Regular and irregular periodic orbits

We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves.     

Distant quasi-periodic orbits around Mercury

In this paper, distant quasi-periodic orbits around Mercury are studied for future Mercury missions. All of these orbits have relatively large sizes, with their altitudes near or above the Mercury sphere of influence. The research is carried out in the framework of the elliptic restricted three-body problem (ER3BP) to account for the planet’s non-negligible orbital eccentricity. Retrograde and prograde quasi-periodic trajectories in the planar ER3BP are generalized from periodic orbits in the CR3BP by the homotopy algorithm, and the shape evolution of such quasi-periodic trajectories around Mercury is investigated. Numerical simulations are performed to evaluate the stability of these distant orbits in the long term. These two classes of orbits present different characteristics: retrograde orbits can maintain shape stability with a large size, although the trajectories in some regions may oscillate with larger amplitudes; for prograde orbits, the range of existence is much smaller, and their trajectories easily move away from the vicinity of Mercury when the orbits become larger. Distant orbits can be used to explore the space environment in the vicinity of Mercury, and some orbits can be taken as transfer orbits for low-cost Mercury return missions or other programs for their high maneuverability.     

Artificial Martian frozen orbits and Sun-Synchronous orbits using continuous low-thrust control

This paper analyses three types of artificial orbits around Mars pushed by continuous low-thrust control: artificial frozen orbits, artificial Sun-Synchronous orbits and artificial Sun-Synchronous frozen orbits. These artificial orbits have similar characteristics to natural frozen orbits and Sun-Synchronous orbits, and their orbital parameters can be selected arbitrarily by using continuous low-thrust control. One control strategy to achieve the artificial frozen orbit is using both the transverse and radial continuous low-thrust control, and another to achieve the artificial Sun-Synchronous orbit is using the normal continuous low-thrust control. These continuous low-thrust control strategies consider J ₂, J ₃, and J ₄ perturbations of Mars. It is proved that both control strategies can minimize characteristic velocity. Relevant formulas are derived, and numerical results are presented. Given the same initial orbital parameters, the control acceleration and characteristic velocity taking into account J ₂, J ₃, and J ₄ perturbations are similar to those taking into account J ₂ perturbations for both Mars and the Earth. The control thrust of the orbit around Mars is smaller than that around the Earth. The magnitude of the control acceleration of ASFOM-4 (named as Artificial Sun-Synchronous Frozen Orbit Method 4) is the lowest among these strategies and the characteristic velocity within one orbital period is only 0.5219 m/s for the artificial Sun-Synchronous frozen orbit around Mars. It is evident that the relationship among the control thrusts and the primary orbital parameters of Martian artificial orbits is always similar to that of the Earth. Simulation shows that the control scheme extends the orbital parameters’ selection range of three types of orbits around Mars, compared with the natural frozen orbit and Sun-Synchronous orbit.     

Hyper-elliptic orbits

A survey of classes of elliptical orbits outside the usual ones due to the attractive direct power and inverse square force laws reveals some extremely interesting orbits with surprising dynamical characteristics. Particular elliptical orbits of interest in celestial mechanics are discussed.     

Bifurcations of triple-periodic orbits

We consider families of periodic orbits in potentials symmetric with respect to thex-axis. The characteristics of triple-periodic orbits (i.e. orbits intersecting thex-axis three times) that bifurcate from the central characteristic do not have their maximum or minimum energy (or perturbation) at the point of intersection. We explain theoretically that this happens only for triple-periodic orbits and not for any other type of resonant periodic orbits and verify this fact by numerical calculations.     

The pendulum dilemma of fish orbits

The shape of a galaxy is constrained both by mechanisms of formation (dissipational versus dissipationless) and by the available orbit families (the shape and amount of regular and stochastic orbits). It is shown that, despite the often very flattened shapes of banana and fish orbits, these boxlet orbits generally do not fit a triaxial galaxy in detail because, similar to loop orbits, they spend too little time at the major axis of the model density distribution. This constraint from the shape of fish orbits is relaxed at (large) radii where the density profile of a galaxy is steep.     

Periodic orbits

Recent results on periodic orbits are presented. Planetary systems can be studied by the model of the general 3-body problem and also some satellite systems and asteroid orbits can be studied by the model of the restricted 3-body problem. Triple stellar systems and planetary systems with two Suns are close to periodic systems. Finally, the motion of stars in various types of galaxies can be studied by finding families of periodic orbits in several galactic models.     

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60^° of mutual planetary inclination, but in most families, the stability does not exceed 20^°–30^°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively.     

Irregular periodic orbits

We study the peculiarities of irregular periodic orbits, i.e. orbits belonging to families not connected with the main families or their bifurcation, of Hamiltonian systems of two degrees of freedom. Families of irregular periodic orbits appear in triplets which are either closed or extend to infinity. If these triplets form an infinite sequence they surround an escape region. It seems probable that in general regions covered by irregular families are of high degree of stochasticity.     

Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.     

Numerical determination of three-dimensional periodic orbits generated from vertical self-resonant satellite orbits

The mechanism by which ‘vertical’ branches consisting of symmetric, three-dimensional periodic orbits bifurcate from families of plane orbits at ‘veertical self-resonant’ orbits is discussed, with emphasis on the relationship between symmetry properties and multiplicity, and methods for the numerical determination of such branches are described. As examples, eight new families of all symmetry classes which branch vertically from the familyf of retrograde satellite orbits in the Sun-Jupiter case of the restricted problem (μ=0.000 95), are given in their entirety; these branches are found, as expected, to occur in pairs, each pair arising from the same self-resonant orbit, and their symmetry properties following the predicted pattern. The stability and other properties of the branch orbits are discussed.     

Displaced non-Keplerian orbits using impulsive thrust

This paper investigates new families of displaced, highly non-Keplerian orbits in the two-body problem and artificial equilibria in the circular restricted three-body problem. The families of orbits presented extend prior work by using periodic impulses to generate displaced orbits rather than continuous thrust. The new displaced orbits comprise a sequence of individual Keplerian arcs whose intersection is continuous in position, with discontinuities in velocity removed using impulses. For frequent impulses the new families of orbits approximate continuous thrust non-Keplerian orbits found in previous studies. To generate approximations to artificial equilibria in the circular restricted three-body problem, periodic impulses are used to generate a sequence of connected three-body arcs which begin and terminate at a fixed position in the rotating frame of reference. Again, these families of orbits reduce to the families of artificial equilibria found using continuous thrust.     

Inhomogeneous potentials producing homogeneous orbits

We prove that, in general, a given two-dimensional inhomogeneous potential V(x,y) does not allow for the creation of homogeneous families of orbits. Yet, depending on the case at hand, if the given potential satisfies certain conditions, this potential is compatible either with one (or two) monoparametric homogeneous families of orbits or at most with five such familes. The orbits are then found on the grounds of the given potential.     

Long-term capture orbits for low-energy space missions

This research aims at ascertaining the existence and characteristics of natural long-term capture orbits around a celestial body of potential interest. The problem is investigated in the dynamical framework of the three-dimensional circular restricted three-body problem. Previous numerical work on two-dimensional trajectories provided numerical evidence of Conley’s theorem, proving that long-term capture orbits are topologically located near trajectories asymptotic to periodic libration point orbits. This work intends to extend the previous investigations to three-dimensional paths. In this dynamical context, several special trajectories exist, such as quasiperiodic orbits. These can be found as special solutions to the linear expansion of the dynamics equations and have already been proven to exist even using the nonlinear equations of motion. The nature of long-term capture orbits is thus investigated in relation to the dynamical conditions that correspond to asymptotic trajectories converging into quasiperiodic orbits. The analysis results in the definition of two parameters characterizing capture condition and the design of a capture strategy, guiding a spacecraft into long-term capture orbits around one of the primaries. Both the results are validated through numerical simulations of the three-dimensional nonlinear dynamics, including fourth-body perturbation, with special focus on the Jupiter–Ganymede system and the Earth–Moon system.     

Polar orbits around binary stars

Oks proposes the existence of a new class of stable planetary orbits around binary stars, in the shape of a helix on a conical surface whose axis of symmetry coincides with the interstellar axis, and rotates with the same orbital frequency as the binary pair. We show that this claim relies on the inappropriate use of an effective potential that is only applicable when the stars are held motionless. We also present numerical evidence that the only planetary orbits whose planes are initially orthogonal to the interstellar axis that remain stable on the time scale of the stellar orbit are ordinary polar orbits around one of the stars, and that the perturbations due to the binary companion do not rotate the plane of the orbit to maintain a fixed relationship with the axis.     

Natural and sail-displaced doubly-symmetric Lagrange point orbits for polar coverage

This paper proposes the use of doubly-symmetric, eight-shaped orbits in the circular restricted three-body problem for continuous coverage of the high-latitude regions of the Earth. These orbits, for a range of amplitudes, spend a large fraction of their period above either pole of the Earth. It is shown that they complement Sun-synchronous polar and highly eccentric Molniya orbits, and present a possible alternative to low thrust pole-sitter orbits. Both natural and solar-sail displaced orbits are considered. Continuation methods are described and used to generate families of these orbits. Starting from ballistic orbits, other families are created either by increasing the sail lightness number, varying the period or changing the sail attitude. Some representative orbits are then chosen to demonstrate the visibility of high-latitude regions throughout the year. A stability analysis is also performed, revealing that the orbits are unstable: it is found that for particular orbits, a solar sail can reduce their instability. A preliminary design of a linear quadratic regulator is presented as a solution to stabilize the system by using the solar sail only. Finally, invariant manifolds are exploited to identify orbits that present the opportunity of a ballistic transfer directly from low Earth orbit.     

Symmetric doubly-asymptotic periodic orbits at collinear equilibria

A systematic search for periodic orbits doubly-asymptotic to the collinear equilibrium points of the restricted three-body problem is carried out and many such orbits are found, each of them existing for a specific value of the mass parameter. These may be useful as reference orbits and seem to be special limit orbits representing period discontinuities in the evolution of the families of periodic orbits.     

On meteorite orbits

The new method of calculation of meteorite orbits based on the analysis of meteorite showers scattering patterns is suggested. Elements of orbits of 14 meteorites are listed. Correlations between oribt type and petrological type of meteorite are absent.     

Non-Keplerian orbits for electric sails

An electric sail is capable of guaranteeing the fulfilment of a class of trajectories that would be otherwise unfeasible through conventional propulsion systems. In particular, the aim of this paper is to analyze the electric sail capabilities of generating a class of displaced non-Keplerian orbits, useful for the observation of the Sun’s polar regions. These orbits are characterized through their physical parameters (orbital period and solar distance) and the spacecraft propulsion capabilities. A comparison with a solar sail is made to highlight which of the two systems is more convenient for a given mission scenario. The optimal (minimum time) transfer trajectories towards the displaced orbits are found with an indirect approach.     

The present status of periodic orbits

Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits.