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.     

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.     

Post-Newtonian satellite orbits

The first post-Newtonian approximation of general relativity is used to account for the motion of solar system bodies and near-Earth objects which are slow moving and produce weak gravitational fields. The \(n\)-body relativistic equations of motion are given by the Einstein-Infeld-Hoffmann equations. For \(n=2\), we investigate the associated dynamics of two-body systems in the first post-Newtonian approximation. By direct integration of the associated planar equations of motion, we deduce a new expression that characterises the orbit of test particles in the first post-Newtonian regime generalising the well-known Binet equation for Newtonian mechanics. The expression so obtained does not appear to have been given in the literature and is consistent with classical orbiting theory in the Newtonian limit. Further, the accuracy of the post-Newtonian Binet equation is numerically verified by comparing secular variations of known expression with the full general relativistic orbit equation.     

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.     

Admissible orbits in the Oort cloud and velocities on such orbits

In this paper we deal with determinations of: admissible orbits and ranges of orbital velocity in the cloud, extremal velocities at the distance r from the Sun. Moreover, in velocity space we consider the _r region in which there are located tips of velocity vectors for comets moving on admissible orbits.     

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.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

Higher order resonant orbits

I present higher-order resonances of the tube and boomerang orbits in the co-rotating meridional plane of an axisymmetric galactic potential. Additionally, mirror forms of these orbits are given.     

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.     

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.     

Lunar frozen orbits revisited

Lunar frozen orbits, characterized by constant orbital elements on average, have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third-body perturbation exerted by the Earth. The resulting mean orbital elements must be converted to osculating elements to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic \(J_2\), the sectorial harmonic \(C_{22}\), and the Earth third-body effect. In the current paper, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of circumlunar orbits, and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include \(C_{22}\), and hence does not affect the equilibria leading to frozen orbits, \(C_{22}\) is present in the mean-to-osculating transformation, hence affecting the initialization of the physical circumlunar orbit. Simulations show that by using the newly-derived transformation, frozen orbits exhibit better behavior in terms of long-term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.     

Stability of particle orbits

The correspondence between the stability of particle orbits in synchrotrons and storage rings and the standard maps of modern nonlinear dynamics is discussed. The problem is exposed in the language of particle accelerator physics. Examples are given how the equations for synchrotron and betatron oscillations can be brought into the form of standard maps.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

Asteroids in cometary orbits

Orbital integrations are presented for a total of 14 asteroids with perihelia inside 1.7 AU and with aphelion distances in excess of 4 AU, 10 of which were discovered in 1979–1984. The integrations were normally extended over approximately ±1000 years in a three-body model (Sun-Jupiter-asteroid). The effects of uncertainties of starting orbits are not treated in this work, and as far as the real asteroids are concerned, the results should be regarded mostly as preliminary indications. A wide variety of orbital evolutions is found, and some of them evidently belong to the cometary, chaotic type. Three such cases are identified with certainty (1983 SA, 1983 XF, and 1984 BC) and two or three more with various degrees of likelihood. An asteroidal motion is found for the well-observed object 1979 VA. A stable libration around the $\text{2}\text{1}$ resonance is found for 1981 FD, which obviously adds to the Griqua group. A long-lasting libration around the $\text{5}\text{3}$ resonance performed by 1982 YA is probably unstable. Temporary librations are also found for 1983 SA ($\text{4}\text{3}$ resonance) and 1983 XF ($\text{2}\text{1}$ resonance), but these objects appear to transit into irregular motions with close approaches to Jupiter (less than 0.01 AU for 1983 XF). A very rapid large-amplitude ω libration around 90° is found in the future motion of 1983 VA. If this will indeed occur for the real asteroid, the object will oscillate with a period of only 750 years between a main-belt orbit of very high inclination and a low-inclination Apollo-type orbit.     

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.     

Satellite orbits and ephemerides

This review considers theoretical work leading to the actual knowledge of the motions of all planetary satellites, except the Moon. It covers recent calculations of inequalities of satellites' motions and the determination of their orbits on the basis of existing observational data. The results are presented and some of them discussed in views of possible future developments.     

Formulation of the restricted three-body problem with two-center problem orbits as intermediate orbits

The restricted three-body Hamiltonian is partitioned into a two-center type principal part and its accompanying perturbational part. The mathematical analysis, involving the Jacobian elliptic functions, is adapted for the case of figure-eight orbits winding around the two given mass points. For many such orbits the elliptic function modulusk is small and can serve as a small parameter.Fourier expansions in terms of a parameter related tot are obtained for the intermediate orbit functions which provide representations in terms of elementary functions.     

Pre-capture orbits of the moon

If the mass of the Earth was not considerably larger than at present, the pre-capture orbit of the Moon was in the range 0.9–1.1 A.U. Capture occurred within several 10⁸ years after formation of the Moon.