Statistical mechanics of Keplerian orbits

Perturbations and gravitational encounters are incorporated into the statistical theory of Keplerian orbits. The birth of planets is discussed as an application of the theory. It turns out to be a consequence of the combined action of collisions, differential rotation and gravitational interaction.     

Statistical mechanics of Keplerian orbits

A statistical theory of Keplerian orbits is constructed for a system of particles, which are subject to partially elastic collisions. If the elasticity decreases with collisional velocity, the system shows an increased tendency to form condensations. Near the central body they are concentric rings, which are separated by gaps void of matter. At larger distances outside the Roche limit, the condensations probably form larger bodies. An application to Saturn's rings suggests that at least rings A and C would consist of separate ringlets.     

Metric spaces of Keplerian orbits

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.     

Statistical mechanics of Keplerian orbits

Theoretical predictions agree with computer simulations at least for those collisional systems in which the restitution coefficient is independent of impact velocity. An uncertainty principle for the orbits restricts the validity of the theory and its predictions. Discussion of the whole theory and of computer simulations shows that a velocity-dependent restitution coefficient provides the only astronomically interesting applications of the collisional processes. The Saturnian and Uranian ring systems correspond very well to theoretical expectations if the restitution coefficient is of this type.     

Keplerian periodic orbits in the isosceles problem

The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision. The basic tool used is the study of the Poincaré maps associated with the periodic solutions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Metrics in Keplerian orbits quotient spaces

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.     

On linking coefficient of two Keplerian orbits

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.     

Collision criterion for two satellites on Keplerian orbits

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.     

Collisions of two solid bodies in Keplerian orbits; new orbits and systematic effects

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.     

A revised flux vector for the collisional evolution of Keplerian orbits

In statistical Keplerian systems the disordered component of collisionally induced motion of matter introduces new terms into the flux vector. This contribution, which is calculated from a transport equation, tends to reduce the density gradient and causes the expansion which is observed in computer simulations of collisional systems. A quantitative comparison with Trulsen's (1972) simulations confirms the revised expression of the flux vector.     

On the Distance Function Between Two Keplerian Elliptic Orbits

The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the minimum distance between two Keplerian orbits with a common focus

The methods used so far for determination of the closest approach between two orbits are discussed, and corrected versions of two of them are presented.     

Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization

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.     

The rotation number of bounded orbits in a central field

After reviewing known results on bounded motions in a central field and on the angle between two consecutive apocenters, we investigate the limit of the orbits as the pericenter tends to the origin. A ‘deflection’ of the almost rectilinear orbits is found which depends on the shape of the potential at the origin. We then study the rotation numbers of all, bounded motions in a field homogeneous inr. Finally on a few examples we show how to represent the set of bounded motions in an inhomogeneous field.     

Triple close approach in the three-body problem: A limit for the bounded orbits

Using the results of Sundman and Birkhoff as also the studies on the generalized Hill's curves, we develop a new method for computing a lower bound of the moment of inertia for the bounded orbits in the general three-body problem.Thus we obtain much higher values than in the classical results. We show for instance, that when the integral of the energy goes to zero, this lower bound goes to the minimum moment of inertia of the corresponding parabolic Euler motion of the same angular momentum: it is then the greatest lower bound.

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Résumé En utilisant les résultats de Sundman et de Birkhoff ainsi que les études sur les courbes de Hill généralisées, on développe une nouvelle méthode pour calculer un minorant du moment d'inertie pour les orbites bornées dans le problème des trois corps.On obtient ainsi des valeurs beaucoup plus élevées que dans les résultats classiques. On montre en particulier que lorsque I'intégrale de I'énergie tend vers zéro ce minorant tend vers le minimum du moment d'inertie du mouvement parabolique d'Euler correspondant de même moment cinétique: c'est alors la limite inférieure.

     

Collision risk against space debris in Earth orbits

?pik’s formulae for the probability of collision are applied to the analysis of the collision risk against space debris in Low-Earth Orbit (LEO) and Medium Earth Orbit. The simple analytical formulation of ?pik’s theory makes it applicable to complex dynamical systems, such as the interaction of the ISS with the whole debris population in LEO The effect of a fragmentation within a multiplane constellation can also be addressed. The analysis of the evolution of the collision risk in Earth orbit shows the need of effective mitigation measures to limit the growth of the collision risk and of the fragmentation debris in the next century.     

An Algebraic Method to Compute the Critical Points of the Distance Function Between Two Keplerian Orbits

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.      

Exploiting Earth horseshoe orbits for space missions

We discuss the usefulness of unstable Earth horseshoe orbits to access interplanetary space, and in particular when low-velocity escape from Earth is involved. The application to a solar stereoscopic mission is described.     

Long-term capture orbits for low-energy space missions

This research aims at ascertaining the existence and characteristics of natural long-term capture orbits around a celestial body of potential interest. The problem is investigated in the dynamical framework of the three-dimensional circular restricted three-body problem. Previous numerical work on two-dimensional trajectories provided numerical evidence of Conley’s theorem, proving that long-term capture orbits are topologically located near trajectories asymptotic to periodic libration point orbits. This work intends to extend the previous investigations to three-dimensional paths. In this dynamical context, several special trajectories exist, such as quasiperiodic orbits. These can be found as special solutions to the linear expansion of the dynamics equations and have already been proven to exist even using the nonlinear equations of motion. The nature of long-term capture orbits is thus investigated in relation to the dynamical conditions that correspond to asymptotic trajectories converging into quasiperiodic orbits. The analysis results in the definition of two parameters characterizing capture condition and the design of a capture strategy, guiding a spacecraft into long-term capture orbits around one of the primaries. Both the results are validated through numerical simulations of the three-dimensional nonlinear dynamics, including fourth-body perturbation, with special focus on the Jupiter–Ganymede system and the Earth–Moon system.