Rigid body dynamics in the restricted ring problem of N + 1 bodies

The dynamic behavior of a small tri-axial body acted upon by the Newtonian forces of N major bodies of spherical symmetry which forma planar ring configuration is studied in this paper. The equations of the translational-rotational motion of the minor body are derived and its equilibrium states as well as their stability are investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multiple Periodic Orbits in the Ring Problem: Families of Triple Periodic Orbits

The ring problem deals with the motion of a small body which is subjected to the combined gravitational attraction of N massive bodies arranged in an annular configuration. In this paper we study the distribution of the triple periodic orbits in the phase space of the initial conditions and we discuss their evolution and their principal features. This revised version was published online in July 2006 with corrections to the Cover Date.     

A proper choice of variables for the study of a satellite's close approaches in a ring assembly of N massive bodies

Close approaches occur very often in celestial mechanics, giving rise to troubles in the integration process of the equations of motion. To encounter the so-appearing difficulties in the restricted ring problem of N+1 bodies, we propose new dependent and independent variables which allow the less-massive body to come very close to the central primary as well as to any peripheral primary.     

Collision mechanics and the structure of planetary ring edges

A numerical simulation of collisional evolution, originally developed to model planetary accretion processes, is applied to a hypothetical ring with parameters modeled after Saturn's rings in order to study changes in radial structure near ring edges. The tendency of rings to spread so as to conserve angular momentum while energy is dissipated in collisions is confirmed if random motion is in equilibrium. Even with no energy loss (coefficient of restitution in velocity ε = 1), spreading occurs becase random motion is increasing. With a moderately side-scattering collisional model, characteristic of collisions of nonrotating spheres (the slippery “billiard-ball” model), random motion increases for ε > 0.63, in agreement with analytical models. For isotropic scattering, which may be more realistic given particle rotation, damping dominates for ε up to 0.83. As long as random motion is damped, ring edges may contract rather than spread, producing concentrations of material just inside the ring edges reminiscent of results of earlier stimulation which did not precisely conserve angular momentum.     

A Planar Case of the n + 1 Body Problem: the 'Ring’ Problem

Our intention in this article is to present a new model for the investigation of the motion of a particle of negligible mass in a multibody surrounding. The proposed general planar configuration consists of ν = n-1 primaries arranged in equal arcs on an ideal ring and a central body of different mass located at the centre of mass of the system. We formulate the general equations of motion and we study the stationary solutions and the zero-velocity contours for various values of ν. This revised version was published online in July 2006 with corrections to the Cover Date.     

Collisionless Saturnian rings

We show that particles orbiting a central body ($\text{i.e.}$, Saturn's rings) can be assembled into one or more dense ($\text{i.e.}$, opaque) independent rings without interparticle collisions taking place despite the inevitable particle oscillations about the ring plane. The resultant apparent bulk motion is a slow “rolling” motion of the ring, as it orbits, the individual rings describing a “helical” motion. Such rings would only evolve due to external perturbations or (slow) internal gravitational perturbations, since the particles need never collide. This picture opens up the possibility of having hollow rings, for example. Moreover, it is possible that an initially uniform disk of randomly moving particles may spontaneously separate into a series of such rings. The consequence would be a striated disk having virtually zero internal viscosity.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

A model for the formation of spokes in Saturn's ring

We suggest that spokes consist of charged micron-sized dust particles elevated from the rings by radially moving dense plasma columns created by meteor impacts on the ring. Dense plasma causes electrostatic wall-sheaths at the ring and charging of the ring with electric fields strong enough to overcome the gravitational force on small dust particles. Under “ordinary” conditions only very few dust particles will be elevated as the probability of a dust particle having at least one excess electronic charge is very low. Dense plasma raises this probability significantly. The radial motion of the plasma column is due to an azimuthal polarization electric field built up by the relative motion between the corotating plasma and the negatively charged dust particles which move with a Keplerian speed.     

Extension of Cassini's Laws

We investigate the Cassini's laws which describe the rotational motion in a 1:1 spin-orbit resonance. When this rotational motion follows the conventional Cassini's laws, the figure axis coincides with the angular momentum axis. In this case we underline the differences between the rotational Hamiltonian for a 'slow rotating' body like the Moon and for a 'fast rotating' body like Phobos. Then, we study a more realistic rotational Hamiltonian where the angle J between the figure axis and the angular momentum axis could be different from zero. This Hamiltonian has not been studied before. We have found a new particular solution for this Hamiltonian which could be seen as an extension of the Cassini's laws. In this new solution the angle J is constant, which is not zero, and the precession of the angular momentum plane is equal to the mean motion of the argument of pericenter of the rotating body. This type of rotational motion is only possible when the orbital eccentricity of the rotating body is not zero. This new law enables describing in particular, the Moon mean rotational motion for which the mean value of the angle J is found to be equal to 103.9±0.7 s of arc.     

On the damping of the bending waves in Saturn’s ring

We study numerically the motion of a single particle in the bending wave of finite thickness in Saturn’s ring. We include the forcing due to the planet, a moon, the coriolis force and the self gravity of the ring. In particular, we compute the variation of the velocity arising due to the variation of the amplitude and the phase of the epicyclic motion across the local vertical height of the ring. We suggest that the dissipation of energy due to the collision of ring particles in this shear layer damps out the bending wave of Saturn’s ring at the 5:3 vertical resonance of Mimas within a distance of 150 km from the site of its launching as is observed in Voyager data.     

A multilayer model for thermal infrared emission of Saturn’s rings II: Albedo, spins, and vertical mixing of ring particles inferred from Cassini CIRS

Since the Saturn orbit insertion of the Cassini spacecraft in mid-2004, the Cassini composite infrared spectrometer (CIRS) measured temperatures of Saturn’s main rings at various observational geometries. In the present study, we apply our new thermal model (Morishima, R., Salo, H., Ohtsuki, K. [2009]. Icarus 201, 634-654) for fitting to the early phase Cassini data (Spilker, L.J., and 11 colleagues [2006]. Planet. Space Sci. 54, 1167-1176). Our model is based on classical radiative transfer and takes into account the heat transport due to particle motion in the azimuthal and vertical directions. The model assumes a bimodal size distribution consisting of small fast rotators and large slow rotators. We estimated the bolometric Bond albedo, A_V, the fraction of fast rotators in cross section, f_fast, and the thermal inertia, Γ, by the data fitting at every radius from the inner C ring to the outer A ring. The albedo A_V is 0.1-0.4, 0.5-0.7, 0.4, 0.5 for the C ring, the B ring, the Cassini division, and the A ring, respectively. The fraction f_fast depends on the ratio of scale height of fast rotators to that of slow rotators, h_r. When h_r = 1, f_fast is roughly half for the entire rings, except for the A ring, where f_fast increases from 0.5 to 0.9 with increasing saturnocentric radius. When h_r increases from 1 to 3, f_fast decreases by 0.2-0.4 for the B and A rings while no change in f_fast is seen for the optically thin C ring and Cassini division. The large f_fast seen in the outer A ring probably indicates that a large number of small particles detach from large particles in high velocity collisions due to satellite perturbations or self-gravity wakes. The thermal inertia, Γ, is constrained from the efficiency of the vertical heat transport due to particle motion between the lit and unlit faces, and is coupled with the type of vertical motion. We found that in most regions, except for the mid B ring, sinusoidal vertical motion without bouncing is more reasonable than cycloidal motion assuming bouncing at the midplane, because the latter motion gives too large Γ as compared with previous estimations. For the mid B ring, where the optical depth is highest in Saturn’s rings, cycloidal vertical motion is more reasonable than sinusoidal vertical motion which gives too small Γ.     

Equivalent problems in rigid body dynamics — I

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.     

Classes of ring galaxies generated by dynamical friction

Dynamical friction is incorporated in the numerical modeling of colliding ring galaxies. We show that dynamical friction may lead to the capture of an initially unbound companion into a damped oscillatory orbit before ultimately merging with the target disk. In the late stages of its motion the companion may be virtually merged with the main galaxy while rings are still propagating in the disk. This raises the possibility that there exist (at least) two classes of ring galaxies: a class of ring galaxies with separate detached companions in which dynamical friction is too weak to have captured the collider, and a second class with no apparent companion in which the latter has either merged with the target galaxy or has disrupted. These two classes could possibly be identified with O-type and P-type ring galaxies of Few and Madore (1986).     

Application of a Fictitious Attracting Center to the Study of the Motion of Minor Bodies Approaching Major Planets

The motion of minor Solar System bodies having close encounters with major planets is described using the model of motion within the framework of the perturbed restricted three-body problem. The actual motion of a minor body is represented as a combination of two motions, namely, the motion of a fictitious attracting center with a variable mass and the motion with respect to the fictitious center. The position and mass of the fictitious center are chosen so that, when the minor body collides with any of the primaries, the fictitious center carries into the center of inertia of the colliding body and the mass of the fictitious center becomes identical to the mass of this body. The regularizing KS-transformation and Sundman’s time transformation were applied to coordinates and velocities. As a result, a system of differential equations of motion that are quasilinear within the nearest vicinity of each of the primary attracting bodies was obtained. These equations are characterized by a numerical behavior during the encounters of the minor body with the primaries that is essentially better than that of the initial equations of motion. The motion of comets Brooks 2 and Gehrels 3, which have fairly close encounters with Jupiter, is simulated.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 272–280.Original Russian Text Copyright © 2005 by Shefer.     

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses be considered. The general solution of the equations of motion of the tri-axial body be obtained in which the motion of the spherical bodies is determined by the classic nonsteady Gyldén-Meshcherskii problem.     

The Voyager 1/Saturn encounter and the cosmogonic shadow effect

If an electrically conducting medium (e.g. a dusty plasma) rotates around a gravitating central body, which possesses an axisymmetric dipole field, the medium is supported to two-thirds by the centrifugal force and to one-third by electromagnetic forces under the condition that the magnetic field is strong enough to controll the motion. If the electromagnetic forces disappear — e.g. by a de-ionisation of the dusty plasma — the medium will fall down to two-thirds of its original central distance. The result of this process will be a cosmogonic shadow effect which is described in some detail.The Voyager 1/Saturn results demonstrate that the macro-structure of the Saturnian ring system can be explained as a result of this effect working at the formation of the system. The agreement between the theoretical results and the observations is better than a few percent.A similar analysis of the asteroidal belt shows that its macro-structure can also be explained by the cosmogonic shadow effect. The agreement between theory and observations is perhaps even better than in the Saturnian ring system.The observational results demonstrate that during their formation both the Saturnian ring and the asteroidal belt passed a plasma state dominated by electromagnetic effects.     

On self–killing and self–creating dynamos

Numerical studies with a spherical dynamo model have shown two remarkable phenomena. The model consists of a spherical body of an electrically conducting incompressible uid surrounded by free space. In addition to a rotation of the body an inner motion due to a given forcing is considered which satisfies a no–slip condition at the boundary. The full interaction of magnetic field and motion is taken into account. Starting from a fluid motion capable of dynamo action and a very weak magnetic field it was observed that the growing magnetic field destroys the dynamo property of the motion and then decays, and that the system ends up in a state with another motion incapable of dynamo action and zero magnetic field. In another case with a motion unable to prevent small magnetic fields from decay it proved to be possible that stronger magnetic fields deform it so that a dynamo starts to work which enables the system to approach a steady state with a finite magnetic field.     

Relativistic dynamical friction in a collisional fluid

The dynamical friction force experienced by a body moving at relativistic speed in a gaseous medium is examined. This force, which arises due to the gravitational interaction of the body with its own gravitationally-induced wake, is calculated for straight-line motion and circular motion, generalizing previous results by several authors. Possible applications to the study of extreme mass-ratio inspirals around strongly accreting supermassive black holes are suggested.