Chaos in Hill's generalized problem: from the solar system to black holes

We generalize the well‐known Hill's circular restricted three‐body problem by assuming that the primary generates a Schwarzschild‐type field of the form U = A/r + B/r³. The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two‐body problem primary‐particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three‐body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A High Order Perturbation Analysis of the Sitnikov Problem

The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré–Lindstedt. The enormous amount of necessary computations were performed by extensive use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.     

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r ₁, r ₂, and r ₁₂, where r ₁ and r ₂ are the distances of two bodies which form an initial binary from the four body systems centre of mass andr ₁₂ is the separation between the two bodies.     

Libration points in the generalised elliptic restricted three body problem

In this paper the existence of collinear as well as equilateral libration points for the generalised elliptic restricted three body problem has been studied distinct from Kondurar and Shinkarik (1972) where a study has been made for the generalised circular restricted three body problem. Here the coordinates of the libration points have been found to be functions of timet.     

Continuation of periodic orbits: Three-dimensional circular restricted to the general three-body problem

It is proved that a periodic orbit of the three-dimensional circular restricted three-body problem can be continued analytically, when the mass of the third body is sufficiently small, to a periodic orbit of the three dimensional general three-body problem in a rotating frame. The above method is not applicable when the period of the periodic orbit of the restricted problem is equal to 2k (k any integer), in the usual normalized units. Several numerical examples are given.     

Periodic motions generated by Lagrangian solutions of the circular restricted three-body problem

The predictor-corrector method is described for numerically extending with respect to the parameters of the periodic solutions of a Lagrangian system, including recurrent solutions. The orbital stability in linear approximation is investigated simultaneously with its construction.The method is applied to the investigation of periodic motions, generated from Lagrangian solutions of the circular restricted three body problem. Small short-period motions are extended in the plane problem with respect to the parameters h, µ (h = energy constant, µ = mass ratio of the two doninant gravitators); small vertical oscillations are extended in the three-dimensional problem with respect to the parameters h, µ. For both problems in parameter's plane h, µ domaines of existince and stability of derived periodic motions are constructed, resonance curves of third and fourth orders are distinguished.     

Infinitesimal Orbits Around the Libration Points in the Elliptic Restricted Three Body Problem

The objective of the present work is to develope explicit analytical expressions for the small amplitude orbits of the infinitesimal mass about the equilibrium points in the elliptic restricted three body problem. To handle this dynamical problem, the Hamiltonian for the elliptic problem is formed with the true anomaly and then with the eccentric anomaly as independent variables. The origin is then transformed to a fixed point and the Hamiltonian is developed up to O(4) in the eccentricity, e, (which plays the role of the small parameter of the problem) of the primaries. The integration of the model problem under consideration is undertaken by means of a perturbation technique based on Lie series developments, which leads to the solution of the canonical equations of motion.     

Global Bifurcations of Periodic Solutions of the Restricted Three Body Problem

We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L ₄, L ₅. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date.     

Orbit determination with the two-body integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.     

Non-linear stability in the generalized restricted three-body problem

The non-linear stability of the triangular equilibrium point L ₄ in the generalized restricted three-body problem has been examined. The problem is generalized in the sense that the infinitesimal body and one of the primaries have been taken as oblate spheroids. It is found that the triangular equilibrium point is stable in the range of linear stability except for three mass ratios.     

Study of the transfer from the Earth to a halo orbit around the equilibrium pointL 1

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium pointL ₁ of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total v required, the figures obtained are similar to the ones given by the standard procedures of optimization.     

Collinear relative equilibria of the planarN-body problem

The following theorem is proved. THEOREM.For any n2, the set of collinear relative equilibria classes of the n-body problem generates by analytical continuation a total of n!(n+3)/2 relative equilibria classes of the n+1 body problem.Together with Arenstorf's results we state a general theorem for the 4 body problem with 3 arbitrary masses and 1 inferior mass.Research supported in part by NSF grant MCS-78-00395 A01.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

Formal Integrals and Nekhoroshev Stability in a Mapping Model for the Trojan Asteroids

A symplectic mapping model for the co-orbital motion (Sándor et al., 2002, Cel. Mech. Dyn. Astr. 84, 355) in the circular restricted three body problem is used to derive Nekhoroshev stability estimates for the Sun–Jupiter Trojans. Following a brief review of the analytical part of Nekhoroshev theory, a direct method is developed to construct formal integrals of motion in symplectic mappings without use of a normal form. Precise estimates are given for the region of effective stability based on the optimization of the size of the remainder of the formal series. The stability region found for t=10¹⁰ yrs corresponds to a libration amplitude D_p=10.6°. About 30% of asteroids with accurately known proper elements (Milani, 1993, Cel. Mech. Dyn. Astron. 57, 59), at low eccentricities and inclinations, are included within this region. This represents an improvement with respect to previous estimates given in the literature. The improvement is due partly to the choice of better variables, but also to the use of a mapping model, which is a simplification of the circular restricted three body problem.     

Orbit determination with the two-body integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.     

Processes in the solar system: On one of the versions of the strict theory of periodic reactions in space and time

The present study develops a previously suggested physico-chemical theory, according to which the regular structure of planetary and satellite systems is explained on the basis of notions regarding periodic condensation of gaseous matter in space and time during formation of the centre body. An accurate solution is provided for the problem on the periodic chemical reaction in space and time at diffusion of reacting substances from two unlimited-capacity sources separated by a definite distance. A precise theory for the above phenomenon has been developed for the case when the critical concentration of the resultant substance (A) is estimated from the condition min(a, b)=A, wherea andb are the reacting component concentrations.     

Remarks on the period variation of HS Herculis and SW Cygni

In order to get a satisfactory understanding of the periodic variation of the orbital period in the binary system HS Herculis, the study of this problem is resumed. Using recently observed primary and secondary minima, it is evident that after 1955 (E > -2000) the corresponding O – C diagram reflects the effect of apsidal motion. Any assumption on the presence of a third body is rejected, at least as long as the current aspect of the O–C diagram is concerned. For the interpretation of the sinusoidal period variation of the semi-detached system SW Cygni, 130 primary minima were compiled form the literature. Though it is considered as very likely that this variation of the period is primarily caused by apsidal motion, the hypothesis of a third body is analysed too. Further precise photometric and spectroscopic observations are recommended.     

Equations of motion of elliptic restricted problem of three bodies with variable mass

The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force.     

Instability of the 2+2 body problem

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m₁ and m₂ in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm ₁,m ₂ and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.     

On the role and the properties ofn body central configurations

The role central configurations play in the analysis ofn body systems is outlined. Emphasis is placed on collision orbits, expanding gravitational systems, andn body zero radial velocity surfaces. In the second half of the paper, properties of cenral configurations are discussed. Here, emphasis is placed on describing a different approach to analyze these configurations, one which is related to the classical problem of the weightedsth mean values of a vector. This approach is illustrated by discussing the non-degeneracy of central configurations and by describing central configurations in various dimensions. This is a written version of a talk given at Oberwolfach, August, 1978.proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August 1978.This research was supported in part by an NSF Grant.     

Celestial n-Body Coupling in the Lunar Orbit Theory

Making use of the fact that, in the solar system, the angular momentum is carried predominantly by the planets while the mass is beared almost entirely by the Sun, an iterative scheme is devised to solve approximately the n-body contributions of the lunar orbit problem. The scheme envisages the Moon-Earth-Sun three-body subsystem as being nested in the grand Earth-Jupiter-Sun system. In the planetocentric representation, the orbital motion of the Sun about the solar system center of mass is transmitted to the third body via the second primary body in both the grand and nested three-body systems. This revised version was published online in July 2006 with corrections to the Cover Date.