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1.
Roman V. Baluyev Konstantin V. Kholshevnikov 《Celestial Mechanics and Dynamical Astronomy》2005,91(3-4):287-300
In the paper by Kholshevnikov and Vassilie, 1999, (see also references therein) the problem of finding critical points of
the distance function between two confocal Keplerian elliptic orbits (hence finding the distance between them in the sense
of set theory) is reduced to the determination of all real roots of a trigonometric polynomial of degree eight. In non-degenerate
cases a polynomial of lower degree with such properties does not exist. Here we extend the results to all possible cases of
ordered pairs of orbits in the Two–Body–Problem. There are nine main cases corresponding to three main types of orbits: ellipse,
hyperbola, and parabola. Note that the ellipse–hyperbola and hyperbola–ellipse cases are not equivalent as we exclude the
variable marking the position on the second curve. For our purposes rectilinear trajectories can be treated as particular
(not limiting) cases of elliptic or hyperbolic orbits. 相似文献
2.
Jared M. Maruskin 《Celestial Mechanics and Dynamical Astronomy》2010,108(3):265-274
In this paper we derive an explicit, analytic formula for the geodesic distance between two points in the space of bounded
Keplerian orbits in a particular topology. The specific topology we use is that of a cone passing through the direct product
of two spheres. The two spheres constitute the configuration manifold for the space of bounded orbits of constant energy.
We scale these spheres by a factor equal to the semi-major axis of the orbit, forming a linear cone. This five-dimensional
manifold inherits a Riemannian metric, which is induced from the Euclidean metric on
\mathbbR6{\mathbb{R}^6}, the space in which it is embedded. We derive an explicit formula for the geodesic distance between any two points in this
space, each point representing a physical, gravitationally bound Keplerian orbit. Finally we derive an expression for the
Riemannian metric that we used in terms of classical orbital elements, which may be thought of as local coordinates on our
configuration manifold. 相似文献
3.
Giovanni F. Gronchi 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):295-329
We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits
(either bounded or unbounded) with a common focus. The critical values of this function are important for different purposes,
for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets. Our algorithm is based
on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th
degree univariate polynomial whose real roots give us one component of the critical points. We discuss also some degenerate
cases and show several examples, involving the orbits of the known asteroids and comets.
相似文献
4.
R. Armellin P. Di Lizia M. Berz K. Makino 《Celestial Mechanics and Dynamical Astronomy》2010,107(3):377-395
A novel method to compute all critical points of the distance function between two Keplerian orbits (either bounded or unbounded)
with a common focus is presented. The problem is attacked as a global optimization problem, solved by a rigorous global optimizer
based on Taylor models. Thus, thigh enclosures of the stationary points are obtained. The embedded capability of the method
of delivering high-order Taylor expansions is then used to analyze how uncertain orbital parameters affect the position of
the stationary points and the associated distance values. Sample orbital sets and Apophis asteroid are used as test cases. 相似文献
5.
Konstantin V. Kholshevnikov 《Celestial Mechanics and Dynamical Astronomy》2008,100(3):169-179
Several metric spaces of Keplerian orbits and a set of their most important subspaces, as well as a factor space (not distinguishing
orbits with the same longitudes of nodes and pericentres) are constructed. Topological and metric properties of them are established.
Simple formulae to calculate the distance are deduced. Applications to a number of problems of Celestial Mechanics are discussed. 相似文献
6.
Konstantin V. Kholshevnikov Nikolay N. Vassiliev 《Celestial Mechanics and Dynamical Astronomy》1999,75(1):67-74
We define a function of the set of pairs of Keplerian ellipses so that the sign of the function will be a topological invariant
of their configuration. The sign is negative if and only if the related ellipses are linked. Two modifications of the coefficient
which are more reliable in the case of closed to coplanar orbits are proposed. Explicit formulae representing the linking
coefficients as functions of orbital elements are deduced. Extension in the case of unbounded orbits is obtained. We suggest
different ways to use these coefficients for determining intersections of pairs of osculating Keplerian orbits. If we study
dynamical behaviour of geometric configuration of pairs of Keplerian orbits, we can fix the moments of their intersections.
These moments correspond exactly to the vanishing of linking coefficients.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
7.
Claudio Bombardelli 《Celestial Mechanics and Dynamical Astronomy》2014,118(2):99-114
The paper deals with the problem of impulsive collision avoidance between two colliding objects in three dimensions and assuming elliptical Keplerian orbits. Closed-form analytical expressions are provided that accurately predict the relative dynamics of the two bodies in the encounter b-plane following an impulsive delta-V manoeuvre performed by one object at a given orbit location prior to the impact and with a generic three-dimensional orientation. After verifying the accuracy of the analytical expressions for different orbital eccentricities and encounter geometries the manoeuvre direction that maximises the miss distance is obtained numerically as a function of the arc length separation between the manoeuvre point and the predicted collision point. The provided formulas can be used for high-accuracy instantaneous estimation of the outcome of a generic impulsive collision avoidance manoeuvre and its optimisation. 相似文献
8.
This paper analyzes the collision possibility for two satellites on Keplerian orbits. Coplanar and noncoplanar cases are considered,
respectively. For each case, the problem of collision possibility analysis can be solved through two steps: First, to determine
whether there is any intersection point of the two orbits; if there is no intersection point, the conclusion can be given
directly that collision never happens. Secondly, if the two orbits do intersect, the collision possibility in the given time-scale
can be studied in terms of the relationship between the two orbital periods and time. Numerical simulations for both cases,
each including several situations, are given to prove the validity of the proposed collision criterion. 相似文献
9.
Quotient spaces of Keplerian orbits are important instruments for the modelling of orbit samples of celestial bodies on a large time span. We suppose that variations of the orbital eccentricities, inclinations and semi-major axes remain sufficiently small, while arbitrary perturbations are allowed for the arguments of pericentres or longitudes of the nodes, or both. The distance between orbits or their images in quotient spaces serves as a numerical criterion for such problems of Celestial Mechanics as search for common origin of meteoroid streams, comets, and asteroids, asteroid families identification, and others. In this paper, we consider quotient sets of the non-rectilinear Keplerian orbits space \(\mathbb H\). Their elements are identified irrespective of the values of pericentre arguments or node longitudes. We prove that distance functions on the quotient sets, introduced in Kholshevnikov et al. (Mon Not R Astron Soc 462:2275–2283, 2016), satisfy metric space axioms and discuss theoretical and practical importance of this result. Isometric embeddings of the quotient spaces into \(\mathbb R^n\), and a space of compact subsets of \(\mathbb H\) with Hausdorff metric are constructed. The Euclidean representations of the orbits spaces find its applications in a problem of orbit averaging and computational algorithms specific to Euclidean space. We also explore completions of \(\mathbb H\) and its quotient spaces with respect to corresponding metrics and establish a relation between elements of the extended spaces and rectilinear trajectories. Distance between an orbit and subsets of elliptic and hyperbolic orbits is calculated. This quantity provides an upper bound for the metric value in a problem of close orbits identification. Finally the invariance of the equivalence relations in \(\mathbb H\) under coordinates change is discussed. 相似文献
10.
Vacheslav Emel'yanenko 《Celestial Mechanics and Dynamical Astronomy》2002,84(4):331-341
A new symplectic algorithm is developed for cometary orbit integrations. The integrator can handle both high-eccentricity orbits and close encounters with planets. The method is based on time transformations for Hamiltonians separated into Keplerian and perturbation parts. The adaptive time-step of this algorithm depends on the distance from a centre and the magnitude of perturbations. The explicit leapfrog technique is simple and efficient. 相似文献
11.
J. A. Docobo 《Celestial Mechanics and Dynamical Astronomy》1985,36(2):143-153
This article studies the existence of periodic Keplerian orbits for visual double stars whose corresponding apparent orbits are to pass through three selected points. The analytical results provide the basis for a new method of calculating orbits which does not require prior calculation of the areal constant. This method is applied to the binary 04404 N 4313. 相似文献
12.
Natural Metrics in the Spaces of Elliptic Orbits 总被引：1，自引：1，他引：0
Konstantin V. Kholshevnikov Nikolay N. Vassiliev 《Celestial Mechanics and Dynamical Astronomy》2004,89(2):119-125
Different natural metrizations by Hölder type on the five dimensional space of Keplerian elliptic orbits are introduced. Certain applications of topological and metrical properties of the space of Keplerian elliptic orbits to several problems of Celestial Mechanics are discussed. 相似文献
13.
It has been shown in various papers dealing with systems of colloding bodies in a Keplerian field that the dynamical evolution does not depend only on the initial orbital conditions. This is a consequence of the wide range of orbits generated by the collision process. From the study of a few pairs of orbits we examine what factors which produce that variety of orbits, and search for systematic effects. The role of the positions along the orbits, of inelasticity, of size, of mass and of relative inclination is emphasized. 相似文献
14.
A canonical transformation in phase space and a rescaling of time are proposed to reduce a Keplerian system with a time-dependent Gaussian parameter to a perturbed Keplerian system with a constant Gaussian parameter. When the time variation is slow, the perturbation through second order in the reduced problem is conservative, and, as a result, the orbital space of the averaged system is a sphere on which the phase flow causes a differential rotation representing a circulation of the line of apsides. The flow presents two isolated singularities corresponding to circular orbits travelled respectively in the direct and in the retrograde sense, and a degenerate manifold of fixed points corresponding to the collision orbits. Normalization beyond order two does not break the degeneracy. Adiabatic invariants, which are conservative functions, may be computed from the normalized Hamiltonian evaluated here to the fourth order. Nonetheless so high an approximation gives little information because the normalizing Lie transformation depends explicitly on the time through mixed secular-periodic terms. As an application, an estimate is offered for the apsidal rotation that a second order time derivative in the mass of the sun would induce on planetary orbits. This suggests an observational method for determining the latter parameter for the solar wind, but the predicted motions are too slow for the current level of observational precision. 相似文献
15.
H. Ma G. Baù D. Bracali Cioci G. F. Gronchi 《Celestial Mechanics and Dynamical Astronomy》2018,130(10):70
We propose a method to account for the Earth oblateness effect in preliminary orbit determination of satellites in low orbits with radar observations. This method is an improvement of the one described in Gronchi et al. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), which uses a pure Keplerian dynamical model. Since the effect of the Earth oblateness is strong at low altitudes, its inclusion in the model can sensibly improve the initial orbit, giving a better starting guess for differential corrections and increasing the chances to obtain their convergence. The input set consists of two tracks of radar observations, each one composed of at least four observations taken during the same pass of the satellite. A single observation gives the topocentric position of the satellite, where the range is very accurate, while the line-of-sight direction is poorly determined. From these data, we can compute by a polynomial fit the values of the range and range rate at the mean epochs of the two tracks. In order to obtain a preliminary orbit, we wish to compute the angular velocity, which is the rate of change of the line of sight. In the same spirit of Gronchi et al. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), we also wish to correct the values of the angular measurements, so that they fit the selected dynamical model if the same holds for the radial distance and velocity. The selected model is a perturbed Keplerian dynamics, where the only perturbation included is the secular effect of the \(J_2\) term of the geopotential. 相似文献
16.
Sándor Molnár 《Celestial Mechanics and Dynamical Astronomy》1981,25(1):81-88
It is shown, that the potential obtained from Joukovsky's formula, corresponding to a given family of orbits is a general solution of Szebehely's equation. Then it is shown how a general solution of Szebehely's equation can be obtained from its particular solution. This method is applied to several examples. Potentials generating families of concentric elliptic orbits and families of orbits of conic sections are determined. Finally, the inverse Keplerian problem is solved using Szebehely's equation in polar coordinates. 相似文献
17.
John D. Hadjidemetriou George Voyatzis 《Celestial Mechanics and Dynamical Astronomy》2011,111(1-2):179-199
We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star. 相似文献
18.
Colin R. McInnes 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):199-215
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. 相似文献
19.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2017,127(3):285-300
This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions. 相似文献
20.
V. A. Shefer 《Celestial Mechanics and Dynamical Astronomy》2002,82(1):19-59
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. 相似文献