Dynamics around non-spherical symmetric bodies – I. The case of a spherical body with mass anomaly (original) (raw)
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Spin-Orbit Coupling and Chaotic Rotation for Coorbital Bodies in Quasi-Circular Orbits
The Astrophysical Journal, 2013
Coorbital bodies are observed around the Sun sharing their orbits with the planets, but also in some pairs of satellites around Saturn. The existence of coorbital planets around other stars has also been proposed. For close-in planets and satellites, the rotation slowly evolves due to dissipative tidal effects until some kind of equilibrium is reached. When the orbits are nearly circular, the rotation period is believed to always end synchronous with the orbital period. Here we demonstrate that for coorbital bodies in quasi-circular orbits, stable non-synchronous rotation is possible for a wide range of mass ratios and body shapes. We show the existence of an entirely new family of spin-orbit resonances at the frequencies n ± kν/2, where n is the orbital mean motion, ν the orbital libration frequency, and k an integer. In addition, when the natural rotational libration frequency due to the axial asymmetry, σ, has the same magnitude as ν, the rotation becomes chaotic. Saturn coorbital satellites are synchronous since ν σ, but coorbital exoplanets may present non-synchronous or chaotic rotation. Our results prove that the spin dynamics of a body cannot be dissociated from its orbital environment. We further anticipate that a similar mechanism may affect the rotation of bodies in any mean-motion resonance.
Regular and chaotic test particle motion in gravitational fields due to stellar bodies with quadrupolar and octupolar deformation are studied using Poincaré surfaces of section. In first instance, we analyze the purely Newtonian case and we find that the octupolar term induces a distortion in the KAM curves corresponding to regular trajectories as well as an increase in chaoticity, even in the case corresponding to oblate deformation. Then we examine the effect of the first general relativistic corrections, provided by the post Newtonian approach. For typical values of the post Newtonian multipoles we find that the phase-space structure practically remains the same as in the classical case, whereas that for certain larger values of these multipoles the chaoticity vanishes. This important fact provides an interesting example of a situation where a non-integrable dynamical system becomes integrable through the introduction of a large perturbation. PACS numbers: 95.10.Fh, 05.45.-a, 04.25.Nx
Birth of resonances in the spin-orbit problem of Celestial Mechanics
The behaviour of resonances in the spin-orbit coupling in Celestial Mechanics is investigated. We introduce a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. A parametric representation of periodic orbits is presented. We provide explicit formulae to compute the Taylor series expansion in the perturbing parameter of the function describing this parametrization. Then we compute approximately the radius of convergence providing an indication of the stability of the periodic orbit. This quantity is used to describe the different probabilities of capture into resonance. In particular, we notice that for low values of the orbital eccentricity the only significative resonance is the synchronous one. Higher order resonances (including 1:2, 3:2, 2:1) appear only as the orbital eccentricity is increased.
Dynamics of multiple bodies in a corotation resonance
AAS/Division of Dynamical Astronomy Meeting, 2018
The interactions among objects in a mean motion resonance are important for the orbital evolution of satellites and rings, especially Saturn's ring arcs and associated moons. In this work, we examine interactions among massive bodies in the same corotation eccentricity resonance site that affect the orbital evolution of those bodies using numerical simulations. During these simulations, the bodies exchange angular momentum and energy during close encounters, altering their orbits. This energy exchange, however, does not mean that one body necessarily moves closer to exact corotation when the other moves away from it. Indeed, if one object moves towards one of these sites, the other object is equally likely to move towards or away from it. This happens because the timescale of these close encounters is short compared to the synodic period between these particles and the secondary mass (i.e., the timescale where corotation sites can be treated as potential maxima). Because the timescale of a gravitational encounter is comparable to the timescale of a collision, we could expect energy to be exchanged in a similar way for collisional interactions. In that case, these findings could be relevant for denser systems like the arcs in Neptune's Adams ring and how they can be maintained in the face of frequent inelastic collisions.
Behavior of nearby synchronous rotations of a Poincaré–Hough satellite at low eccentricity
Celestial Mechanics and Dynamical Astronomy, 2012
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 systems, 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.
Sixth International Symposium on Classical and Celestial Mechanics (CCMECH6)
2007
In this work we consider a 1:-1 non semi-simple resonant periodic orbit of a three-degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, it is proved the branching off a two-parameter family of two-dimensional invariant tori of the normalised system, whose normal behaviour depends intrinsically on the coefficients of its low-order terms. Thus, only elliptic or elliptic together with parabolic and hyperbolic tori may detach form the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and, respectively, inverse quasiperiodic Hopf bifurcation. In this report we focus on the direct case, which has many applications in several fields of science. Thus, we present here a summary of the results, obtained in the framework of KAM theory, concerning the persistence of most of the (normally) elliptic tori of the normal form, when the whole Hamiltonian is taken into account, and to give a very precise characterisation o...
Journal of Physics A: Mathematical and Theoretical, 2019
I consider a self-gravitating, N-body system assuming that the N constituents follow regular orbits about the center of mass of the cluster, where a central massive object may be present. I calculate the average over a characteristic timescale of the full, N-body Hamiltonian including all kinetic and potential energy terms. The resulting effective system allows for the identification of the orbital planes with N rigid, disk-shaped tops, that can rotate about their fixed common centre and are subject to mutual gravitational torques. The time-averaging imposes boundaries on the canonical generalized momenta of the resulting canonical phase space. I investigate the statistical mechanics induced by the effective Hamiltonian on this bounded phase space and calculate the thermal equilibrium states. These are a result of the relaxation of spins' directions, identified with orbital planes' orientations, which is called vector resonant relaxation. I calculate the dependence of spins' angular velocity dispersion on temperature and calculate the velocity distribution functions. I argue that the range of validity of the gravitational phase transitions, identified in the special case of zero kinetic term by Roupas, Kocsis & Tremaine, is expanded to non-zero values of the ratio of masses between the cluster of N-bodies and the central massive object. The relevance with astrophysics is discussed focusing on stellar clusters. The same analysis performed on an unbounded phase space accounts for continuous rigid tops. timescales. The orbital angular momentum's vectors' directions (the orbital planes' orientations) relax in several, realistic circumstances independently from their magnitudes, in which case the process is called Vector Resonant Relaxation (VRR). The relaxation of orbital angular momentum's magnitudes is called Scalar Resonant Relaxation. Resonant Relaxation has been studied in astrophysical settings [32-36] especially with numerical simulations [37-40], but also on a kinetic theory basis [41-45]. The method of time-averaging of gravitational orbits and their approximation with rigid wires was introduced by Gauss and has been extensively used in planetary dynamics [46]. In Ref. [47], the time-averaging was applied in a VRR system without any reference to a kinetic energy term. A dynamics of non-canonical variables (the components of orbital planes' direction vector) satisfying the SO(3) algebra on a non-canonical phase space is induced by solely the effective potential energy term of VRR. For this dynamics, Roupas, Kocsis & Tremaine [48] identified gravitational phase transitions in VRR. They calculated the spacial distribution of orbital planes' orientation vectors at thermodynamic equilibrium. In this work, I will again apply the time-averaging method over the apsidal precession's timescale , but now on the full N-body Hamiltonian, with all kinetic terms consistently included. The resulting "rigid-body decomposition" of the effective energy accounts for three terms determining the evolution; namely, a rotational, normal kinetic term accounting for the orbital planes' precession and nutation, a spin kinetic term accounting for the in-plane rotation and the gravitational interaction term at quadrapole and higher order. This effective Hamiltonian describes rigid, disk-shaped, spinning tops allowed to rotate about any of their diameters crossing the common fixed centre, in direct analogy with rigid body dynamics [49] Torques on each disk develop due to mutual gravitational attraction. The general dynamical equations of motion of VRR are calculated in the rigid-body decomposition. They naturally induce new physical parameters, which connect the physical properties of the effective system (rigid annular disks) with these of the implicit system (orbiting point masses). These parameters are the moments of inertia and spin magnitudes of the effective rigid disks. They are connected with the averaging timescale and the ratio ε of the mass of the cluster to that of the central object. The gravitational couplings mediate the two views-implicit and effective-of the system and allow for such relations to emerge. The aforementioned SO(3) evolution induced by a zero kinetic term turns out to be the approximation of the special limit ε = 0 at zeroth order. More importantly, the identified relations between properties of the implicit and effective systems allow for the generalization of the dynamics and the validation and further generalization of the gravitational phase transitions in the cases that the clusters' mass is comparable to that of the central massive object. I specify the dynamical conditions for which such generalization may be valid. Last, but not least, I calculate the dependence of the dispersion of disks' precession and nutation on temperature. It depends on ε and moments of inertia in a non-trivial way. Due to the later dependence, it is possible that different families of bodies acquire different dispersions, even at orders of magnitude. Note that VRR resembles mathematically in certain aspects the Hamiltonian mean-field model (HMF) [50, 51] and the interested reader might find instructive the analogy. In the next section 2 I time-average the self-gravitating N-body Hamiltonian, demonstrate the equations of motion that emerge and calculate the boundaries of the effective, canonical phase space. In section 3 I develop in detail the statistical mechanics of the system. I formally define the microcanonical, the canonical and the Gibbs-canonical ensembles and consider a thermodynamic limit. In section 4, I discuss the inequivalence of ensmbles. In section 5 I review, validate and generalize the VRR gravitational phase transitions. In section 6 I inspect the kinetic energy term and calculate the dependence of the velocity dispersion on temperature. In section 7 I briefly modify the analysis to account for continuous rigid bodies. In the final section 8 I discuss the results.
A generalization of the Lagrangian points: Studies of resonance for highly eccentric orbits
The Astronomical Journal, 2004
We develop a framework based on energy kicks for the evolution of higheccentricity long-period orbits with Jacobi constant close to 3 in the restricted circular planar three-body problem where the secondary and primary masses have mass ratio µ ≪ 1. We use this framework to explore mean-motion resonances between the test particle and the secondary mass. This approach leads to (i) a redefinition of resonance orders to reflect the importance of interactions at periapse; (ii) a pendulum-like equation describing the librations of resonance orbits; (iii) an analogy between these new fixed points and the Lagrangian points as well as between librations around the fixed points and the well known tadpole and horseshoe orbits; (iv) a condition a ∼ µ −2/5 for the onset of chaos at large semimajor axis a; (v) the existence of a range µ < ∼ 5 × 10 −6 in secondary mass for which a test particle initially close to the secondary cannot escape from the system, at least in the planar problem; (vi) a simple explanation for the presence of asymmetric librations in exterior 1 : N resonances.
Online International Interdisciplinary Research Journa, 2016
In this paper we discuss the position of the libration points and we demonstrate the relationship between curves defined by the Jacobi constant C and the orbital path of the object. This work focuses on the dynamics of a small object (typically an asteroid, a comet, a spacecraft, or just a particle) near the libration points of the Earth-Moon system and a system with the mass ratio µ = 0.0385 (this value is critical). We calculate the position and the Jacobi constant of the libration points for this systems and the threedimensional nonlinear equations of motion were numerically integrated to find the orbital path of the object in three-dimensional Euclidian space. Same orbits are stable, same orbits are unstable and same orbits are chaotic. Chaotic orbits are unpredictable on the long run.
The non-resonant, relativistic dynamics of circumbinary planets
Monthly Notices of the Royal Astronomical Society, 2010
We investigate a non-resonant, three-dimensional (spatial) model of a hierarchical system composed of a point-mass stellar (or substellar) binary and a low-mass companion (a circumbinary planet or a brown dwarf). We take into account the leading relativistic corrections to the Newtonian gravity. The secular model of the system relies on the expansion of the perturbing Hamiltonian in terms of the ratio of semi-major axes α, averaged over the mean anomalies. We found that a low-mass object in a distant orbit may excite a large eccentricity of the inner binary when the mutual inclination of the orbits is larger than about 60 •. This is related to the strong instability caused by a phenomenon that acts similarly to the Lidov-Kozai resonance (LKR). The secular system may be strongly chaotic and its dynamics unpredictable over long-term timescales. Our study shows that in the Jupiter-or brown-dwarf-mass regime of the low-mass companion, the restricted model does not properly describe the long-term dynamics in the vicinity of the LKR. The relativistic correction is significant for the parametric structure of a few families of stationary solutions in this problem, in particular for direct orbit configurations (with mutual inclination less than 90 •). We found that the dynamics of hierarchical systems with small α ∼ 0.01 may be qualitatively different in the realms of Newtonian (classic) and relativistic models. This holds true even for relatively large masses of the secondaries.