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.     

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus’ equilateral equilibrium points is investigated with respect to the inclination of the asteroid’s orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L ₄ and L ₅, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 × 10⁹ years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 ≤ a ≤ 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids’ orbit was varied in the range 0° < i < 60° and the semimajor axes were fixed. It turned out that only four ‘windows’ of stable orbits survive: these are the orbits for the initial inclinations 0° < i < 7°, 9° < i < 13°, 31° < i < 36° and 38° < i < 50°. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.     

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.     

Newtonian and relativistic periodic orbits around two fixed black holes

We compare families of simple periodic orbits of test particles in the Newtonian and relativistic problems of two fixed centers (black holes). The Newtonian problem is integrable, while the relativistic problem is highly non-integrable.The orbits are calculated on the meridian plane through the fixed centersM ₁ (atz=+1) andM ₂ (atz=–1) for energies smaller than the escape energyE=1. We use prolate spheroidal coordinates (, , =const) and also the variables =cosh and =–cos . The orbits are inside a curve of zero velocity (CZV). The Newtonian orbits are also limited by an ellipse and a hyperbola, or by two eillipses. There are 3 main types of periodic orbits (1) elliptic type (around both centers), (2) hyperbolic-type, and (3) resonant-type.The elliptic type orbits are stable in the Newtonian case and both stable and unstable in the relativistic case. From the stable orbits bifurcate double period orbits both symmetric and asymmetric with respect to thez-axis. There are also higher order bifurcations. The hyperbolic-type orbits are unstable. The Newtonian resonant orbits are defined by the ratiot _µ/t =n/m of oscillations along and during one period, and they are all marginally unstable. The corresponding relativistic orbits are stable, or unstable. The main families are figure eight orbits aroundM ₁, or aroundM ₂ (3/1 orbits); gamma, or inverse gamma orbits (4/2); higher resonant families 5/1,7/1,...,8/2,12/2,...;, more complicated orbits, like 5/3, and bifurcations from the above orbits. Satellite orbits aroundM ₁, orM ₂, and their bifurcations (e.g. double period) exist in the relativistic case but not in the Newtonian case. The characteristics of the various families are quite different in the Newtonian and the relativistic cases. The sizes of the orbits and their stabilities are also quite different in general. In the Appendix we study the various types of straight line orbits and prove that some subcases introduced by Charlier (1902) are impossible.     

“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.     

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.     

Periodic orbits in the general three-body problem and the relationship between them

We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ₁ and φ₂, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ₁, φ₂∈(0, π) at steps of δk=0.01; δφ₁=δφ₂=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ₁, φ₂). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup.     

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.     

Asymptotic orbits in the (<Emphasis Type="Italic">N</Emphasis>+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m ₀=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L ₁ and L ₂. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

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…).     

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.     

Asymmetric Periodic Solutions in the Restricted Problem of Three Bodies

A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections, that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

On a Property of Gravitational Systems

If one attributes to each component i of a gravitational system a dimensionless parameter ψ _i equal to the ratio of its relative mass (with respect to the mass of the system) to its relative position (with respect to a generally defined radius) and sums up the ψ _i values of all components outside the central core, one obtains a mass distribution index Σψ of the order of unity irrespective of the size or the type of the system. In the case of spiral galaxies (and probable other galactic systems) this property applies not only to the whole galaxy, but also to the matter inside any radius larger than the core radius. The mass distribution index in these systems has a maximum Σψ_* at a certain radius r _*, which strongly correlates with the surface brightness at r _* in galaxies with similar mass to light ratio. The gravitational acceleration of all galaxies at r _* divided by (Σψ_*)² is constant and approximately equal to MOND acceleration parameter. Also, at this radius all galaxies have a surface temperature of the order of the temperature of the cosmic microwave background radiation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Orbital evolution of Uranus’s new outer satellites

Data on three recently discovered satellites of Uranus are used to determine basic evolutional parameters of their orbits: the extreme eccentricities and inclinations, as well as the circulation periods of the pericenter arguments and of the longitudes of the ascending nodes. The evolution is mainly investigated by analytically solving Hill’s double-averaged problem for the Uranus-Sun-satellite system, in which Uranus’s orbital eccentricity e _U and inclination i _U to the ecliptic are assumed to be zero. For the real model of Uranus’s evolving orbit with e _U≠0 and i _U≠0, we refine the evolutional parameters of the satellite orbits by numerically integrating the averaged system. Having analyzed the configuration and dynamics of the orbits of Uranus’s five outer satellites, we have revealed the possibility of their mutual crossings and obtained approximate temporal estimates.     

Critical symmetric periodic orbits in the photogravitational restricted three-body problem

In this paper, we determine series of horizontally critical symmetric periodic orbits of the six basic families, f,g,h,i,l,m, of the photogravitational restricted three-body problem, and computetheir vertical stability. We restrict our study in the case where only the first primary is radiating, namely q ₁≠1 andq ₂=1. We also compare our results with those of Hénon and Guyot (1970) so as to study the effect of radiation to this kind of orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bifurcations of plane to 3D periodic orbits in the photogravitational restricted three-body problem

We consider bifurcation of 3D periodic orbits from the plane ofmotion of the primaries in the photogravitational restricted three-bodyproblem. The simplest periodic 3D orbits branch from the plane periodicorbits of indifferent vertical stability. We compute the first few suchorbits of the basic families l, m, i, h, a, b, c forvarying mass parameter and for varying radiation coefficient of thelarger primary. The horizontal stability of the orbits is also computedleading to predictions about possible stability of the 3D orbits.     

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.