Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

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.     

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.     

Calculation of the intermediate perturbed orbit from three or more positions of a small body on the celestial sphere

Two new methods are described for finding the orbit of a small celestial body from three or more pairs of angular measurements and the corresponding time points. The methods are based on, first, the approach that has been developed previously by the author to the determination, from a minimum number of observations, of intermediate orbit considering most of the perturbations in the bodies’ motion and, second, Herget’s algorithmic procedure enabling the introduction of additional observations. The errors of orbital parameters calculated by the proposed methods are two orders of magnitude smaller than the corresponding errors of the traditional approach based on the construction of an unperturbed Keplerian orbit. The thus-calculated orbits of the minor planets 1566 Icarus, 2002 EC1, and 2010 TO48 are used to compare the results of Herget’s multiposition procedure and the new methods. The comparison shows that the new methods are highly effective in the study of perturbed motion. They are particularly beneficial if high-precision observational data covering short orbital arcs are available.     

A new method for the determination of an intermediate orbit using three positions of a small body on the celestial sphere

We propose a new method for the determination of the preliminary orbit of a small celestial body using three pairs of its angular coordinates in three moments of time. The method is based on the use of the intermediate orbit we constructed earlier using three position vectors and the corresponding time moments. This intermediate orbit accounts for the main part of the perturbations of the motion of the body under study. We compare the results obtained by the classical Lagrange-Gauss method, Herrick-Gibbs method, generalized Herrick-Gibbs method, and the new method by the examples of the determination of the orbit of the small planet 1566 Icarus. The comparison showed that the new method is a highly efficient tool for the study of perturbed motion. It is especially efficient when applied to high-precision observational data covering short arcs of the orbit.     

A Method for the Determination of an Intermediate Orbit from Three Positions of the Small Body on the Celestial Sphere

A new method is suggested for finding the preliminary orbit from three complete measurements of the angular coordinates of a celestial body developed by analogy with the classic Lagrange–Gauss method. The proposed method uses the intermediate orbit that we had constructed in an earlier paper based on two position vectors and the corresponding time interval. This intermediate orbit allows for most of the perturbations in the motion of the body. Using the orbital motion of asteroid 1566 Icarus as an example, we compare the results obtained by applying the classic and the new method. The comparison shows the new method to be highly efficient for studying perturbed motion. It is especially efficient if applied to high-precision observational data covering short orbital arcs.     

A method of intermediate orbit determination based on four positions of a minor body on the celestial sphere

A new method of computing the preliminary orbit of a celestial body based on four pairs of angle measurements has been suggested. The method makes use of preliminary orbit previously constructed by the author based on two position vectors and a corresponding time interval, taking into account the main part of the perturbations in the motion of the body under study. Using the example of constructing the orbit of the minor planet 1383 Limburgia, the results obtained using a four-position procedure of the Gaussian type based on the solution of a two-body problem have been compared with those of the new method. The comparison showed the new method to be highly efficient for perturbed motion studies. It is especially advantageous in the case of high-accuracy observation data on small orbital arcs.     

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.     

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.     

An exact solution to determination of an open orbit

We present an exact solution of the equations for orbit determination of a two body system in a hyperbolic or parabolic motion. In solving this problem, we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary system in an elliptic orbit. The solutions applicable to each of elliptic, hyperbolic and parabolic orbits are obtained by the new approach, and they are all expressed in an explicit form, remarkably, only in terms of elementary functions. We show also that the solutions for an open orbit are recovered by making a suitable transformation of the AAK solution for an elliptic case.     

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.     

The intermediate orbit of resonance asteroids of the Hecuba family constructed on the basis of solution of the internal variant of the generalized problem of the three fixed centres

On the basis of the solution of the internal variant of the generalized problem of three fixed centres (taking as an example a minor planet of 108 Hecuba) an intermediate orbit of the resonance asteroids of the Hecuba family has been constructed. Comparison of the results obtained from the formulae with observations and also with the results of Isaeva (1976) showed that in investigating the motion of the celestial bodies it would be reasonable to use the orbits of the internal variant of the generalized problem of three fixed centres.     

Periodic orbits of the elliptic restricted problem for the Sun-Jupiter-Saturn system

A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable.     

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.     

Hierarchy of periodic solutions families of spatial Hill’s problem

Many modern space projects require the knowledge of orbits with certain properties. Most of these projects assume the motion of a space vehicle in the neighborhood of a celestial body, which in turn moves in the field of the Sun or another massive celestial body. A good approximation of this situation is Hill’s problem. This paper is devoted to the investigation of the families of spatial periodic solutions to the three-dimensional Hill’s problem. This problem is nonintegrable; therefore, periodic solutions are studied numerically. The Poincare theory of periodic solutions of the second kind is applied; either planar or vertical impact orbits are used as generating solutions.     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

Numerical models for the study of motion of lunar satellites

The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.     

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.     

Quick numerical evaluation of the probability that an asteroid will collide with a planet

We propose a numerical method for quick evaluation of the probability that an asteroid will collide with a planet. The method is based on linear mappings of an expected moment of a close approach of the asteroid to the planet and the detection of collisions of the virtual objects with the massive body. The standard way for solving the problem of estimating the collision probability consists in simulating the evolution of the uncertainty cloud numerically based on the stepwise integration of virtual orbits. This is naturally associated with huge processor time costs. The proposed method is tested using the examples of the 2011 AG5 and 2007 VK184 asteroids that are presently in the top of the list of the most dangerous celestial objects. The test results show that linear mappings allow one to obtain the estimates of probabilities quicker by several orders than numerical integration of all virtual orbits.