Qualitative study of the planar isosceles three-body problem

We consider the particular case of the planar three body problem obtained when the masses form an isosceles triangle for all time. Various authors [1, 2, 12, 8, 9, 13, 10] have contributed in the knowledge of the triple collision and of several families of periodic orbits in this problem. We study the flow on a fixed level of negative energy. First we obtain a topological representation of the energy manifold including the triple collision and infinity as boundaries of that manifold. The existence of orbits connecting the triple collision and infinity gives some homoclinic and heteroclinic orbits. Using these orbits and the homothetic solutions of the problem we can characterize orbits which pass near triple collision and near infinity by pairs of sequences. One of the sequences describes the regions visited by the orbit, the other refers to the behaviour of the orbit between two consecutive passages by a suitable surface of section. This symbolic dynamics which has a topological character is given in an abstract form and after it is applied to the isosceles problem. We try to keep globality as far as possible. This strongly relies on the fact that the intersection of some invariant manifolds with an equatorial plane (v=0) have nice spiraling properties. This can be proved by analytical means in some local cases. Numerical simulations given in Appendix A make clear that these properties hold globally.     

Analytical formulation of impulsive collision avoidance dynamics

The paper deals with the problem of impulsive collision avoidance between two colliding objects in three dimensions and assuming elliptical Keplerian orbits. Closed-form analytical expressions are provided that accurately predict the relative dynamics of the two bodies in the encounter b-plane following an impulsive delta-V manoeuvre performed by one object at a given orbit location prior to the impact and with a generic three-dimensional orientation. After verifying the accuracy of the analytical expressions for different orbital eccentricities and encounter geometries the manoeuvre direction that maximises the miss distance is obtained numerically as a function of the arc length separation between the manoeuvre point and the predicted collision point. The provided formulas can be used for high-accuracy instantaneous estimation of the outcome of a generic impulsive collision avoidance manoeuvre and its optimisation.     

A Research on the Collision Probability Calculation of Space Debris for Nonlinear Relative Motions†

The calculation of collision probability is the foundation of collision detection and avoidance maneuver for space objects. Now an assumption of linear relative motion is usually applied in the calculation of collision probability and then the complex 3-dimensional problem can be reduced to a 2-dimensional integral of probability density function over the area of circle. However, if the relative velocity value is very small, the term of linear relative motion is not valid. So it is necessary to consider the calculation of collision probability for nonlinear relative motions. The method used to calculate collision probability for nonlinear relative motion is studied, and test cases are designed to justify the validity of this method. It is applicable to collision probability problems involving relative velocity and error covariance varying with time. The results indicate that it is necessary to calculate collision probability with this nonlinear method under certain circumstances. For example, for elliptical relative motions in Satellite Formation Flying, when the relative velocity is below 100 m/s, the relative error between the linear method and the nonlinear method exceeds 5%; for the problem of conjunction analysis of two satellites with circular orbits, when the relative velocity is below 10 m/s, the relative error is also larger than 1%. Some significant conclusions are obtained for the collision detection system of our country.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

The Tetrahedral 4‐Body Problem with Rotation

We consider four bodies in space with same masses forming two binaries, each one symmetric with respect to a fixed axis and moving under Newtonian gravitation in opposite directions about this axis. It is given a direct proof that all singularities of this model are due to collisions, and it is proved that the singularities due to simultaneous double collisions are regularizable. The set of equilibrium points on the total collision manifold is studied as well as the possible connections among them. We show that the set of initial conditions on a given energy surface going to quadruple collision is a union of twenty submanifolds: twelve of them have dimension 2 and the others have dimension 3. Similarly for ejection orbits from quadruple collision. This revised version was published online in July 2006 with corrections to the Cover Date.     

A numerical study of the orbits of second species of the planar circular RTBP

We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs.     

Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Families of periodic collision orbits in the general three-body problem

In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

Structural stability of homothetic solutions of the collinearn-body problem

We investigate specific homothetic solutions of then-body problem which both begin and end in a simultaneous collision of all of the particles. Under a suitable change of variables, these solutions become heteroclinic orbits, i.e., they lie in the intersection of the stable and unstable manifolds of distinct equilibrium points. Our main result is that these manifolds intersect transversely along these orbits. This proves that the homothetic solutions are structurally stable.Partially supported by NSF Grant MCS 77-00430.     

A Hopf variables view on the libration points dynamics

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.     

Two‐Body Problems with Drag or Thrust: Qualitative Results

We consider two‐body problems in which the drag is proportional to the velocity divided by the square of the distance and whose radial and tangential components have distinct coefficients. For all parameters, we study the flow of the system obtained by suitable coordinate and time transformations and draw conclusions about the qualitative behavior of solutions. In each case, we examine the existence of collision–ejection, collision–escape, capture–collision, capture–escape, and oscillatory rectilinear orbits, study the motion near collision, and show that if periodic orbits exist they must be limit cycles. This revised version was published online in July 2006 with corrections to the Cover Date.     

A search for collision orbits in the free-fall three-body problem I. Numerical procedure

A numerical procedure is devised to find binary collision orbits in the free-fall three-body problem. Applying this procedure, families of binary collision orbits are found and a sequence of triple collision orbits are positioned. A property of sets of binary collision orbits which is convenient to search triple collision orbits is found. Important numerical results are formulated and summarized in the final section.     

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L₄ terminates at the other triangular point L₅ while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L₃. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations.     

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J ₃ has the same sign as J ₂; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.     

EJECTION–COLLISION ORBITS AND INVARIANT PUNCTURED TORI IN A RESTRICTED FOUR‐BODY PROBLEM

We study the motion of an infinitesimal mass point under the gravitational action of three mass points of masses μ, 1–2μ and μ moving under Newton's gravitational law in circular periodic orbits around their center of masses. The three point masses form at any time a collinear central configuration. The body of mass 1–2μ is located at the center of mass. The paper has two main goals. First, to prove the existence of four transversal ejection–collision orbits, and second to show the existence of an uncountable number of invariant punctured tori. Both results are for a given large value of the Jacobi constant and for an arbitrary value of the mass parameter 0<μ≤1/2. This revised version was published online in July 2006 with corrections to the Cover Date.     

Circular and zero-inclination solutions for optical observations of Earth-orbiting objects

Situational awareness of Earth-orbiting particles is important for human extraterrestrial activities. Given an optical observation, an admissible region can be defined over the topocentric range/range-rate space, with each point representing a possible orbit for the object. However, based on our understanding of Earth orbiting objects, we expect that certain orbits in that distribution, such as circular or zero-inclination orbits, would be more likely than others. In this research, we present an analytical approach for describing the existence of such special orbits for a given observation pass, and investigate topological features of the range/range-rate space by means of singularities in orbital elements.     

Chaos-induced resistivity of collisionless magnetic reconnection in the presence of a guide field

One of the most puzzling problems in astrophysics is to understand the anomalous resistivity in collisionless magnetic reconnection that is believed extensively to be responsible for the energy release in various eruptive phenomena. The magnetic null point in the reconnecting current sheet, acting as a scattering center, can lead to chaotic motions of particles in the current sheet, which is one of the possible mechanisms for anomalous resistivity and is called chaos-induced resistivity. In many interesting cases, however, instead of the magnetic null point, there is a nonzero magnetic field perpendicular to the merging field lines, usually called the guide field, whose effect on chaos-induced resistivity has been an open problem. By use of the test particle simulation method and statistical analysis, we investigate chaos-induced resistivity in the presence of a constant guide field. The characteristics of particle motion in the reconnecting region, in particular, the chaotic behavior of particle orbits and evolving statistical features, are analyzed. The results show that as the guide field increases, the radius of the chaos region increases and the Lyapunov index decreases. However, the effective collision frequency, and hence the chaos-induced resistivity, reach their peak values when the guide field approaches half of the characteristic strength of the reconnection magnetic field. The presence of a guide field can significantly influence the chaos of the particle orbits and hence the chaos-induced resistivity in the reconnection sheet, which decides the collisionless reconnection rate. The present result is helpful for us to understand the microphysics of anomalous resistivity in collisionless reconnection with a guide field.