Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.     

Meridional Motion and Reynolds Stress from Debrecen Photoheliographic Data

The Debrecen Photoheliographic Data catalogue is a continuation of the Greenwich Photoheliographic Results providing daily positions of sunspots and sunspot groups. We analyse the data for sunspot groups focussing on meridional motions and transfer of angular momentum towards the solar equator. Velocities are calculated with a daily shift method including an automatic iterative process of removing the outliers. Apart from the standard differential rotation profile, we find meridional motion directed towards the zone of solar activity. The difference in measured meridional flow in comparison to Doppler measurements and some other tracer measurements is interpreted as a consequence of different flow patterns inside and outside of active regions. We also find a statistically significant dependence of meridional motion on rotation velocity residuals confirming the transfer of angular momentum towards the equator. Analysis of horizontal Reynolds stress reveals that the transfer of angular momentum is stronger with increasing latitude up to about \(40^{\circ}\), where there is a possible maximum in absolute value.     

The structure of motion in a 4-component galaxy mass model

We use a composite galaxy model consisting of a disk-halo, bulge, nucleus and dark-halo components in order to investigate the motion of stars in ther-z plane. It is observed that high angular momentum stars move in regular orbits. The majority of orbits are box orbits. There are also banana-like orbits. For a given value of energy, only a fraction of the low angular momentum stars — those going near the nucleus — show chaotic motion while the rest move in regular orbits. Again one observes the above two kinds of orbits. In addition to the above one can also see orbits with the characteristics of the 2/3 and 3/4 resonance. It is also shown that, in the absence of the bulge component, the area of chaotic motion in the surface of section increases, significantly. This suggests that a larger number of low angular momentum stars are in chaotic orbits in galaxies with massive nuclei and no bulge components.     

Tidal Angular Momentum in Close Binary Systems in presence of Wind Driven Non-conservative Mass Transfer with Uniform Mass Accretion Rate

A very well-known property of close binary stars is that they usually rotate slowly than a similar type single star. Massive stars in close binary systems are supposed to experience an exchange of mass and angular momentum via mass transfer and tidal interaction, and thus the evolution of binary stars becomes more complex than that of individual stars. In recent times, it has become clear that a large number of massive stars interact with binary companions before they die. The observation also reveals that in close pairs the rotation tends to be synchronized with the orbital motion and the companions are naturally tempted to invoke tidal friction. We here introduce the effect of tidal angular momentum in the model of wind driven non-conservative mass transfer taking mass accretion rate as uniform with respect to time. To model the angular momentum evolution of a low mass main sequence companion star can be a challenging task. So, to make the present study more interesting, we have considered initial masses of the donor and gainer stars at the proximity of bottom-line main sequence stars and they are taken with lower angular momentum. We have produced a graphical profile of the rate of change of tidal angular momentum and the variation of tidal angular momentum with respect to time under the present consideration.     

Angular momentum and gyroscope in flat space-time theory of gravitation

The study of a previously proposed theory of gravitation in flat space-time (Petry, 1981a) is continued. A conservation law for the angular momentum is derived. Additional to the usual form, there must be added a term coming from the spin of the gravitational field. The equations of motion and of spin angular momentum for a spinning test particle in a gravitational field are given. An approximation of the equations of the spin angular momentum in the rest frame of the test particle is studied. For a gyroscope in an orbit of a rotating massive body (e.g., the Earth) the precession of the spin axis agrees with the result of Einstein's general theory of relativity.     

The axial angular momentum of the general circulation of the oceans

With the Hamburg geostrophic ocean general circulation model, the temporal variation of the angular momentum of the oceans is studied. The input wind stresses consist of climatological winds plus the deviations of the actual winds 1981–1989. The motion part of the angular momenta of the two hemispheres resembles the (counter-phase) behaviour of the corresponding atmospheric angular momentum. The dominating matter part however shows no phase shift between the hemispheres. The sum of the angular momenta of the solid Earth and the atmosphere is deduced from observations; in principle it should be inversely correlated to the oceanic angular momentum. A first attempt to recognize the correlation between these empirical data and the theoretical values is described.     

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.     

Kinematical modeling of the Earth rotation,focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.     

On linear equations of variation in dynamical problems

The linear equations of variation, associated with a motion of a particle moving in a plane under a field of force which admits a first integral of the motion of any form, are drawn up in terms of the tangential and normal displacements. The existence of the first integral implies that the normal displacement satisfies a single second-order differential equation, the tangential displacement being given from the solution of this by a single quadrature. The special cases are examined in which the integral is one of energy, and in which it is one of angular momentum. The extension is made to the motion of two particles moving in a plane under a conservative force-field depending on their positions, which admits also an integral of angular momentum. (The study of the relative motion in the gravitational problem of three bodies in the plane may be put into this form by Jacobi's formulation). An equation is given for finding the non-zero characteristic exponents of a periodic solution of this second problem.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Formation of solar nebula and mass distribution in the planetary system

The formation of the solar nebula and the distribution of mass in its planetary system is studied. The underlying idea is that the protosun, fragmented out from an interstellar cloud as a result of cluster formation, gathered the planetary material and, hence, spin angular momentum by gravitational accretion during its orbital motion around the centre of the Galaxy. The study gives the initial angular momentum of the solar nebula nearly equal to the present value of the solar system.     

Relative motion of satellites exploiting the super-integrability of Kepler’s problem

This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the Sun to visit the Kuiper belt.     

On synchronism between axial rotation and orbital motion in close binary systems

In this paper we examine the possible outcome of the tidal evolution of a close binary system using a method from which the outline has already been given by Counselman (1973). If the value of the total angular momentum of the system is sufficiently large, two equilibrium states corresponding to synchronism between stellar rotation and orbital motion are possible. In one of these states the total energy attains no extreme value. The considered evolution can be visualized geometrically by the motion of a point along a hyperbolic cylinder in three-dimensional space. A comparison with some observational data reveals that most of the synchronously rotating detached systems have attained a stable equilibrium state of minimum total energy for the given value of total angular momentum.     

Dynamical model of binary asteroid systems using binary octahedrons

We used binary octahedrons to investigate the dynamical behaviors of binary asteroid systems. The mutual potential of the binary polyhedron method is derived from the fourth order to the sixth order. The irregular shapes, relative orbits, attitude angles, as well as the angular velocities of the binary asteroid system are included in the model. We investigated the relative trajectory of the secondary relative to the primary, the total angular momentum and total energy of the system, the three-axis attitude angular velocity of the binary system, as well as the angular momentum of the two components. The relative errors of the total angular momentum and the total energy indicate that the calculation has a high precision. It is found that the influence of the orbital and attitude motion of the primary from the gravitational force of the secondary is obvious. This study is useful in understanding the complicated dynamical behaviors of the binary asteroid systems discovered in our Solar system.     

Note on Kepler's problem with variable angular momentum

Based on the equations of planar motion with drag used in point dynamics, a simple derivation is presented that shows how a constant Keplerian (elliptical) orbit can be obtained with a non-central force field and variable angular momentum.     

Oceanic and solid Earth angular momentum

We reconsider two hypotheses used in calculating the transfer of angular momentum between the oceans and the solid Earth: (1) The locked-cean-ypothesis was already given up some time ago; here we provide a simple manner of understanding the relative importance of the motion and matter term. (2) The isolation hypothesis implied the isolation of the whole Earth in short timescales with regard to angular momentum exchange, and consequently, the neglection of the exchange with the tide-enerating body. It is shown that for present accuracy requirements this exchange has to be taken into account.     

Rotational dynamics of celestial bodies: Generalized rotation and considerations of angular momentum

The transformation equations under generalized rotation are obtained for an initially defined reduced velocity tensor governing the motion of a deformable finite material continuum. Then angular momentum considerations lead to relations between flow properties of the continuum and properties of a coordinate system introduced to describe generalized rotation of the continuum. Such relations could define preferable coordinate systems perceiving zero angular momentum for the continuum or referring that it moves according to linear laws.     

Counting relative equilibrium configurations of the full two-body problem

Consider a system of two rigid, massive bodies interacting according to their mutual gravitational attraction. In a relative equilibrium motion, the bodies rotate rigidly and uniformly about a fixed axis in \({\mathbb {R}}^3\). This is possible only for special positions and orientations of the bodies. After fixing the angular momentum, these relative equilibrium configurations can be characterized as critical points of a smooth function on configuration space. The goal of this paper is to use Morse theory and Lusternik–Schnirelmann category theory to give lower bounds for the number of critical points when the angular momentum is sufficiently large. In addition, the exact number of critical points and their Morse indices are found in the limit as the angular momentum tends to infinity.     

Dynamics of the 3/1 planetary mean-motion resonance: an application to the HD60532 b-c planetary system

In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance. The case where the outer planet is more massive than its inner companion is considered. We show that the resonant dynamics can be described using two fundamental parameters, the total angular momentum and the spacing parameter. The topology of the Hamiltonian function describing the resonant behaviour is investigated on a large domain of the phase space without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The families of the Apsidal Corotation Resonances (ACR) parameterized by the planetary mass ratio are obtained and their stability is analyzed. The main dynamical features in the domains around the ACR are also investigated in detail by means of spectral analysis techniques, which allow us to detect the regions of different regimes of motion of resonant systems. The construction of dynamical maps for various values of the total angular momentum shows the evolution of domains of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems.     

Bounded motion for a modified three-body problem

The energy and the angular momentum integral of motion for the planar three point problem cannot assure bounded motion. In this paper it is shown that if one of the points is replaced by a homogeneous sphere then bounded motion can be found. The zero velocity surfaces for this modified problem are found and their evolution is described.     

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.