Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S _i , , which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S ₂ and S ₃, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S ₂- or S ₃-symmetric periodic solutions. The symmetries S ₂ and S ₃ constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).     

The equilibrium solutions of the restricted problem of three rigid bodies with variable mass

In this paper we consider the circular planar restricted problem of three rigid bodiesS _i(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS ₃ is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL ₄ andL ₅ form nearly equilateral tringles with the primaries. The equilibrium pointsL ₁,L ₂,L ₃ remain collinear an ies on the line joining the primaries.     

The elliptic restricted problem at the 3 : 1 resonance

Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I _c , II _c bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I _e , II _e , from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I _c , II _c , but is poor for the families I _e , II _e . A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem.     

Unified iterative methods in orbit determination

The classical problem of Keplerian orbit determination from only three measurements of time and angular coordinates (t _i, _i, _i) has been solved here numerically in two different ways, using Newton's method for non-linear equations in both cases. The first method (Perov, 1989) is based on KS variables, whereas the second emphasizes the fundamental part played by the unified Lambert's equation and the related formulae in that kind of applications. These two methods have been compared and put into practice in various numerical tests based on real asteroid orbits and ficitious Keplerian asteroid, comet and artificial satellite orbits in order to try the stability of these methods for peculiar orbits.     

Stationary Configurations for Co-orbital Satellites with Small Arbitrary Masses

We derive general results on the existence of stationary configurations for N co-orbital satellites with small but otherwise arbitrary masses m _i , revolving on circular and planar orbits around a massive primary. The existence of stationary configurations depends on the parity of N. If N is odd, then for any arbitrary angular separation between the satellites, there always exists a set of masses (positive or negative) which achieves stationarity. However, physically acceptable solutions (m _i > 0 for all i) restrict this existence to sub-domains of angular separations. If N is even, then for given angular separations of the satellites, there is in general no set of masses which achieves stationarity. The case N=3 is treated completely for small arbitrary satellite masses, giving all the possible solutions and their stability, to within our approximations.     

The existence of three-dimensional symmetric orbits in the magnetic-binary problem

We study the existence of three-dimensional symmetric orbits in a magnetic-binary system. We point out that only two kinds of such orbits exist, depending on the orientation of both magnetic momentsM _i,i=1, 2; one with respect to the plane,y=0 and one with respect to thex-axis of the rotating-coordinate system.     

“Stickiness” in mappings and dynamical systems

We present results of a study of the so-called “stickiness” regions where orbits in mappings and dynamical systems stay for very long times near an island and then escape to the surrounding chaotic region. First we investigated the standard map in the form x_i+1 = x_i+y_i+1 and y_i+1 = y_i+K/2π · sin(2πx_i) with a stochasticity parameter K = 5, where only two islands of regular motion survive. We checked now many consecutive points—for special initial conditions of the mapping—stay within a certain region around the island. For an orbit on an invariant curve all the points remain forever inside this region, but outside the “last invariant curve” this number changes significantly even for very small changes in the initial conditions. In our study we found out that there exist two regions of “sticky” orbits around the invariant curves: A small region I confined by Cantori with small holes and an extended region II is outside these cantori which has an interesting fractal character. Investigating also the Sitnikov-Problem where two equally massive primary bodies move on elliptical Keplerian orbits, and a third massless body oscillates through the barycentre of the two primaries perpendicularly to the plane of the primaries—a similar behaviour of the stickiness region was found. Although no clearly defined border between the two stickiness regions was found in the latter problem the fractal character of the outer region was confirmed.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².     

Constraining the relative inclinations of the planets B and C of the millisecond pulsar PSR B1257+12

We investigate on the relative inclination of the planets B and C orbiting the pulsar PSR B1257+12. First, we show that the third Kepler’s law does represent an adequate model for the orbital periods P of the planets, because other Newtonian and Einsteinian corrections are orders of magnitude smaller than the accuracy in measuring P _B/C. Then, on the basis of available timing data, we determine the ratio sin i _C/ sin i _B = 0.92±0.05 of the orbital inclinations i _B and i _C independently of the pulsar’s mass M. It turns out that coplanarity of the orbits of B and C would imply a violation of the equivalence principle. Adopting a pulsar mass range 1 ≲ M ≲ 3, in solar masses (supported by present-day theoretical and observational bounds for pulsar’s masses), both face-on and edge-on orbital configurations for the orbits of the two planets are ruled out; the acceptable inclinations for B span the range 36 deg ≲ i _B ≲ 66 deg, with a corresponding relative inclination range 6 deg ≲ (i _C − i _B) ≲ 13 deg.     

Global stability and the restricted 3-body problem

Global stability regions are found for classi orbits of the circular restricted 3-body problem for primary masses equal and Jacobi constantK>15.5. As this constant decreases, the stability, region shrinks extremely rapidly.     

General three-body problem: Families of vertical critical periodic orbits

In this paper we present four families of vertical critical periodic orbits found by continuation, with respect to the small massm ₃, of the vertical critical periodic orbitsl1v, ilv, mlv, c3v of the circular restricted problem. The periodic orbits refer to a suitably defined rotating frame of reference.     

Periodic orbits around an oblate spheroid

The general properties of certain differential systems are used to prove the existence of periodic orbits for a particle around an oblate spheroid.In a fixed frame, there are periodic orbits only fori=0 andi near /2. Furthermore, the generating orbits are circles.In a rotating frame, there are three families of orbits: first a family of periodic orbits in the vicinity of the critical inclination; secondly a family of periodic orbits in the equatorial plane with 0<e<1; thirdly a family of periodic orbits for any value of the inclination ife=0.     

Numerical exploration of the photogravitational restricted five-body problem

We study numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on the mass parameter β and the radiation factors q _i , i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q _i vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are calculated.     

The Third Integral in a Self-consistent Galactic Model

We apply the theory of the third integral to a self-consistent galactic model, generated by the collapse of a N-body system. The final configuration after the collapse is a stationary triaxial system, that represents an almost prolate non-rotating elliptical galaxy with its longest axis in the z-direction. This system is represented by an axisymmetric potential V plus a small triaxial perturbation V ₁. The orbits in the potential V are of three types: box orbits, tube orbits (corresponding to various resonances), and chaotic orbits.The intersections of the box and tube orbits by a Poincaré surface of section z=0 are closed invariant curves. The main tube orbits are like ellipses and form an island of stability on the (R,R) plane.We calculated the third integral I in the potential V for the general non-resonant case and for various resonant cases. The agreement between the invariant curves of the orbits and the level curves of the third integral is good for the box and tube orbits, if we truncate the third integral at an appropriate level. As expected the third integral fails in the case of chaotic orbits. The most important result is the form of the number density F on the Poincaré surface of section. This function decreases exponentially outwards for the box orbits, like Fexp(–bI), while it is constant, as expected, for the chaotic orbits. In the case of the island of the main tube orbits it has a minimum at the center of the island. This can be explained by the form of the near elliptical orbits that are elongated along R, thus they fail to support a self-consistent galaxy, which is elongated along the z-axis.     

Nonextensive Statistical Mechanics: Some Links with Astronomical Phenomena

A variety of astronomical phenomena appear to not satisfy the ergodic hypothesis in the relevant stationary state, if any. As such, there is no reason for expecting the applicability of Boltzmann–Gibbs (BG) statistical mechanics. Some of these phenomena appear to follow, instead, nonextensive statistical mechanics. In the same manner that the BG formalism is based on the entropy S _BG=?k∑ _i p _i ln p _i, the nonextensive one is based on the form S _q=k(1 ?∑ _i p _i ^q)/(q? 1) (with S ₁=S _BG). The stationary states of the former are characterized by an exponential dependence on the energy, whereas those of the latter are characterized by an (asymptotic) power law. A brief review of this theory is given here, as well as of some of its applications, such as the solar neutrino problem, polytropic self-gravitating systems, galactic peculiar velocities, cosmic rays and some cosmological aspects. In addition to these, an analogy with the Keplerian elliptic orbits versus the Ptolemaic epicycles is developed, where we show that optimizing S _q with a few constraints is equivalent to optimizing S _BG with an infinite number of constraints.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ¹ action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J ₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J ₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J ₂ effect as geometric phases.     

Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces

We explore the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. The location, nature and size of periodic and quasi-periodic orbits are studied using the numerical technique of Poincare surface of sections. The maximum amplitude of oscillations about the periodic orbits is determined and is used as a parameter to measure the degree of stability in the phase space for such orbits. It is found that the orbits around Saturn remain around it and their stability increases with the increase in the value of Jacobi constant C. The orbits around Titan move towards it with the increase in C. At C=3.1, the pericenter and apocenter are 358.2 and 358.5 km, respectively. No periodic or quasi-periodic orbits could be found by the present method around the collinear Lagrangian point L ₁ (0.9569373834…).