Problems Of Celestial Mechanics Research Articles

Symmetry, reduction and relative equilibria of a rigid body in the J2 problem

The J2 problem is an important problem in celestial mechanics, orbital dynamics and orbital design of spacecraft, as non-spherical mass distribution of the celestial body is taken into account. In this paper, the J2 problem is generalized to the motion of a rigid body in a J2 gravitational field. The relative equilibria are studied by using geometric mechanics. A Poisson reduction process is carried out by means of the symmetry. Non-canonical Hamiltonian structure and equations of motion of the reduced system are obtained. The basic geometrical properties of the relative equilibria are given through some analyses on the equilibrium conditions. Then we restrict to the zeroth and second-order approximations of the gravitational potential. Under these approximations, the existence and detailed properties of the relative equilibria are investigated. The orbit–rotation coupling of the rigid body is discussed. It is found that under the second-order approximation, there exists a classical type of relative equilibria except when the rigid body is near the surface of the central body and the central body is very elongated. Another non-classical type of relative equilibria can exist when the central body is elongated enough and has a low average density. The non-classical type of relative equilibria in our paper is distinct from the non-Lagrangian relative equilibria in the spherically-simplified Full Two Body Problem, which cannot exist under the second-order approximation. Our results also extend the previous results on the classical type of relative equilibria in the spherically-simplified Full Two Body Problem by taking into account the oblateness of the primary body. The results on relative equilibria are useful for studies on the motion of many natural satellites, whose motion are close to the relative equilibria.

The dynamics of Mutual Motion Camouflage

In this paper we show that a biologically-plausible pursuit law, designed to implement the motion camouflage strategy, can be used as a building-block for generating collective motion. We introduce the fundamental case of two individuals in mutual pursuit, which we refer to as Mutual Motion Camouflage, and completely characterize its dynamics. The resulting system is of theoretical interest, because of its rich symmetry and a certain similarity with the Kepler problem of celestial mechanics, concerning the dynamics of two point particles under mutual gravitational attraction. It is also of practical interest, because the individual trajectories have useful spatial coverage properties.

Supergeneralized Runge-Lenz vector in the problem of two Coulomb or Newton centers

A so-called two centers problem (TCP) has the following two mathematically equivalent, but physically different embodiments. The first one is the motion of an electron in the field of two stationary Coulomb centers of charges $Z$ and $Z$\ensuremath{'} separated by a distance $R$, which is one of the most fundamental problems in quantum mechanics. The second one is the motion of a planet in the gravitational field of two stationary stars of generally different masses, which is one of the most fundamental problems in celestial mechanics. At least, two groups of authors claimed that for the TCP they derived a supergeneralized Runge-Lenz vector, whose projection on the internuclear (or interstellar) axis is conserved. In the present paper, first, we show that their claims are incorrect: the projection of their supergeneralized Runge-Lenz vector is not conserved. Second, we derive a correct supergeneralized Runge-Lenz vector whose projection on the internuclear or interstellar axis does conserve. Third, since in the literature there are several expressions for the separation constant for the TCP---the expressions not having the form of a projection of any vector on the internuclear or interstellar axis---we provide relations between those expressions and our result.

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

Spherical harmonic series for derivatives of all orders of the gravitational potential of a planet and their application in satellite geodesy and space navigation

Series of spherical harmonics are constructed for derivatives of all orders of the gravitational potential of an arbitrary three-dimensional body, including the Earth, Moon and other planets. These series have a common structure, as simple as the potential itself. They differ from each other and from the series for the potential only by numerical coefficients of the spherical functions, by the degree of a numerical multiplier of the sum of double series, and by the limits of summation. The constructed series can be applied in solving many problems of celestial mechanics, satellite geodesy, and space navigation.

A ratchet trap for Leidenfrost drops

AbstractThe Leidenfrost effect occurs when a drop of liquid (or a sublimating solid) is levitated above a sufficiently hot surface through the action of an insulating vapour layer flowing from its bottom surface. When such a drop is levitated above a surface with parallel, asymmetric sawtooth-shaped ridges it is known to be propelled in a unique direction, or ratcheted, by the interaction of the vapour layer with the surface. Here we exploit this effect to construct a ‘ratchet trap’ for Leidenfrost drops: a surface with concentric circular ridges, each asymmetric in cross-section. A combination of experiment and theory is used to study the dynamics of drops in these traps, whose centre is a stable fixed point. Numerical analysis of the evaporating flows over a ratchet surface suggests new insights into the mechanism of motion rectification that are incorporated into the simplest equations of motion for ratchet-driven motion of a Leidenfrost body; these resemble a central force problem in celestial mechanics with mass loss and drag. A phase-plane analysis of experimental trajectories is used to extract more detailed information about the ratcheting phenomenon. Orbiting drops are found to exhibit substantial deformations; those with large internal angular momentum can even undergo binary fission. Such ratchet traps may thus prove useful in the controlled study of many properties of Leidenfrost drops.

The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.

Spinors and Their Applications to Track near Earth Objects

In this paper Spinors of Euclidean spaces of Geometric Algebras over a real field are defined and their algebraic and rotational properties are discussed. The advantage of using Spinors in solving problems of Celestial Mechanics is illustrated by studying the tracking problem of near Earth objects. It is shown that this technique of using spinors can replace the conventional methods and also provide a richer formalism. {   k s s s se a

Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the $n$-vortex problem specialized to the case where $N$ vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued, and its linear stability is determined by properties of the potential. For $N\geq 3$ there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an $N$-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when $N\geq 4$. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the $N$-gon is the only relative equilibrium for $N$ sufficiently large, and is linearly stable if and only if $N\geq 7$.

A note on the first-order theories of equilibrium figures of celestial bodies

One of the main problems in celestial mechanics is the determination of the shape of the equilibrium configuration of celestial bodies. In this paper, a model of a fluid mass rotating in space like a rigid body will be developed. To this aim, the equipotential surfaces are developed by using the Neumann series with respect to the Clairaut coordinates, and from these developments, the equilibrium equations and the boundary conditions can be obtained. Classical methods involve convergence problems, and in this paper two methods are developed to solve this problem, one based on numerical quadrature methods and the other one based on an analytical development.

Transition state geometry near higher-rank saddles in phase space

We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Russmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.

The Escape Rate of a Molecule

We show existence and give an implicit formula for the escape rate of the n-centre problem of celestial mechanics for high energies. Furthermore we give precise computable estimates of this rate. This exponential decay rate plays an important role especially in semiclassical scattering theory of n-atomic molecules. Our result shows that the diameter of a molecule is measurable in a (classical) high-energy scattering experiment.

Simulation of subdiurnal variation in the earth’s rotation

On the basis of the celestialmechanical approach, the mathematical model of subdiurnal tidal variation in the rotation of the deformable Earth is developed taking into account the gravity-tidal lunar and solar disturbances, which is adequate for astrometric mea� surements of the International Earth Rotation Service (IERS) (1). The subdiurnal and nearly diurnal varia� tions of the Earth's axial rotation are investigated using the third Euler-Liouville equation and the results of these measurements within a day (IGS). The model parameters were identified by the least squares method, and the features of the Earth's axial rotation within a day were revealed. The combinational fre� quencies confirmed by the IERS observations were found. For solving very actual problems of celestial mechanics and astrometry, it is necessary to create a precision model of the motion of Earth'srotation with respect to the center of mass. The analysis of equations of motion (2, 3) and observation data (1, 4) indicates the necessity of taking into account the dis� turbing moments of forces of various physical natures and the considerable deformability of the Earth's fig� ure. The gravity-tidal moments of forces from the Sun and the Moon exert the major effect. However, a sig� nificant correction of the model results in the compli� cation of processes of the filtration of unknown parameters including the estimation of the initial data. In a number of important problems of astrometry, navigation, and geophysics, the precision forecast of the Earth's rotation is of substantial significance for relatively short time intervals. For applications, it is the extreme precision forecast for the intervals from 1-2 to 10-30 days of length that can be of interest. Analysis of the theoretical model indicates the fact that an accuracy of about 10 -4 -10 -5 s can be achieved. It is necessary to accomplish the construction of an ade� quate set of basic functions, the choice of optimum length of the interpolation interval, and the adjust� ment of the filtration algorithm by the method of "weighed" least squares. The error in the IERS data (1) amounts to several μs. The achieved very high accuracy of measurements is sufficient for construct� ing an adequate model of the Earth's rotation (1-3). 1. The universal time (UT1) related to the Earth's rotation is a reasonably important value, which requires continuous measurements. Because the aver� age solar time, and consequently, the UT1 is not suffi� ciently accurate time scale, the atomic time scale (TAI) having a relative stability of 10 -14 can be used as such a scale on reasonably short intervals (several years). The creation of the unified atomic time scale accepted as the international standard, precisely, the international atomic time (TAI), made it possible to accept it as the practical timescale standard. It is adapted for supporting the relation to the scale UT1 determined by the Earth's rotation and is known as the coordinated universal time (UTC).

Non-Integrability of Some Few Body Problems in Two Degrees of Freedom

The basic theory of Differential Galois and in particular Morales–Ramis theory is reviewed with focus in analyzing the non-integrability of various problems of few bodies in Celestial Mechanics. The main theoretical tools are: Morales–Ramis theorem, the algebrization method and Kovacic’s algorithm. Morales–Ramis states that if a Hamiltonian system is completely integrable with meromorphic first integrals in involution in a neighborhood of a specific solution, then the differential Galois group of the normal variational equations is abelian. In case of two degrees of freedom, completely integrable means to have an additional first integral in involution with the Hamiltonian. The algebrization method permits under general conditions to recast the variational equation in a form suitable for its analysis by means of Kovacic’s algorithm. We apply these tools to various examples of few body problems in Celestial Mechanics: (a) the elliptic restricted three body in the plane with collision of the primaries; (b) a general Hamiltonian system of two degrees of freedom with homogeneous potential of degree −1; here we perform McGehee’s blow up and obtain the normal variational equation in the form of an hypergeometric equation. We recover Yoshida’s criterion for non-integrability. Then we contrast two methods to compute the Galois group: the well known, based in the Schwartz–Kimura table, and the lesser based in Kovacic’s algorithm. We apply these methodology to three problems: the rectangular four body problem, the anisotropic Kepler problem and two uncoupled Kepler problems in the line; the last two depend on a mass parameter, but while in the anisotropic problem it is integrable for only two values of the parameter, the two uncoupled Kepler problems is completely integrable for all values of the masses.

A method to improve the computation of tidal potential in the equilibrium configuration of a close binary system

A classical problem in celestial mechanics is the study of equilibrium configuration figures of extended deformable bodies. A particularly interesting case of this problem is the study of the figures of equilibrium of close binary systems. In this paper an improved theory of tidal potential using Clairaut coordinates has been developed by two methods, one based in analytical developments and one iterative.

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.

Random walk in celestial mechanics

The process of random walk can be demonstrated to occur in the three-body problem. This is shown to be the case for a family of horseshoe orbits. This offers a possible alternate approach to studying special motions of this problem and other problems in celestial mechanics.

Optimal capture trajectories using multiple gravity assists

Graph theoretic methods of optimal control in the presence of uncertainty are applied to a celestial mechanics problem. We find a fuel-efficient spacecraft trajectory which starts at infinity and is captured by the smaller member of a binary system, e.g., a moon of Jupiter, using multiple gravity assists.

Improved grid search method: an efficient tool for global computation of periodic orbits

We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases; a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification (clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a perturbed Hill problem, a species of families which are difficult to find by continuation methods.