First-order resonant in periodic orbits

In the frame work of Saturn–Titan system, the resonant orbits of first-order are analyzed for three different families of periodic orbits, namely, interior resonant orbits, exterior resonant orbits and [Formula: see text]-Family orbits. This analysis is developed by considering Saturn as a spherical and oblate body. The initial position, semi-major axis, eccentricity, orbital period and order of resonant orbits of these families are investigated for different values of Jacobi constant and oblateness parameter.

The effect of zonal harmonics on dynamical structures in the circular restricted three-body problem near the secondary body

The circular restricted three-body model is widely used for astrodynamical studies in systems where two major bodies are present. However, this model relies on many simplifications, such as point-mass gravity and planar, circular orbits of the bodies, and limiting its accuracy. In an effort to achieve higher-fidelity results while maintaining the autonomous simplicity of the classic model, we employ zonal harmonic perturbations since they are symmetric about the z-axis, thus bearing no time-dependent terms. In this study, we focus on how these perturbations affect the dynamic environment near the secondary body in real systems. Concise, easily implementable equations for gravitational potential, particle motion, and modified Jacobi constant in the perturbed model are presented. These perturbations cause a change in the normalized mean motion, and two different formulations are addressed for assigning this new value. The shifting of collinear equilibrium points in many real systems due to $$J_2$$ J 2 of each body is reported, and we study how families of common periodic orbits—Lyapunov, vertical, and southern halo—shift and distort when $$J_2$$ J 2 , $$J_4$$ J 4 , and $$J_6$$ J 6 of the primary and $$J_2$$ J 2 of the secondary body are accounted for in the Jupiter–Europa and Saturn–Enceladus systems. It is found that these families of periodic orbits change shape, position, and energy, which can lead to dramatically different dynamical behavior in some cases. The primary focus is on moons of the outer planets, many of which have very small odd zonal harmonic terms, or no measured value at all, so while the developed equations are meant for any and all zonal harmonic terms, only even terms are considered in the simulations. Early utilization of this refined CR3BP model in mission design will result in a more smooth transition to full ephemeris model.

The role of the mass ratio in ballistic capture

ABSTRACT A massless particle can be naturally captured by a celestial body with the aid of a third body. In this work, the influence of the mass ratio on ballistic capture is investigated in the planar circular restricted three-body problem (CR3BP) model. Four typical dynamical environments with decreasing mass ratios, that is, the Pluto–Charon, Earth–Moon, Sun–Jupiter, and Saturn–Titan systems, are considered. A generalized method is introduced to derive ballistic capture orbits by starting from a set of initial conditions and integrating backward in time. Particular attention is paid to the backward escape orbits, following which a test particle can be temporarily trapped by a three-body gravity system, although the particle will eventually deviate away from the system. This approach is applied to the four candidate systems with a series of Jacobi constant levels to survey and compare the capture probability (quantitatively) and capture capability (qualitatively) when the mass ratio varies. Capture mechanisms inducing favourable ballistic capture are discussed. Moreover, the possibility and stability of capture by secondary celestial bodies are analysed. The obtained results may be useful in explaining the capture phenomena of minor bodies or in designing mission trajectories for interplanetary probes.

Orbit Taxonomy in an Electromagnetic Binary System of Two Magnetic Dipoles

We elucidate the orbital dynamics of a binary system of two magnetic dipoles, by utilizing the grid classification method. Our target is to unveil how the total energy (expressed through the Jacobi constant), as well as the ratio of the magnetic moments affect the character of the trajectories of the test particle. By integrating numerically large data sets of starting conditions of trajectories in different types of 2D maps, we manage to reveal the basins corresponding to bounded, close encounter and escape motion, along with the respective time scales of the phenomena.

Orbital element distribution of invariant manifolds associated with Lyapunov family of periodic orbits around L1 and L2

This study explores the initial configurations that lead to an eventual approach to a given planet, particularly Jupiter, using the invariant manifold of Lyapunov orbits around Lagrangian points L1 or L2. Reachability to the vicinity of planets would provide information on developing a process for capturing irregular satellites, which are small bodies orbiting around a giant planet with a high eccentricity that are considered to have been captured by the mother planet, rather than formed in situ. A region several times the Hill radius is often used for determining reachability, combined with checking the velocity of the planetesimal with respect to the mother planet. This kind of direct computation is inapplicable when the Hill sphere is widely open and its boundary cannot be clearly recognized. Here, we thus employ Lyapunov periodic orbits (LOs) as a formal definition of the vicinity of Jupiter and numerically track the orbital distribution of the invariant manifold of an LO while varying the Jacobi constant, CJ. Numerical tracking of the manifold is carried out directly via repeated Poincaré mapping of points initially allocated densely on a fragment of the manifold near the fixed points, with the assistance of multi-precision arithmetic using the Multiple Precision Floating-Point Reliable Library. The numerical computations show that the distribution of the semi-major axis of points on the manifolds is quite heavily tailed and that its median spans roughly 1–2 times the Jovian orbital radius. The invariant manifold provides a distribution profile of a that is similar to that obtained using a direct method.

Improving the accuracy of simulated chaotic N-body orbits using smoothness

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

Building CX peanut-shaped disk galaxy profiles

Context. We present and discuss the orbital content of a rather unusual rotating barred galaxy model, in which the three-dimensional (3D) family, bifurcating from x1 at the 2:1 vertical resonance with the known “frown-smile” side-on morphology, is unstable. Aims. Our goal is to study the differences that occur in the phase space structure at the vertical 2:1 resonance region in this case, with respect to the known, well studied, standard case, in which the families with the frown-smile profiles are stable and support an X-shaped morphology. Methods. The potential used in the study originates in a frozen snapshot of an N-body simulation in which a fast bar has evolved. We follow the evolution of the vertical stability of the central family of periodic orbits as a function of the energy (Jacobi constant) and we investigate the phase space content by means of spaces of section. Results. The two bifurcating families at the vertical 2:1 resonance region of the new model change their stability with respect to that of most studied analytic potentials. The structure in the side-on view that is directly supported by the trapping of quasi-periodic orbits around 3D stable periodic orbits has now an infinity symbol (i.e. ∞-type) profile. However, the available sticky orbits can reinforce other types of side-on morphologies as well. Conclusions. In the new model, the dynamical mechanism of trapping quasi-periodic orbits around the 3D stable periodic orbits that build the peanut, supports the ∞-type profile. The same mechanism in the standard case supports the X shape with the frown-smile orbits. Nevertheless, in both cases (i.e. in the new and in the standard model) a combination of 3D quasi-periodic orbits around the stable x1 family with sticky orbits can support a profile reminiscent of the shape of the orbits of the 3D unstable family existing in each model.

Orbital dynamics in the post-Newtonian planar circular restricted Sun–Jupiter system

The theory of the post-Newtonian (PN) planar circular restricted three-body problem is used for numerically investigating the orbital dynamics of a test particle (e.g. a comet, asteroid, meteor or spacecraft) in the planar Sun–Jupiter system with a scattering region around Jupiter. For determining the orbital properties of the test particle, we classify large sets of initial conditions of orbits for several values of the Jacobi constant in all possible Hill region configurations. The initial conditions are classified into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Using the smaller alignment index (SALI) chaos indicator, we further classify bounded orbits into regular, sticky or chaotic. In order to get a spherical view of the dynamics of the system, the grids of the initial conditions of the orbits are defined on different types of two-dimensional planes. We locate the different types of basins and we also relate them with the corresponding spatial distributions of the escape and collision time. Our thorough analysis exposes the high complexity of the orbital dynamics and exhibits an appreciable difference between the final states of the orbits in the classical and PN approaches. Furthermore, our numerical results reveal a strong dependence of the properties of the considered basins with the Jacobi constant, along with a remarkable presence of fractal basin boundaries. Our outcomes are compared with the earlier ones regarding other planetary systems.

Analysis of Periodic Orbits with Smaller Primary As Oblate Spheroid

We have studied closed periodic orbits with loops for two systems – Sun – Mars and Sun – Earth systems – using Poincare surface section (PSS) technique. Perturbation due to oblateness for the second primary (Mars or Earth) is taken in to consideration and obtained orbits with loops varying from one to five around both primaries. It is found that the oblateness coefficient A2 and Jacobi constant C has non- negligible effect on the position of the orbits. The model may be useful for designing space mission for low – energy trajectories.