Rotational evolution of a symmetric gyrostat with visco-elastic bars around the center of mass in a circular orbit

The motion of a gyrostat in a circular orbit in a Newtonian field of force is considered. The gyrostat has four homogeneous viscoelastic bars attached to it. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Bending deformations of the bars, accompanied by dissipation of energy, are the cause of the evolution of the system's rotational motion. Approximate equations describing this evolution are derived, together with averaged equations in Andoyer variables.     

Dynamics and control of dual-spin gyrostat spacecraft with changing structure

We study the motion of the free dual-spin gyrostat spacecraft that consists of the platform with a triaxial ellipsoid of inertia and the rotor with a small asymmetry with respect to the axis of rotation. The system with perturbations caused by a small asymmetry of the rotor and the time-varying moments of inertia of the rotor is considered. The dimensionless equations of the system are written in Serret–Andoyer canonical variables. The system’s phase space is described. It is shown that changes in the moments of inertia of the gyrostat leads to the deformation of the phase space. The internal torque control law is proposed that keeps the system at the center point in the phase space. The effectiveness of the control is shown through a numerical simulation. It’s shown that the uncontrolled gyrostat can lose its axis orientation. Proposed internal torque keeps the initial angle between the axis of the gyrostat and the total angular momentum vector.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the motion around the centre of mass of a viscously elastic sphere in the restricted elliptic three-body problem

In the restricted three-body problem we consider the motion of a viscously elastic sphere (planet) with its centre of mass moving in a conditionally-periodic orbit. The approximate equations describing the rotational motion of the sphere in terms of the Andoyer variables are obtained by the method of the separation of motion and averaging; the evolution of the motion is also analysed.     

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.     

Exact solution of a triaxial gyrostat with one rotor

The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves.     

Some Equilibrium of the Restricted Three-Body Problem with Axial Symmetric Primaries and a Gyrostat as Infinitesimal Mass

The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

Orbits supporting bars within bars

High-resolution observations of the inner regions of barred disc galaxies have revealed many asymmetrical, small-scale central features, some of which are best described as secondary bars. Because orbital time-scales in the galaxy centre are short, secondary bars are likely to be dynamically decoupled from the main kiloparsec-scale bars. Here we show that regular orbits exist in such doubly barred potentials, and that they can support the bars in their motion. We find orbits in which particles remain on loops : closed curves which return to their original positions after two bars have come back to the same relative orientation. Stars trapped around stable loops could form the building blocks for a long-lived, doubly barred galaxy. Using the loop representation, we can find which orbits support the bars in their motion, and the constraints on the sizes and shapes of self-consistent double bars. In particular, it appears that a long-lived secondary bar may exist only when an inner Lindblad resonance is present in the primary bar, and that it would not extend beyond this resonance.     

Gravity-gradient stabilization of gyrostat satellites with rotor axes in principal planes

Three special classes of equilibrium orientations of gyrostat satellites subject to gravitational torques have been treated in the literature. Here we find the set of all equilibria for a restricted class of gyrostat configurations. Those configurations for which the internal angular momentum vector (or the rotor axis) is aligned with a principal axis have been treated in a separate work, where it is shown that at one, and only one, rotor speed there exists a continuum of equilibrium orientations. When the rotor axis is moved away from a principal axis in such a way that it is contained in a plane formed by two principal axes, it is shown that the continuum disappears, and we have a new set of eight equilibrium orientations which have not previously been described. The stability of these orientations is then investigated using the Hamiltonian as a Liapunov testing function. For properly chosen satellite inertia ratios some of these orientations are stable, and might be used in future gravitygradient stabilized satellites.This research is sponsored by the United States Air Force under Project RAND-Contract No. F44620-67-C-0045-monitored by the Directorate of Operational Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq. USAF. Views or conclusions contained in this study should not be interpreted as representing the official opinion or policy of the United States Air Force. The material presented here was originally published in RAND Corporation Memorandum RM-5921-PR. The author wishes to acknowledge his indebtedness to Dr. R. E. Roberson for helpful discussions, and for suggesting a research area, part of which is treated here.     

Equivalent problems in rigid body dynamics—II

The problem of motion of a dynamically symmetric gyrostat acted upon by non-symmetric potential forces admitting a cyclic integral is considered. This problem is brought into equivalence with another one concerning the motion of a similar gyrostat under the action of axisymmetric potential forces. Using this analogy, new integrable cases of each of the two problems are obtained from the known cases of the other. The equations of motion are reduced to a single equation of the second order.     

The Influence of Reactive Torques on Comet Nucleus Rotation

Reactive torques, due to anisotropic sublimation on a comet nucleus surface, produce slow variations of its rotation. In this paper the secular effects of this sublimation are studied. The general rotational equations of motion are averaged over unperturbed fast rotation around the mass center (Euler-Poinsot motion) and over the orbital comet motion. We discuss the parameters that define typical properties of the rotational evolution and discover different classifications of the rotational evolution. As an example we discuss some possible scenarios of rotational evolution for the nuclei of the comets Halley and Borrelly.     

Dynamical Evolution of Barred Galaxies

Angular momentum redistribution within barred galaxies drives their dynamical evolution. Angular momentum is emitted mainly by near-resonant material in the bar region and absorbed by resonant material mainly in the outer disc and in the halo. This exchange determines the strength of the bar, the decrease of its pattern speed, as well as its morphology. If the galaxy has also a gaseous component and/or a companion or satellite, then these also take part in the angular momentum exchange. During the evolution a bar structure forms in the inner parts of the halo as well. This bar is shorter and fatter than the disc bar and stays so all through the simulation, although its length grows considerably with time. Viewed edge-on, the bar in the disc component acquires a boxy or peanut shape. I describe the families of periodic orbits that explain such structures and review the observations showing that boxy/peanut ‘bulges’ are in fact just bars seen edge-on.     

Origin and structure of the Galactic disc(s)

We examine the chemical and dynamical structure in the solar neighbourhood of a model Galaxy that is the endpoint of a simulation of the chemical evolution of the Milky Way in the presence of radial mixing of stars and gas. Although the simulation's star formation rate declines monotonically from its unique peak and no merger or tidal event ever takes place, the model replicates all known properties of a thick disc, as well as matching special features of the local stellar population such as a metal-poor extension of the thin disc that has high rotational velocity. We divide the disc by chemistry and relate this dissection to observationally more convenient kinematic selection criteria. We conclude that the observed chemistry of the Galactic disc does not provide convincing evidence for a violent origin of the thick disc, as has been widely claimed.     

Evolution and destruction of bars

We summarize here some ongoing work by our group on the formation and evolution of barred galaxies. We first discuss the evolution of bars in isolated disc galaxies. Then we address the problem of bar destruction during interactions and mergers. Finally we describe work on generation and regeneration of bars in interactions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Evidence for a rotational velocity field about the white dwarf of V347 Puppis

We present time-resolved optical spectroscopy of the deeply eclipsing nova-like binary V347 Pup, which provides evidence for a rotational velocity field in the low-excitation lines, as expected from accretion disc emission. The low-excitation line profiles are a composite of the disc and the irradiated inner-face of the donor star. This picture is remarkably simple compared to the other members of the eclipsing nova-like family. Unsatisfactory radial velocity solutions for the accreting object provided by standard methods encourage us to apply a technique developed to measure stellar motion from tomographic reconstructions of the time-dependent line profiles. We find some evidence for a spiral pattern over the disc and for disc-overflow accretion. If the emission distribution proves to be a long-term feature, V347 Pup will provide an opportunity to study disc kinematics with greater confidence than allowed by the other eclipsing nova-like objects.     

Numerical simulations of interacting gas-rich barred galaxies: vertical impact of small companions

We investigate the dynamical effects of an interaction between an initially barred galaxy and a small spherical companion using an N -body/smoothed-particle-hydrodynamics algorithm. In the models described here the small companion passes through the disc of the larger galaxy nearly perpendicular to its plane. The impact positions and times are varied with respect to the phase of the bar and the dynamical evolution of the disc.
The interactions produce expanding ring structures, offset bars, spokes and other asymmetries in the stars and gas. These characteristic signatures of the interaction are present in the disc for about 1 Gyr. We find that in some cases it is possible to destroy the bar while keeping the disc structure. In general, the central impacts cause larger damage to the bar and the disc than the peripheral ones. The interaction tends to accelerate the transition from a strongly-barred galaxy to a weakly- or non-barred galaxy. The final disc morphology is determined more by the impact position relative to the bar rather than the impact time.     

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.     

Attitude stability of a gravity-stabilized gyrostat satellite

This paper treats analytically the problem of the stability of the attitude motions of a gravity-stabilized gyrostat satellite that is in a circular orbit around a spherical planet. The vehicle considered consists of a body with no special symmetries that has any number of rotors attached to it. The internal angular momentum vector due to these rotors is parallel to one of the principal axes of the entire satellite; this axis is aligned with (or close to) the normal to the orbit plane. Both the cases in which each rotor is driven by a motor at a constant spin rate relative to the main body of the vehicle and the one in which each rotor is rotating freely, without any friction, are treated. Stability (both infinitesimal and in the sense of Liapunov) of the attitude motions of the vehicle can be quickly predicted by using the results derived here, which are summarized in the form of a continuous, three-dimensional, stability diagram.National Research Council Post-Doctoral Research Associate at NASA-Ames Research Center, Moffett Field, Calif. Parts of this research were included in [12].     

On the migration of a system of protoplanets

The evolution of a system consisting of a protoplanetary disc with two embedded Jupiter-sized planets is studied numerically. The disc is assumed to be flat and non-self-gravitating; this is modelled by the planar (two-dimensional) Navier–Stokes equations. The mutual gravitational interaction of the planets and the star, and the gravitational torques of the disc acting on the planets and the central star are included. The planets have an initial mass of one Jupiter mass M _Jup each, and the radial distances from the star are one and two semimajor axes of Jupiter, respectively.
During the evolution a joint wide annular gap is created by the planets. Both planets increase their mass owing to accretion of gas from the disc: after about 2500 orbital periods of the inner planet it has reached a mass of 2.3  M _Jup, while the outer planet has reached a mass of 3.2  M _Jup. The net gravitational torques exerted by the disc on the planets result in an inward migration of the outer planet on time-scales comparable to the viscous evolution time of the disc. The semimajor axis of the inner planet remains constant as there is very little gas left in its vicinity to induce any migration. When the distance of close approach eventually becomes smaller than the mutual Hill radius, the eccentricities increase strongly and the system may become unstable.
If disc depletion occurs rapidly enough before the planets come too close to each other, a stable system similar to our own Solar system may remain. Otherwise the orbits may become unstable and produce systems like υ And.