On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits (original) (raw)
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On the stability of the three classes of Newtonian three-body planar periodic orbits
Science China Physics, Mechanics & Astronomy, 2014
Currently, the fifteen new periodic solutions of Newtonian three-body problem with equal mass were reported byŠuvakov and Dmitrašinović (PRL, 2013) [1]. However, using a reliable numerical approach (namely the Clean Numerical Simulation, CNS) that is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision, it is found that at least seven of them greatly depart from the periodic orbits after a long enough interval of time. Therefore, the reported initial conditions of at least seven of the fifteen orbits reported byŠuvakov and Dmitrašinović [1] are not accurate enough to predict periodic orbits. Besides, it is found that these seven orbits are unstable. According to H. Poincaré, orbits of the famous threebody problem [2] are not integrable in general cases. Although chaotic orbits of three-body problems widely exist, three families of periodic orbits were found:
On the Bifurcation and Continuation of Periodic Orbits in the Three Body Problem
International Journal of Bifurcation and Chaos, 2011
We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem asymmetric orbits may exist not only
The dynamics of the elliptic Hill problem: periodic orbits and stability regions
Celestial Mechanics and Dynamical Astronomy, 2012
The motion of a satellite around a planet can be studied by the Hill model, which is a modification of the restricted three body problem pertaining to motion of a satellite around a planet. Although the dynamics of the circular Hill model have been extensively studied in the literature, only few results about the dynamics of the elliptic model were known up to now, namely the equations of motion and few unstable families of periodic orbits. In the present study we extend these results by computing a large set of families of periodic orbits and their linear stability and classify them according to their resonance condition. Although most of them are unstable, we were able to find a considerable number of stable ones. By computing appropriate maps of dynamical stability, we study the effect of the planetary eccentricity on the stability of satellite orbits. We see that, even for large values of the planetary eccentricity, regular orbits can be found in the vicinity of stable periodic orbits. The majority of irregular orbits are escape orbits.
The rectilinear three-body problem as a basis for studying highly eccentric systems
Celestial Mechanics and Dynamical Astronomy
The rectilinear elliptic restricted Three Body Problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity e ′ = 1, but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter µ = 0.5 (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke's computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to µ and e ′ < 1. Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.
Instability of the periodic hip-hop orbit in the 2N-body problem with equal masses
Discrete and Continuous Dynamical Systems, 2013
The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with equal masses, or a uniformly rotating planar 2N+1 body relative equilibrium with a large central mass [18]. We study the question of stability or instability for symmetric periodic solutions of the equal mass 2N-body problem without perturbation methods. The hip-hop family is a family of Z2-symmetric action minimizing solutions, investigated by [7, 23], and is shown to be generically hyperbolic on its reduced energy-momentum surface. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing hip-hop orbit to develop conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.
Local Motion, Stability, and Evolution of Three Planet Periodic Orbits
Dreft
Using periodic orbit and Floquet theory, we examine the stability and local motion about a three planet periodic orbit. The N planet problem is derived, and its integrals of the motion enumerated. Periodic orbits are constructed for the inner three Jovian moons, and the similar system Gliese 876, although in the latter the eccentricity of the smallest planet is found to be part of the periodic orbit. Then the Floquet solution for motion near the periodic orbit is carefully constructed in canonical variables. Extreme care is detailed for the numerical calculations. Local motion consists of oscillatory modes describing free but coupled eccentricities and inclinations, a global 'tilt' mode of the periodic orbit plane, as well as two degenerate modes. These latter modes' presence is due to the energy and angular momentum integrals, and their construction is detailed. It is argued that external perturbations (e.g. the tidal interaction observed in the Jovian system) will drive the local motion directly to the periodic orbit itself, eliminating drift in the so called "critical argument". Longer term evolutin of local motion is studied by the method of adiabatic invariants. This shows the system will remained 'pinned' on the periodic orbit, until the latter should perhaps change its stability.
Astronomy and Astrophysics, 2002
The possibility of dynamic chaos in the spin motion of minor natural planetary satellites is studied numerically and analytically. A satellite is modelled as a tri-axial rigid body in a fixed elliptic orbit. The Lyapunov characteristic exponents (LCEs) are used as indicators of the degree of chaos of the motion. For a set of real satellites (i.e. satellites with actual values of inertial and orbital parameters), the full Lyapunov spectra of the chaotic rotation are computed by the HQR-method of von Bremen et al. (1997). A more traditional "shadow trajectory" method for the computation of maximum LCEs is also used. Numerical LCEs obtained in the spatial and planar cases of chaotic rotation are compared to analytical estimates obtained by the separatrix map theory in the model of nonlinear resonance (here: synchronous spin-orbit resonance) as a perturbed nonlinear pendulum (Shevchenko 2000a, 2002). Further evidence is given that the agreement of the numerical data with the separatrix map theory in the planar case is very good. It is shown that the theory developed for the planar case is most probably still applicable in the case of spatial rotation, if the dynamical asymmetry of the satellite is sufficiently small or/and the orbital eccentricity is relatively large (but, for the dynamical model to be valid, not too large). The theoretical implications are discussed, and simple statistical dependences of the components of the LCE spectrum on the parameters of the problem are derived.
Stability of S-type Orbits in Binaries
2002
This numerical investigation is concerned with the stability of planets moving around one component of a double star system. Since the discovery of four extra solar planets moving in such orbits, there is a growing interest of stability studies thereto. We determined the stable regions in the elliptic restricted three body problem, for the whole range of mass-ratios from 0.1 to 0.9, by means of the Fast Lyapunov Indicators. The computations have been carried out for eccentricities of the binary and of the planet in the range 0-0.5. Therefore, we present for the first time the variation of the stable regions when the initial eccentricity of the planet is increased. We have found a correlation between the reduction of the stable zones if the eccentricity of the planet or of the binary is increased -of course the latter one is more effective.
In this note we present s o m e n umerical results about a family of periodic orbits of the Restricted Three Body Problem (RTBP). The family considered is one of the Lyapunov families related to the equlibrium point L 5. More concretely, w e deal with the family related to the vertical oscillations around this point. Here we present a study of the normal behaviour of this family for several values of the mass parameter. We focus on the case in which tends to zero (note that = 0 is a degenerate case), and we identify the orbits for = 0 (they are Keplerian orbits around the primary) that give rise to the vertical family when 6 = 0 .