Determination of an intermediate perturbed orbit from three position vectors

We suggest a new approach to solving the problem of finding the orbit of a celestial body from its three spatial position vectors and the corresponding times. It allows most of the perturbations in the motion of a celestial body to be taken into account. The approach is based on the theory of intermediate orbits that we developed previously. We construct the orbit the motion along which is a combination of two motions: the motion of a fictitious attracting center whose mass varies according to Mestschersky’s first law and the motion relative to the fictitious center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden-Mestschersky problem. The constructed orbit has such parameters that their limiting values at any reference epoch define a superosculating intermediate orbit with a fourth-order tangency. We have performed a numerical analysis to estimate the accuracy of approximating the perturbed motion of two minor planets, 145 Adeona and 4179 Toutatis, by the orbits computed using two-position procedures (the classical Gauss method and the method that we suggested previously), a three-position procedure based on the Herrick-Gibbs equation, and the new method. Comparison of the results obtained suggests that the latter method has an advantage.     

Osculating and Superosculating Intermediate Orbits and their Applications

A method of construction of intermediate orbits for approximating the real motion of celestial bodies in the initial part of trajectory is proposed. The method is based on introducing a fictitious attracting centre with a time-variable gravitational parameter. The variation of thisparameter is assumed to obey the Eddington–Jeans mass-variationlaw. New classes of orbits having first-, second-, and third-order tangency to the perturbed trajectory at the initial instant of time are constructed. For planar motion, the tangency increases by one or two orders. The constructed intermediate orbits approximate the perturbed motion better than the osculating Keplerian orbit and analogous orbits of otherauthors. The applications of the orbits constructed in Encke's methodfor special perturbations and in the procedure for predicting themotion in which the perturbed trajectory is represented by a sequenceof short arcs of the intermediate orbits are suggested.The use of the constructed orbits is especially advantageous in the investigation of motion under the action of large perturbations.     

The Determination of an Intermediate Perturbed Orbit from Two Position Vectors

Based on the theory of intermediate orbits developed earlier by the author of this paper, a new approach is proposed to the solution of the problem of finding the orbit of a celestial body with the use of two position vectors of this body and the corresponding time interval. This approach makes it possible to take into account the main part of perturbations. The orbit is constructed, the motion along which is a combination of two motions: the uniform motion along a straight line of a fictitious attracting center, whose mass varies according to the first Meshchersky law, and the motion around this center. The latter is described by the equations of the Gylden–Meshchersky problem. The parameters of the constructed orbit are chosen so that their limiting values at any reference epoch determine a superosculating intermediate orbit with third-order tangency. The accuracy of approximation of the perturbed motion by the orbits calculated by the classical Gauss method and the new method is illustrated by an example of the motion of the unusual minor planet 1566 Icarus. Comparison of the results obtained shows that the new method has obvious advantages over the Gauss method. These advantages are especially prominent in cases where the angular distances between the reference positions are small.     

New Numerical/Analytical Methods for Solving Equations of Orbital Motion

New methods are proposed for solving equations of motion of celestial bodies. The methods are based on the use of superosculating orbits with second- and third-order tangency to the trajectory of the real motion of a body. The construction of these orbits is related to the concept of a fictitious attracting center, whose mass varies in accordance with the first Meshchersky law. In the original reference methods, the perturbed trajectory is represented by a sequence of small arcs of superosculating orbits. The order of accuracy of the reference methods coincides with the order of tangency of the superosculating orbit used in calculations. Using Runge's rule and Richardson's extrapolation scheme leads to the methods of higher order. The efficiency of the new methods in comparison with the numerical integration of equations of motion based on the well-known fourth- and seventh-order Runge–Kutta–Fehlberg methods is illustrated by examples of the calculation of perturbed orbits of some asteroids.     

Application of a Fictitious Attracting Center to the Study of the Motion of Minor Bodies Approaching Major Planets

The motion of minor Solar System bodies having close encounters with major planets is described using the model of motion within the framework of the perturbed restricted three-body problem. The actual motion of a minor body is represented as a combination of two motions, namely, the motion of a fictitious attracting center with a variable mass and the motion with respect to the fictitious center. The position and mass of the fictitious center are chosen so that, when the minor body collides with any of the primaries, the fictitious center carries into the center of inertia of the colliding body and the mass of the fictitious center becomes identical to the mass of this body. The regularizing KS-transformation and Sundman’s time transformation were applied to coordinates and velocities. As a result, a system of differential equations of motion that are quasilinear within the nearest vicinity of each of the primary attracting bodies was obtained. These equations are characterized by a numerical behavior during the encounters of the minor body with the primaries that is essentially better than that of the initial equations of motion. The motion of comets Brooks 2 and Gehrels 3, which have fairly close encounters with Jupiter, is simulated.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 272–280.Original Russian Text Copyright © 2005 by Shefer.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

On the accuracy of approximation of motion of a small celestial body by intermediate perturbed orbits calculated on the basis of three position vectors and three observations

Intermediate perturbed orbits, which were proposed earlier by the first author and are calculated based on three position vectors and three measurements of angular coordinates of a small celestial body, are examined. Provided that the reference time interval encompassing the measurements is short, these orbits are close in the accuracy of approximation of actual motion to an orbit with fourth-order tangency. The shorter the reference time interval is, the better is the approximation. The laws of variation of the errors of methods for constructing such intermediate orbits with the length of the reference time interval are formulated. According to these laws, the rate of convergence of the methods to an exact solution in the process of shortening of the reference time interval is, in general, three orders of magnitude higher than that of conventional methods relying on an unperturbed Keplerian orbit. The considered orbits are among the most accurate of their class that is defined by the order of tangency. The obtained theoretical results are verified by numerical experiments on determining the orbit of 99942 Apophis.     

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.     

Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found.     

Some properties of the dumbbell satellite attitude dynamics

The dumbbell satellite is a simple structure consisting of two point masses connected by a massless rod. We assume that it moves around the planet whose gravity field is approximated by the field of the attracting center. The distance between the point masses is assumed to be much smaller than the distance between the satellite’s center of mass and the attracting center, so that we can neglect the influence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Our aim is to study the satellite’s attitude dynamics. When the center of mass moves on a circular orbit, one can find a stable relative equilibrium in which the satellite is permanently elongated along the line joining the center of mass with the attracting center (the so called local vertical). In case of elliptic orbits, there are no stable equilibrium positions even for small values of the eccentricity. However, planar periodic motions are determined, where the satellite oscillates around the local vertical in such a way that the point masses do not leave the orbital plane. We prove analytically that these planar periodic motions are unstable with respect to out-of-plane perturbations (a result known from numerical investigations cf. Beletsky and Levin Adv Astronaut Sci 83, 1993). We provide also both analytical and numerical evidences of the existence of stable spatial periodic motions.     

Discussions of the internal motions and mass segregation of M67 on the basis of accurate proper motions

The radius and virial mass of the old open cluster M67 are presented. The internal motion and mass segregation of the cluster are also discussed on the basis of accurate stellar proper motions obtained combining three independent proper motion catalogues of the cluster. Increases of the mean proper motion and the intrinsic dispersion of member stars with radial distance from the cluster center might suggest that the stars are escaping from the cluster. The stars in both inner and outer regions appear to be in isotropic orbits. At last, it is found that both space and velocity mass segregations exist for the old open cluster due to the dynamical evolution.     

Periodic orbits in the general three-body problem and the relationship between them

We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ₁ and φ₂, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ₁, φ₂∈(0, π) at steps of δk=0.01; δφ₁=δφ₂=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ₁, φ₂). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup.     

Celestial-mechanical peculiarities of Uranus’s satellite system

We consider the structural peculiarities of Uranus’s satellite system associated with its separation into two groups: inner equatorial satellites moving in nearly circular orbits and distant irregular satellites with retrograde motion in highly elliptical orbits. The intermediate region is free from satellites in a wide range of semimajor axes. By analyzing the evolution of satellite orbits under the combined effect of solar attraction and Uranus’s oblateness, we offer a celestial-mechanical explanation for the absence of equatorial satellites in this region. M.L. Lidov’s studies during 1961–1963 have served as a basis for our analysis.     

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

On the stability of circumbinary planetary systems

The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits.     

Optimal transfers between unstable periodic orbits using invariant manifolds

This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.     

Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

Where are the Uranus Trojans?

The area of stable motion for fictitious Trojan asteroids around Uranus’ equilateral equilibrium points is investigated with respect to the inclination of the asteroid’s orbit to determine the size of the regions and their shape. For this task we used the results of extensive numerical integrations of orbits for a grid of initial conditions around the points L ₄ and L ₅, and analyzed the stability of the individual orbits. Our basic dynamical model was the Outer Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations of motion of fictitious Trojans in the vicinity of the stable equilibrium points for selected orbits up to the age of the Solar system of 5 × 10⁹ years. One experiment has been undertaken for cuts through the Lagrange points for fixed values of the inclinations, while the semimajor axes were varied. The extension of the stable region with respect to the initial semimajor axis lies between 19.05 ≤ a ≤ 19.3 AU but depends on the initial inclination. In another run the inclination of the asteroids’ orbit was varied in the range 0° < i < 60° and the semimajor axes were fixed. It turned out that only four ‘windows’ of stable orbits survive: these are the orbits for the initial inclinations 0° < i < 7°, 9° < i < 13°, 31° < i < 36° and 38° < i < 50°. We postulate the existence of at least some Trojans around the Uranus Lagrange points for the stability window at small and also high inclinations.     

The rectilinear three-body problem

The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.