of the ring particles by the planetary satellites. In this paper, we study the possibility of collisionless motion of the whole aggregate of particles forming a thin continuous self-gravitating ring in the gravity field of a motionless massive center without influence of other external factors (neighboring rings, satellites, etc.). The study revealed rings possessing the above effects.

Thermodynamics of the self-gravitating ring model: Analyses with new iterative method

In order to obtain the stable stationary mass distribution, we apply a new iterative method, inspired by a previous one used in 2D turbulence, which ensures entropy increase and, hence, convergence towards an equilibrium state. Applying new iterative method, we analyze the phase transition and the difference between microcanonical and canonical ensemble in an intermediate energy region.