The Two Fixed Centers: An Exceptional Integrable System

It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures, a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space. We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation number’) as a function of the two integrals of motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Secular orbit-orbit resonance between two satellites with non-zero masses

After the mean anomaly has been removed from the perturbations, the reduced Hamiltonian becomes a function over the Lie algebra determined by the infinitesimal generators associated with the dynamical symmetries of an unperturbed Keplerian system. The phase space being now the group SO(3), average motions consist of rotations, and the normalized Hamiltonian serves as a Morse function whose critical points determine the intrinsic topology of the perturbed system.     

The Lissajous transformation IV. Delaunay and Lissajous variables

Relations are established between the Delaunay variables defined over a phase space E in four dimensions and the Lissajous variables defined over a four-dimensional phase space F when the latter is mapped onto E by a parabolic canonical transformation.     

Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.     

Quasi-critical orbits for artificial lunar satellites

We study the problem of critical inclination orbits for artificial lunar satellites, when in the lunar potential we include, besides the Keplerian term, the J ₂ and C ₂₂ terms and lunar rotation. We show that, at the fixed points of the 1-D averaged Hamiltonian, the inclination and the argument of pericenter do not remain both constant at the same time, as is the case when only the J ₂ term is taken into account. Instead, there exist quasi-critical solutions, for which the argument of pericenter librates around a constant value. These solutions are represented by smooth curves in phase space, which determine the dependence of the quasi-critical inclination on the initial nodal phase. The amplitude of libration of both argument of pericenter and inclination would be quite large for a non-rotating Moon, but is reduced to <0°.1 for both quantities, when a uniform rotation of the Moon is taken into account. The values of J ₂, C ₂₂ and the rotation rate strongly affect the quasi-critical inclination and the libration amplitude of the argument of pericenter. Examples for other celestial bodies are given, showing the dependence of the results on J ₂, C ₂₂ and rotation rate.     

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.     

Precession/nutation solution consistent with the general planetary theory

In the present paper the equations of the translatory motion of the major planets and the Moon and the Poisson equations of the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation is presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients.     

A unified state model of orbital trajectory and attitude dynamics

The trajectory and attitude dynamics of an orbital spacecraft are defined by a unified state model, which enables efficient and rapid machine computation for mission analysis, orbit determination and prediction, satellite geodesy and reentry analysis. The state variables are momenta — a general form for attitude, and a parametric form for orbital motion. The orbital parameters are the velocity state characteristics of the orbital hodograph. The coordinate variables are sets of four Euler parameters, which define the rotation transformation by the quaternion algebra. The unified state model possesses many analytical properties which are invaluable for dynamical system synthesis, numerical analysis and machine solution: regularization, unified matrix algebra, state graphs and transforms. The analytic partials of position and velocity with the state and coordinate variables are presented, as well as representative perturbation functions such as air drag, gravitational potential harmonics, and propulsion thrust.     

On constructing the general Earth’s rotation theory

In the present paper the equations of the orbital motion of the major planets and the Moon and the equations of the three–axial rigid Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution). Hence, the theory of the Earth’s rotation can be presented by means of the series in powers of the evolutionary variables with quasi-periodic coefficients with respect to the planetary–lunar mean longitudes. This form of the Earth’s rotation problem is compatible with the general planetary theory involving the separation of the short–period and long–period variables and avoiding the appearance of the non–physical secular terms.     

Gyldén systems: Rotation of pericenters

A canonical transformation in phase space and a rescaling of time are proposed to reduce a Keplerian system with a time-dependent Gaussian parameter to a perturbed Keplerian system with a constant Gaussian parameter. When the time variation is slow, the perturbation through second order in the reduced problem is conservative, and, as a result, the orbital space of the averaged system is a sphere on which the phase flow causes a differential rotation representing a circulation of the line of apsides. The flow presents two isolated singularities corresponding to circular orbits travelled respectively in the direct and in the retrograde sense, and a degenerate manifold of fixed points corresponding to the collision orbits. Normalization beyond order two does not break the degeneracy. Adiabatic invariants, which are conservative functions, may be computed from the normalized Hamiltonian evaluated here to the fourth order. Nonetheless so high an approximation gives little information because the normalizing Lie transformation depends explicitly on the time through mixed secular-periodic terms. As an application, an estimate is offered for the apsidal rotation that a second order time derivative in the mass of the sun would induce on planetary orbits. This suggests an observational method for determining the latter parameter for the solar wind, but the predicted motions are too slow for the current level of observational precision.     

The critical inclination in artificial satellite theory

Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.A delicacy so very susceptible of offence...—Hester Lynch PIOZZI,Observations and Reflections made in the Course of a Journey through France, Italy and Germany (1789)NAS/NRC Postgraduate Research Associate in 1984–1985.     

Symmetric and asymmetric librations in extrasolar planetary systems: a global view

We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits.     

An analysis of close binaries (CB) based on photometric measurements

The subject of the paper is the problem of stellar differntial rotation in close binaries (CB) ofRS CV n type. The differential-rotation parameters we find on the basis of the migration of the depression in the light curves caused by the spot effect over the orbital phase. For that purpose, a simple model (Bussoet al., 1985) and inverse-problem procedure, based on the Marquardt (1963) algorithm, are used. To verify the obtained solutions, the SIMPLEX algorithm (Torczon, 1991) is applied, suitable for the nonlinear parameter optimisation. This algorithm enables a correct solution of the nonlinear equation system describing the differential rotation. The procedure is applied in the determination of the parameters of differential rotation forCV Cam, VV Mon andSS Boo binaries.     

The Lissajous transformation II. Normalization

Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria.     

Construction of invariant tori for the spin-orbit problem in the mercury-sun system

The stability of spin-orbit resonances, namely commensurabilities between the periods of rotation and revolution of an oblate satellite orbiting around a primary body, is investigated using perturbation theory. We reduce the system to a model described by a one-dimensional, time-dependent Hamiltonian function. By means of KAM theory we rigorously construct bidimensional invariant surfaces, which separate the three dimensional phase space. In particular with a suitable choice of the rotation numbers of the invariant tori we are able to trap the periodic orbit associated with a given resonance in a finite region of the phase space. This technique is applied to the Mercury-Sun system. A connection with the probability of capture in a resonance is also provided.     

Dynamics of rotation of super-Earths

We numerically investigate the dynamics of rotation of several close-in terrestrial exoplanet candidates. In our model, the rotation of the planet is disturbed by the torque of the central star due to the asymmetric equilibrium figure of the planet. We model the shape of the planet by a Jeans spheroid. We use surfaces of section and spectral analysis to explore numerically the rotation phase space of the systems adopting different sets of parameters and initial conditions close to the main spin–orbit resonant states. One of the parameters, the orbital eccentricity, is critically discussed here within the domain of validity of orbital circularization timescales given by tidal models. We show that, depending on some parameters of the system like the radius and mass of the planet, eccentricity etc., the rotation can be strongly perturbed and a chaotic layer around the synchronous state may occupy a significant region of the phase space. 55 Cnc e is an example.     

Control of chaos in the vicinity of the Earth‐Moon L5 Lagrangian point to keep a spacecraft in orbit

The concept of Space Manifold Dynamics is a new method of space research. We have applied it along with the basic idea of the method of Ott, Grebogi, and York (OGY method) to stabilize the motion of a spacecraft around the triangular Lagrange point L₅ of the Earth‐Moon system. We have determined the escape rate of the trajectories in the general three‐ and four‐body problem and estimated the average lifetime of the particles. Integrating the two models we mapped in detail the phase space around the L₅ point of the Earth‐Moon system. Using the phase space portrait our next goal was to apply a modified OGY method to keep a spacecraft close to the vicinity of L₅. We modified the equation of motions with the addition of a time dependent force to the motion of the spacecraft. In our orbit‐keeping procedure there are three free parameters: (i) the magnitude of the thrust, (ii) the start time, and (iii) the length of the control. Based on our numerical experiments we were able to determine possible values for these parameters and successfully apply a control phase to a spacecraft to keep it on orbit around L₅. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

THE PHASE SPACE STRUCTURE OF THE EXTENDED SITNIKOV-PROBLEM

In this article we treat the 'Extended Sitnikov Problem' where three bodies of equal masses stay always in the Sitnikov configuration. One of the bodies is confined to a motion perpendicular to the instantaneous plane of motion of the two other bodies (called the primaries), which are always equally far away from the barycenter of the system (and from the third body). In contrary to the Sitnikov Problem with one mass less body the primaries are not moving on Keplerian orbits. After a qualitative analysis of possible motions in the 'Extended Sitnikov Problem' we explore the structure of phase space with the aid of properly chosen surfaces of section. It turns out that for very small energies H the motion is possible only in small region of phase space and only thin layers of chaos appear in this region of mostly regular motion. We have chosen the plane ( ) as surface of section, where r is the distance between the primaries; we plot the respective points when the three bodies are 'aligned'. The fixed point which corresponds to the 1 : 2 resonant orbit between the primaries' period and the period of motion of the third mass is in the middle of the region of motion. For low energies this fixed point is stable, then for an increased value of the energy splits into an unstable and two stable fixed points. The unstable fixed point splits again for larger energies into a stable and two unstable ones. For energies close toH = 0 the stable center splits one more time into an unstable and two stable ones. With increasing energy more and more of the phase space is filled with chaotic orbits with very long intermediate time intervals in between two crossings of the surface of section. We also checked the rotation numbers for some specific orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Properties of a spherical galaxy with exponential energy distribution

Some analytical relations for the phase space functions of a self-consistent spherical stellar system are derived. The integral constraints on the distribution function by imposing a given (r) density distribution andN(E) fractional energy distribution are determined. For the case of radially-anisotropic velocity distribution in theE0 limit the constraint by an exponentialN(E) implies thatf(E, J ²) tends to zero in the order (–E)^3/2. This lends analytical support to the use of the Stiavelli and Bertin (1985) distribution function for modeling elliptical galaxies. Maximum phase space density constraint confirms the necessity of high collapse factors to produce such a distribution function. Limits on the steepness of an exponentialN(E) for the case when (r) resembles the emissivity law of ellipticals are also derived.     

Horseshoes and nonintegrability in the restricted case of a spinless axisymmetric rigid body in a central gravitational field

The purpose of this paper is to study the motion of a spinless axisymmetric rigid body in a Newtonian field when we suppose the motion of the center of mass of the rigid body is on a Keplerian orbit. In this case the system can be reduced to a Hamiltonian system with configuration space of a two-dimensional sphere. We prove that the restricted planar motion is analytical nonintegrable and we find horseshoes due to the eccentricity of the orbit. In the caseI ₃/I ₁>4/3, we prove that the system on the sphere is also analytical nonintegrable.On leave from the Polytechnic Institute of Bucharest, Romania.