Problems Of Celestial Mechanics Research Articles

The Two-Body Problem in Celestial Mechanics

The Two-Body Problem in Celestial Mechanics

Intrinsic nonlinearity and method of disturbed observations in inverse problems of celestial mechanics

Orbit determination from a small sample of observations over a very short observed orbital arc is a strongly nonlinear inverse problem. In such problems an evaluation of orbital uncertainty due to random observation errors is greatly complicated, since linear estimations conventionally used are no longer acceptable for describing the uncertainty even as a rough approximation. Nevertheless, if an inverse problem is weakly intrinsically nonlinear, then one can resort to the so-called method of disturbed observations (aka observational Monte Carlo). Previously, we showed that the weaker the intrinsic nonlinearity, the more efficient the method, i.e. the more accurate it enables one to simulate stochastically the orbital uncertainty, while it is strictly exact only when the problem is intrinsically linear. However, as we ascertained experimentally, its efficiency was found to be higher than that of other stochastic methods widely applied in practice. In the present paper we investigate the intrinsic nonlinearity in complicated inverse problems of Celestial Mechanics when orbits are determined from little informative samples of observations, which typically occurs for recently discovered asteroids. To inquire into the question, we introduce an index of intrinsic nonlinearity. In asteroid problems it evinces that the intrinsic nonlinearity can be strong enough to affect appreciably probabilistic estimates, especially at the very short observed orbital arcs that the asteroids travel on for about a hundredth of their orbital periods and less. As it is known from regression analysis, the source of intrinsic nonlinearity is the nonflatness of the estimation subspace specified by a dynamical model in the observation space. Our numerical results indicate that when determining asteroid orbits it is actually very slight. However, in the parametric space the effect of intrinsic nonlinearity is exaggerated mainly by the ill-conditioning of the inverse problem. Even so, as for the method of disturbed observations, we conclude that it practically should be still entirely acceptable to adequately describe the orbital uncertainty since, from a geometrical point of view, the efficiency of the method directly depends only on the nonflatness of the estimation subspace and it gets higher as the nonflatness decreases.

Poincaré–Treshchev Mechanism in Multi-scale, Nearly Integrable Hamiltonian Systems

This paper is a continuation to our work (Xu et al. in Ann Henri Poincare 18(1):53–83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Russmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincare non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincare 10(8):1419–1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincare non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincare–Treshchev mechanism.

The Classical N-body Problem in the Context of Curved Space

AbstractWe provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature κ, for all κ ∊ ℝ. In previous studies, the equations of motion made sense only for κ ≠ 0. The system derived here does more than just include the Euclidean case in the limit κ → 0; it recovers the classical equations for κ = 0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus oòers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.

Two algorithms to construct a consistent first order theory of equilibrium figures of close binary systems

One of the main problems in celestial mechanics is the study of the shape adopted by extended deformable celestial bodies in its equilibrium configuration. In this paper, a new point of view about classical theories on equilibrium figures in close binary systems is offered.Classical methods are based on the evaluation of the self-gravitational, centrifugal and tidal potentials. The most common technique used by classical methods shows convergence problems. To solve this problem up to first order in amplitudes two algorithms has been developed, the first one based on the Laplace method to develop the inverse of the distance and the second one based on the asymptotic properties of the numerical quadrature formulas.

Construction of approximate solutions some problems of celestial mechanics in the form of trigonometric splines

There was suggested a method of constructing approximate solutions of the first boundary problem forordinary differential equations of the second order with variable coefficients in trigonometricpolynomials using the phantom nodes method. Unknown parameters are determined by collocation.An example is given; it is shown that the relative error of the solution has reduced times with theintroduction of phantom nodes

Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem

We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.

The limit passage of space curvature in problems of celestial mechanics with the generalized Kepler and Hooke potentials

The connection between the problems of celestial mechanics: the Kepler problem, the two-center problem and the two body problem in spaces of constant curvature with the generalized Kepler and Hooke potentials is investigated. The limit passage in the two-center and two body problems in the Lobachevsky space and on a sphere is carried out as \(\lambda\to0\) (\(\lambda\) is the curvature of the corresponding space) for the two potentials. The potentials and metrics in spaces under study are written in the gnomonic coordinates. It is shown that as the curvature radius tends to infinity, the generalized gravitational and elastic potentials transform to the Kepler and Hooke forms in the Euclidean space.

A paradox of Newtonian gravitation and Laplace’s solution

Newton’s theory of gravitation solvedmany problems of celestialmechanics but led to some major problems, one of which was that it gave rise to a serious paradox discussed in this article. The paradox has a number of manifestations one of which is presented here. It should be remembered that the root cause of all these paradoxes is fundamentally the same. Laplace attempted an ad hoc solution. However, it could not be mathematically demonstrated that his proposal solves the difficulty. In this article, it is shown that his suggestion removes the problem. A physical phenomenon that can justify Laplace’s suggestion is also mentioned briefly. This article also poses an interesting mathematical problem that can be attempted by interested readers. The computational solution is presented here.

A new bi-parametric family of temporal transformations to improve the integration algorithms in the study of the orbital motion

One of the fundamental problems in celestial mechanics is the study of the orbital motion of the bodies in the solar system. This study can be performed through analytical and numerical methods. Analytical methods are based on the well-known two-body problem; it is an integrable problem and its solution can be related to six constants called orbital elements. To obtain the solution of the perturbed problem, we can replace the constants of the two-body problem with the osculating elements given by the Lagrange planetary equations. Numerical methods are based on the direct integration of the motion equations. To test these methods we use the model of the two-body problem with high eccentricity.In this paper we define a new family of anomalies depending on two parameters that includes the most common anomalies. This family allows one to obtain more compact developments to be used in analytical series and also to improve the efficiency of numerical methods because it defines a more suitable point distribution with the dynamics of the two-body problem.

The force function of two rigid celestial bodies in Delaunay–Andoyer variables

Two new expansions of the force function of two rigid celestial bodies of finite size and arbitrary shape are obtained in Delaunay–Andoyer variables with any degree of accuracy, in the form of a partial sum of an eight dimensional Fourier series. These expansions of the force function contain products of expressions for the momenta and Stokes constants in terms of sines and cosines, whose arguments are linear combinations of the Delaunay and Andoyer angular variables. These representations of the force function are compact and convenient for applications in various problems in celestial mechanics and astrodynamics.

Trigonometric splines and their applications to solve some problems of celestial mechanics

Trigonometric splines and their applications to solve some problems of celestial mechanics

Bayesian statistical approach to binary asteroid orbit determination

AbstractThe problem of binary asteroids orbit determination is of particular interest, given knowledge of the orbit is the best way to derive the mass of the system. Orbit determination from observed points is a classic problem of celestial mechanics. However, in the case of binary asteroids, particularly with a small number of observations, the solution is not evident to derive. In the case of resolved binaries the problem consists in the determination of the relative orbit from observed relative positions of a secondary asteroid with respect to the primary. In this work, the problem is investigated as a statistical inverse problem. Within this context, we propose a method based on Bayesian modelling together with a global optimisation procedure that is based on the simulated annealing algorithm.

Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system

Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist.

Topological Decoupling Near Planar Parabolic Orbits

In two different three body problems, oscillatory orbits have been shown to exist for the three-body problem in Celestial Mechanics: Sitnikov, Alekseev, Moser, and McGehee considered a spatial problem which had one degree of freedom; Easton, McGehee, and Xia considered a planar problem which had at least three degrees of freedom. Both situations involve analyzing the motion as one particle with mass \(m_3\) goes to infinity while the other two masses stay bounded in elliptic motion. Motion with \(m_3\) at infinity corresponds to a periodic orbit in the first problem and the Hopf flow on \(S^3\) in the second problem, both of which are normally degenerately hyperbolic. The proof of the existence of oscillatory orbits uses stable and unstable manifolds for these degenerate cases. In order to get the symbolic dynamics which shows the existence of oscillation, the orbits which go near infinity need to be controlled for an unbounded length of time. In this paper, we prove that the flow near infinity for the Easton–McGehee example with three degrees of freedom is topologically equivalent to a product flow, i.e., a Grobman–Hartman type theorem in the degenerate situation.

Passive magnetic stabilization of the rotational motion of the satellite in its inclined orbit

The problem of perturbed rotational motion of the satellite is one of the most interesting, important and, at the same time, mathematically complex problems of celestial mechanics and space flight dynamics. Among existing stabilization systems, a passive magnetic stabilization systems have a special place, since they have an exceptional reliability and are easy to manufacture. In this paper the problem of passive magnetic stabilization of the rotational motion of the satellite is studied. It is assumed that passive magnetic system provides its orientation along the vector of the geomagnetic field strength ~ H. The geomagnetic field is simulated by the direct dipole model, considering different orbits of inclination. In the considered model an effect of the gravitational torque is taken into account. Results of computational experiments are presented.

An improved algorithm to develop semi-analytical planetary theories using Sundman generalized variables

One of the main problems in celestial mechanics is the construction of analytical theories of planetary motion. The most common solution of this problem is arranged by means of Poisson series developments. These developments depend on the selection of the anomaly to be used as temporal variable. In this paper we develop an improved algorithm in order to use arbitrary anomalies included in the family of the generalized Sundman anomalies as temporal variables.

Avoiding collisions under topological constraints in variational problems coming from celestial mechanics

In a singular potential setting, we generalize a method which allows to show that minimizers under topological constraints of the action functional (or of the Maupertuis one) are collision-free. This methods applies to 3-dimensional problems of celestial mechanics exhibiting a particular cylindrical symmetry, as well as to planar problems of N-centre type, where it gives optimal results.

Studying the behaviour of averaged models in the third body perturbation problem

The main goal of the present paper is to perform a study of the behaviour of the single and double averaged methods used to study the third-body perturbation problem in celestial mechanics. Both models are compared with the full elliptic restricted three-body problem. The study shows the differences between the values of the eccentricity and inclination of the satellite predicted by those models, showing the differences at every instant of time, as well as the integral over the time of the differences between the models. This integral is a new way to look at those differences and can give us a more complete view of the quality of the approximations, instead of just instantaneous views.

Celestial mechanics: Past, present, future

Over all steps of its development celestial mechanics has played a key role in solar system researches and verification of the physical theories of gravitation, space and time. This is particularly characteristic for celestial mechanics of the second half of the 20th century with its various physical applications and sophisticated mathematical techniques. This paper is attempted to analyze, in a simple form (without mathematical formulas), the celestial mechanics problems already solved, the problems that can be and should be solved more completely, and the problems still waiting to be solved.