Escape regions of a quartic potential

We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0.     

Introduction to the restricted Jupiter orbiter problem

We consider a restricted six-body problem, consisting of Jupiter, the four Galilean satellites, and an orbiter. The Galilean satellites' orbits are circular and coplanar; Io, Europa, and Ganymede are in exact resonance; their mean longitudes obey the Laplace relation. We seek periodic orbits which avoid close approaches to any satellite; such orbits are of interest for mission planning. They are approximated as equilibrium points of sets of variational equations associated with time-averaged disturbing functions. Stability of the solutions is also determined. The orbits of greatest interest are:Planar: twice Callisto's period, eccentricity0.6Planar: four times Callisto's period, eccentricity0.75Slightly inclined: twice Callisto's period, eccentricity arbitraryPlanar: 4/5 or 5/4 Europa's period.     

Family <Emphasis Type="Italic">h</Emphasis> of periodic solutions of the restricted problem for small μ

The results of the calculation of the family h of symmetric periodic solutions of the planar restricted three-body problem for four values of μ = 0, 10^?3, 0.1, and 0.2 are presented. This family begins with retrograde circular orbits around the body of bigger mass. Associated with each value of μ are the table of critical orbits, the orbit pictures, graphs of the characteristics of the family in four coordinate systems, and graphs of the period and of traces (planar and vertical). Regularities on the family and its connection to the generating family are observed.     

The rectilinear three-body problem using symbol sequence II: role of the periodic orbits

We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally, periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit.     

A Fine Structure of the Perseid Meteoroid Stream

A fine structure of the Perseid stream in the range of photographic magnitudes is studied using the method of indices. A new completed 2003 version of the IAU Meteor Data Center Catalogue of 4581 photographic orbits is used. The method of indices is used to acquire a basic data set for the Perseids. Subsequently, the method is applied on the chosen Perseids to study their structure. Sixty four percent of chosen Perseids taken into account are attached to one of the 17 determined filaments of orbits. The filaments are not distributed in the space accidentally, but they form a higher structure consisting of at least four well-defined and distinguished “branches”.     

Orbits and manifolds near the equilibrium points around a rotating asteroid

We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka.     

Periodic orbits of an electric charge in a magnetic dipole field

Periodic orbits in the Stormer problem are studied using the symmetry lines of the Poincaré map introduced by De Vogelaere. Many known facts are explained by mean of these lines. The dynamics of four special symmetry lines when the Stormer parameter ₁ changes is presented, and we obtain a clear global view of the structure of the simple periodic orbits and their bifurcations, including the asymmetrical ones. New asymmetrical multiple periodic orbits are obtained.     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

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.     

Constructing ballistic capture orbits in the real Solar System model

A method to design ballistic capture orbits in the real Solar System model is presented, so extending previous works using the planar restricted three-body problem. In this generalization a number of issues arise, which are treated in the present work. These involve reformulating the notion of stability in three-dimensions, managing a multi-dimensional space of initial conditions, and implementing a restricted \(n\) -body model with accurate planetary ephemerides. Initial conditions are categorized into four subsets according to the orbits they generate in forward and backward time. These are labelled weakly stable, unstable, crash, and acrobatic, and their manipulation allows us to derive orbits with prescribed behavior. A post-capture stability index is formulated to extract the ideal orbits, which are those of practical interest. Study cases analyze ballistic capture about Mercury, Europa, and the Earth. These simulations show the effectiveness of the developed method in finding solutions matching mission requirements.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

Periodic Orbits in Rotating Second Degree and Order Gravity Fields

Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates.     

DYNAMICAL RELATION OF METEORIDS TO COMETS AND ASTEROIDS

We tested four criteria used for discrimination between asteroidal and cometary type of orbits: Whipple criterion K, Kresak criterion Pe, Tisserand invariant T and aphelion distance Q. To estimate their reliability, all criteria were applied to classify the 2225 orbits of NEAs and 582 orbits of comets, for several epochs spanning the time interval of 40 thousands years. The Q-criterion produced the smallest number of exceptions and has shown the best stability. The biggest number of exceptions and the biggest variations are obtained for the K-criterion. We applied the Q-criterion to classify meteor orbits from the IAU Meteor Data Center and the video meteor orbits available on the Web sites. Among the sporadic radar orbits, as well as among the mean orbits of meteor streams a strong preponderance of asteroid-type orbits was observed. In case of the photographic and video meteors a weak preponderance of cometary and asteroidal orbits was found, respectively.     

Transversal ejection-collision orbits in Hill's problem forC≫1

In a recent paper [3], Lacomba and Llibre showed numerically the existence of two transversal ejection-collision orbits in Hill's problem for a valueC=5 of the Jacobian constant. This result can be used to prove the non-existence ofC ¹-extendable regular integrals for Hill's problem. Here we give an analytic proof of the existence of four ejection-collision orbits which are transversal for large enough values ofC.     

The Structure of Periodic Solutions in the Restricted Problem of the Three Bodies II

In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations.     

Family <Emphasis Type="Italic">h</Emphasis> of periodic solutions of the restricted problem for big μ

The results of the computation of the family h of symmetric periodic solutions of the circular planar restricted three-body problem for μ = 0.3, 0.4, and 0.5 are presented. This family begins with retrograde circular orbits around a massive body. Associated with each value of μ are the table of critical orbits, the orbit pictures, the graphs of characteristics of the family in four coordinate systems, and the graphs of the period and traces (planar and vertical). Regularities on the family and its evolution as μ increased were observed.     

Orbital evolution of giant comet-like objects

We investigated by numerical integrations the long-term orbital evolution of four giant comets or comet-like objects. They are Chiron, P/Schwassmann-Wachmann 1 (SW1), Hidalgo, and 1992AD (5145), and their orbits were traced for 100–200 thousand years (kyr) toward both the past and the future. For each object, 13 orbits were calculated, one for the nominal orbital elements and other 12 with slightly modified elements based on the rms residual of the orbit determination and on the number of observations. As past studies indicate, their orbital evolution is found to be very chaotic, and thus can be described only in terms of probability. Plots of the semi-major axis (a) and perihelion distance (q) of the objects treated here seem to cross each other frequently, suggesting a possibility of their common evolutionary paths. About a half of all the calculated orbits showedq- ora-decreasing evolution. This indicates that, at least on the time scale in question, the giant comet-like objects are possibly on a dynamical track that can lead to capture from the outer solar system. We could hardly find the orbits with perihelia far outside the orbit of Saturn (q>15 AU). This is perhaps because the evolution of the orbits beyond Saturn is so slow that substantial orbital changes do not take place within 100–200 kyr.     

The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N -body simulations.
The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and the orbit of the star within the satellite . We demonstrate that this last point is critical and suggest using three tidal radii to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power-law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically.
Over short times (≲1–2 Gyr ∼1 satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.     

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.     

Earth–Moon Weak Stability Boundaries in the restricted three and four body problem

This work deals with the structure of the lunar Weak Stability Boundaries (WSB) in the framework of the restricted three and four body problem. Geometry and properties of the escape trajectories have been studied by changing the spacecraft orbital parameters around the Moon. Results obtained using the algorithm definition of the WSB have been compared with an analytical approximation based on the value of the Jacobi constant. Planar and three-dimensional cases have been studied in both three and four body models and the effects on the WSB structure, due to the presence of the gravitational force of the Sun and the Moon orbital eccentricity, have been investigated. The study of the dynamical evolution of the spacecraft after lunar capture allowed us to find regions of the WSB corresponding to stable and safe orbits, that is orbits that will not impact onto lunar surface after capture. By using a bicircular four body model, then, it has been possible to study low-energy transfer trajectories and results are given in terms of eccentricity, pericenter altitude and inclination of the capture orbit. Equatorial and polar capture orbits have been compared and differences in terms of energy between these two kinds of orbits are shown. Finally, the knowledge of the WSB geometry permitted us to modify the design of the low-energy capture trajectories in order to reach stable capture, which allows orbit circularization using low-thrust propulsion systems.