Rotation of a rigid satellite with a fluid component: a new light onto Titan’s obliquity

We revisit the rotation dynamics of a rigid satellite with either a liquid core or a global subsurface ocean. In both problems, the flow of the fluid component is assumed inviscid. The study of a hollow satellite with a liquid core is based on the Poincaré–Hough model which provides exact equations of motion. We introduce an approximation when the ellipticity of the cavity is low. This simplification allows to model both types of satellite in the same manner. The analysis of their rotation is done in a non-canonical Hamiltonian formalism closely related to Poincaré’s “forme nouvelle des équations de la mécanique”. In the case of a satellite with a global ocean, we obtain a seven-degree-of-freedom system. Six of them account for the motion of the two rigid components, and the last one is associated with the fluid layer. We apply our model to Titan for which the origin of the obliquity is still a debated question. We show that the observed value is compatible with Titan slightly departing from the hydrostatic equilibrium and being in a Cassini equilibrium state.     

Bruns' Theorem: The Proof and Some Generalizations

We give here a proof of Bruns’ Theorem which is both complete and as general as possible: Generalized Bruns’ Theorem.In the Newtonian (n+1)-body problem in ℝ ^p with n≥2 and 1≤p≤n+1, every first integral which is algebraic with respect to positions, linear momenta and time, is an algebraic function of the classical first integrals: the energy, the p(p−1)/2 components of angular momentum and the 2p integrals that come from the uniform linear motion of the center of mass. Bruns’ Theorem only dealt with the Newtonian three-body problem in ℝ³; we have generalized the proof to n+1 bodies in ℝ^p with p≤n+1. The whole proof is much more rigorous than the previous versions (Bruns, Painlevé, Forsyth, Whittaker and Hagiara). Poincaré had picked out a mistake in the proof; we have understood and developed Poincaré’s instructions in order to correct this point (see Subsection 3.1). We have added a new paragraph on time dependence which fills in an up to now unnoticed mistake (see Section 6). We also wrote a complete proof of a relation which was wrongly considered as obvious (see Section 3.3). Lastly, the generalization, obvious in some parts, sometimes needed significant modifications, especially for the case p=1 (see Section 4). This revised version was published online in July 2006 with corrections to the Cover Date.     

The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori–Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincaré system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems.     

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in Celestial Mechanics and Astrodynamics, e.g., to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic map. The HOTM is only accurate close to the expansion point and therefore the number of revolutions for which the map is accurate tends to be limited. The proper selection of coordinates is of key importance for improving the performance of the HOTM method. In this paper, we investigate the use of different element sets for expressing the high-order map in order to find the coordinates that perform best in terms of accuracy. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.     

Symmetric and Nonsymmetric Periodic Orbits in the Exterior Mean Motion Resonances with Neptune

We present families of periodic orbits and their stability for the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune in the framework of the planar circular restricted three-body problem. We found that in each resonance there exist two branches of symmetric elliptic periodic orbits with stable and unstable segments. Asymmetric periodic orbits bifurcate from the corresponding symmetric ones. Asymmetric periodic orbits are stable and the motion in their neighbourhood is a libration with respect to the resonant angle variable. In all the families of asymmetric periodic orbits the eccentricity extends to high values. Poincaré sections reveal the changes of the topology in phase space.     

Second Species Periodic Orbits of the Elliptic 3 Body Problem

We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.     

On the global solution in the resonance problem of Poincaré

Poincaré formulated a general problem of resonance in the case of a dynamical system which is reducible to one degree of freedom. He introduced the concept of a global solution; in essence, this means that the domain of the solution(s) covers the entire phase plane, comprising regions of libration and circulation. It is the author's opinion that the technique proposed by Poincaré for the construction of a global solution is impractical. Indeed, in §§201 and 211 ofLes méthodes nouvelles de la méchanique céleste, where he describes the passage from shallow resonance to deep resonance, Poincaré asserts an erroneous conclusion. An alternative procedure, which admits secular terms into the determining function and introduces a regularizing function, is outlined. The latter method has been successfully applied to the Ideal Resonance Problem, which is a special case of the more general problem considered by Poincaré, (Garfinkelet al. (1971); Garfinkel (1972).     

A High Order Perturbation Analysis of the Sitnikov Problem

The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré–Lindstedt. The enormous amount of necessary computations were performed by extensive use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering an increasing number of algebraic terms originating from high order perturbation theory.     

On the stability of periodic solutions in the problem of the motion of a star inside an elliptical galaxy

The problem of the spatial motion of a star inside an inhomogeneous rotating elliptical galaxy with a homothetic density distribution is considered. Periodic solutions are constructed by the method of a small Poincaré parameter. Linear variational equations with periodic coefficients are used to analyze the Lyapunov stability of these solutions.     

An averaging method to study the motion of lunar artificial satellites I: Disturbing Function

We describe a semi-analytical averaging method aimed at the computation of the motion of an artificial satellite of the Moon. In this paper, the first of the two part study, we expand the disturbing function with respect to the small parameters. In particular, a semi-analytic theory of the motion of the Moon around the Earth and the libration of the lunar equatorial plane using different reference frames are introduced. The second part of this article shows that the choice of the canonical Poincaré variables lead to equations in closed form without singularities in e = 0 or I = 0. We introduce new expressions that are sufficiently compact to be used for the study of any artificial satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the long-term motion of Pluto

A modified periodic orbit of the third kind is introduced that is closely related to periodic orbits of the third kind as defined by Poincaré. It is shown that Pluto librates about the periodic orbit with apparent stability. This further explains the librational motion of the resonant argument of Pluto and the avoidance of a Pluto-Neptune close approach as found by Cohen and Hubbard and the long-term motion of Pluto and the librational motion of the perihelion as found by Williams and Benson. With libration about a periodic orbit, the numerical solution of Williams and Benson can be extrapolated to longer times in the past and future.     

Equilibrium points and zero velocity surfaces in the restricted four-body problem with solar wind drag

We have analyzed the motion of an infinitesimal mass in the restricted four-body problem with solar wind drag. It is assumed that the forces which govern the motion are mutual gravitational attractions of the primaries, radiation pressure force and solar wind drag. We have derived the equations of motion and found the Jacobi integral, zero velocity surfaces, and particular solutions of the system. It is found that three collinear points are real when the radiation factor 0<β<0.1 whereas only one real point is obtained when 0.125<β<0.2. The stability property of the system is examined with the help of Poincaré surface of section (PSS) and Lyapunov characteristic exponents (LCEs). It is found that in presence of drag forces LCE is negative for a specific initial condition, hence the corresponding trajectory is regular whereas regular islands in the PSS are expanded.     

Les equations aux variations et la 11e integrale du probleme desN corps

LetH=T(p)?U(q) the Hamiltonian of theN-body problem in cartesian coordinates. In his ‘Nouvelles Méthodes de la Mécanique Céleste’ (tome 1, p. 172) Poincaré observes that the equations leave invariant the differential form $$\alpha = \Sigma (q \delta p + \tfrac{1}{2}p \delta q) - \tfrac{3}{2}t \delta H.$$ This property was rediscovered by Elie Cartan in the framework of his ‘Leçons sur les Invariants Intégraux’ (p. 89, last line). We propose to study some applications of this fact from the point of view of Poincaré, that is: the point of view of the equations of variation.     

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.     

Explicit adaptive symplectic integrators for solving Hamiltonian systems

We consider Sundman and Poincaré transformations for the long-time numerical integration of Hamiltonian systems whose evolution occurs at different time scales. The transformed systems are numerically integrated using explicit symplectic methods. The schemes we consider are explicit symplectic methods with adaptive time steps and they generalise other methods from the literature, while exhibiting a high performance. The Sundman transformation can also be used on non-Hamiltonian systems while the Poincaré transformation can be used, in some cases, with more efficient symplectic integrators. The performance of both transformations with different symplectic methods is analysed on several numerical examples.     

Escape dynamics and fractal basin boundaries in the planar Earth–Moon system

The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth–Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points \(L_1\) and \(L_2\) as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.     

Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

Behavior of nearby synchronous rotations of a Poincaré–Hough satellite at low eccentricity

This paper presents a study of the Poincaré–Hough model of rotation of the synchronous natural satellites, in which these bodies are assumed to be composed of a rigid mantle and a triaxial cavity filled with inviscid fluid of constant uniform density and vorticity. In considering an Io-like body on a low eccentricity orbit, we describe the different possible behaviors of the system, depending on the size, polar flattening and shape of the core. We use for that the numerical tool. We propagate numerically the Hamilton equations of the system, before expressing the resulting variables under a quasi-periodic representation. This expression is obtained numerically by frequency analysis. This allows us to characterise the equilibria of the system, and to distinguish the causes of their time variations. We show that, even without orbital eccentricity, the system can have complex behaviors, in particular when the core is highly flattened. In such a case, the polar motion is forced by several degrees and longitudinal librations appear. This is due to splitting of the equilibrium position of the polar motion. We also get a shift of the obliquity when the polar flattening of the core is small.