Families Of Orbits Research Articles

Periodic orbits around 216-Kleopatra asteroid modelled by a dipole-segment

Asteroid 216-Kleopatra presents a ham-bone shape, that is to say, it is quite elongated body with two protuberances on its end-points, which makes of it the perfect candidate to model its shape and mass distribution by a Dipole-Segment, consisting on a massive segment with two spherical masses at the end-points of the rod and that is rotating about its center of mass with uniform velocity. Once obtained an approximation of the actual values of the involved parameters from recent results of Burov and coworkers, we obtain the complete map of families of symmetric periodic orbits, and describe the orbits of the different families, its parametric evolution, stability and also we compute the heteroclinic orbits connecting unstable isosceles equilibria.

Solar sail orbital motion at the non-autonomous oblate earth-moon system: family of periodic orbits

Solar sail orbital motion at the non-autonomous oblate earth-moon system: family of periodic orbits

Chaos and Sub-Harmonic Bifurcations in a Simple SIR Model with Periodically and Parametrically Nonlinear Incidence Rate

Chaos and sub-harmonic bifurcations of a simple SIR model with periodically and parametrically nonlinear incidence rate are investigated analytically with the Melnikov method and sub-harmonic Melnikov method, respectively. The critical curves of chaos for the model with the periodically and parametrically nonlinear incidence rate are studied analytically for the first time. Some new interesting dynamic phenomena are presented and proved rigorously, including the nonchaotic band, controllable frequencies and even controllable frequency interval, which may exist for some system parameters in this model. The sub-harmonic periodic orbit family is derived. It is presented that sub-harmonic bifurcations with even order and integer order may exist in the system. The sub-harmonic bifurcation conditions are obtained rigorously for two different starting points. Numerical simulations including phase portraits, Poincaré sections, basins of attraction and Lyapunov exponents are given to verify the chaos threshold obtained by the analytical results. It is also shown that the system can undergo chaos through even-order sub-harmonic bifurcations.

Optimization Over Families of Quasi-Periodic Orbits

Quasi-periodic orbit families in astrodynamics are usually studied from a global standpoint without much attention to the specific orbits which are computed. Instead, we focus on the computation of particular quasi-periodic orbits and develop tools to do so. These tools leverage the parametric structure of families of quasi-periodic orbits to treat orbits only as a set of orbit frequencies instead of states in phase space. We develop a retraction on the family of quasi-periodic orbits to precisely navigate through frequency space, allowing us to compute orbits with specific frequencies. The retraction allows for movements in arbitrary directions. To combat the effects of resonances which slice through frequency space we develop resonance avoidance methods which detect resonances during continuation procedures and change the step size accordingly. We also develop an augmented Newton’s method for root-finding and an augmented gradient descent method for unconstrained optimization over a family of quasi-periodic orbits. Lastly, we implement an augmented Lagrangian method to solve constrained optimization problems. We note that many of the tools developed here are applicable to a wider range of solutions defined implicitly by a system of equations, but focus on quasi-periodic orbits.

Identification and analysis of interior and exterior resonant orbits in the Sun–Venus system

With Venus being Earth’s nearest neighbor and it being similar in size to Earth, it is the ideal candidate for certain mission types. These missions range from scientific, such as helio-physics and space weather monitoring, to defense focused, such as monitoring Earth’s orbital path for meteoroid threats. Limited research has been done on the periodic orbits in the Sun–Venus system. The first step to understanding which orbits could be useful for such a mission is to identify families of resonant periodic orbits in the Sun–Venus system and conduct a stability analysis on these orbits. This research work identifies 90 periodic, resonant orbits in the Sun–Venus system and their associated families of orbits. The orbits are found within the Circular Restricted 3 Body Problem (CR3BP) dynamical model with solar radiation pressure included as a perturbation. The periodic orbits are found using Poincaré maps, and the families are generated using a continuation method that steps through different Jacobi constants. The stability for each orbit in the family is calculated and the structure of the eigenvalues for each is assessed to determine when the family has crossed a bifurcation point. This research work seeks to generate a catalog of resonant orbits within the Sun–Venus system while providing stability and bifurcation information for each resonant orbit family.

Orbital dynamics in the GG Tau A system: Investigating its enigmatic disc

Context. GG Tau is one of the most studied young multiple stellar systems: GG Tau A is a hierarchical triple surrounded by a massive disc and its companion, GG Tau B, is also a binary. Despite numerous observational attempts, a comprehensive understanding of the geometry of the GG Tau A system is still elusive. Given the significant role of dynamical interactions in shaping the evolution of these systems, it is relevant to characterise the stellar orbits and the discs’ properties. Aims. To determine the best orbital configuration of the GG Tau A system and its circumtriple disc, we provide new astrometric measures of the system and we run a set of hydrodynamical simulations with two representative orbits to test how they impact a disc composed of dust and gas. Methods. We tested the dynamical evolution of the two scenarios on short and long timescales. We obtained synthetic flux emission from our simulations at different timescales and we compared them with multi-wavelength observations of 1300 µm ALMA dust continuum emission and 1.67 µm SPHERE dust scattering to infer the most likely orbital arrangement. Results. We extend the analysis of the binary orbital parameters using six new epochs from archival data, showing that the current measurements alone (and future observations coming in the next 5–10 yr) are not capable of fully breaking the degeneracy between families of coplanar and misaligned orbits, but finding that a modest misalignment is probable. We find that the timescale for the onset of the disc eccentricity growth, τecc, is a fundamental timescale for the morphology of the system. Results from the numerical simulations obtained using the representative coplanar and misaligned (∆θ = 30°) orbits show that the best match between the position of the stars, the cavity size, and the dust ring size of GG Tau A is obtained with the misaligned configuration on timescales shorter than τecc. The results exhibit an almost circular cavity and dust ring, favouring slightly misaligned (∆θ ~ 10–30°) low-eccentricity (e ~ 0.2–0.4) orbits. However, for both scenarios, the cavity size and its eccentricity quickly grow for timescales longer than τecc and the models do not reproduce the observed morphology anymore. This implies that either the age of the system is shorter than τecc or that the disc eccentricity growth is not triggered or dissipated in the system. This finding raises questions about the future evolution of the GG Tau A system and, more generally, the time evolution of eccentric binaries and their circumbinary discs.

The dynamics about asteroid (162173) Ryugu

The dynamical environment around the asteroid (162173) Ryugu is analyzed in detail using a constant-density polyhedron model based on the measurements from the Hayabusa2 mission. Six exterior equilibrium points (EPs) are identified along the ridge line of Ryugu, and their topological classifications fall into two distinctive categories. The initial periodic orbit (PO) families are computed and analyzed, including distant retrograde/prograde orbit (DRO/DPO) families and fifteen PO families emanating from the exterior EPs. The fifteen PO families are further divided into three categories: seven converge to an EP, seven reach Ryugu’s surface, and one exhibits cyclic behavior during its progression. The existence of initial PO families converging to an EP is analyzed using the bifurcation of a degenerate EP. Connection between these families and similar ones in the circular restricted three-body problem (CRTBP) is also examined. Bifurcated PO families are identified and computed from the initial PO families, including ten families from the DROs, fifteen from the DPOs, and twenty-five associated with the EPs. The bifurcated families are separately analyzed and categorized in terms of their corresponding initial families. Connections established by the same bifurcation points between different bifurcated families are identified. A comparison is made for the dynamical environments of Ryugu and Bennu to evaluate the similarities and differences in the evolution of EPs and the progression of PO families in top-shaped asteroids.

Orbital analysis in the gravitational potential of elongated asteroids

This work studies the motion around irregular elongated asteroids through two approaches. Firstly, it revisits the dipole-segment model, identifying families of periodic orbits for asymmetric mass distribution. Additionally, a new model incorporating variable density for elongated asteroids is introduced and compared to the dipole-segment model. Several families of periodic orbits have been found through continuation of planar orbits and out-of-plane bifurcation processes, obtaining results in agreement with previous studies about the dynamics around irregular asteroids. This highlights the relevance of simple mathematical models in studying asteroid dynamics and the importance of accounting for density and geometric properties. Although the families of periodic orbits studied in this work are not comprehensively sampled, they constitute an example of the variety of orbits that can be followed by a particle orbiting the asteroid, helping us to better understand the dynamics around these elongated bodies.

Periodic orbits in a Hamiltonian system of stellar type

We investigate the existence of periodic orbits in a perturbed Hamiltonian system of stellar type in 1:1 resonance. The perturbation consists of a potential of degree four with two real parameters. We determine six families of periodic orbits using reduction and averaging theories. Also, we characterize the stability of these orbits and their bifurcation curves in terms of the parameters. Finally, we show a complete picture of the choreographies of critical points originating the periodic orbits.

Cislunar [formula omitted] and [formula omitted] axial orbits and their applications

Cislunar L4 or L5 axial orbit families are continuous asymmetrical periodic orbits from the figure-8 orbit around L4 or L5 Lagrange point of the Earth–Moon system to the halo-family around the Moon. They provide special locations of lunar relay satellites for the lunar south pole due to their offset locations. In this paper, we investigate the orbital characters and applications of cislunar L4 and L5 axial orbits. A construction method of cislunar L4 and L5 axial orbit families is proposed according to their topological feature in the torus space. Then, lunar relay performances, including coverage and eclipse, are analyzed for different cislunar L4 and L5 axial orbits. Numerical computation indicates that 3:1 resonant L4 and L5 axial orbits are the optimal options in terms of the stability, coverage and eclipse avoidance. Lunar relay performances of a two-satellite constellation deployed in these orbits are compared with that in a near rectilinear halo orbit (NRHO). Numerical results show that the constellation of axial orbits can provide longer contact duration to the lunar equator and southern hemisphere than that of the NRHO. Finally, a station-keeping strategy and a transfer design of cislunar L4 and L5 axial orbits are developed, and their fuel costs are studied.

Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem

A novel station-keeping strategy leveraging periodic revolutions of homeomorphic orbits in the Elliptic Restricted Three-Body Problem within the pulsating frame is presented. A systemic approach founded on arc-length continuation is presented for the discovery, computation, and classification of periodic revolutions that morph from their traditional circular restricted three-body counterparts to build an a priori dataset. The dataset is comprehensive in covering all possible geometric architectures of the restricted problem. Shape similarity is quantified using Hausdorff distance and works as a filter for the station-keeping algorithm in relation to appropriate target conditions. Finally, an efficient scheme to quantify impulsive orbit maintenance maneuvers that minimize the total fuel cost is presented. The proposed approach is salient in its generic applicability across any elliptic three-body system and any periodic orbit family. Finally, average annual station-keeping costs using the described methodology are quantified for selected “orbits of interest” in the cis-lunar and the Sun–Earth systems. The robustness and efficacy of the approach instill confidence in its applicability for realistic mission design scenarios.

Orbital analysis of stars in the nuclear stellar disc of the Milky Way

Context. While orbital analysis studies were so far mainly focused on the Galactic halo, it is possible now to do these studies in the heavily obscured region close to the Galactic Centre. Aims. We aim to do a detailed orbital analysis of stars located in the nuclear stellar disc (NSD) of the Milky Way allowing us to trace the dynamical history of this structure. Methods. We integrated orbits of the observed stars in a non-axisymmetric potential. We used a Fourier transform to estimate the orbital frequencies. We compared two orbital classifications, one made by eye and the other with an algorithm, in order to identify the main orbital families. We also compared the Lyapunov and the frequency drift techniques to estimate the chaoticity of the orbits. Results. We identified several orbital families as chaotic, z-tube, x-tube, banana, fish, saucer, pretzel, 5:4, and 5:6 orbits. As expected for stars located in a NSD, the large majority of orbits are identified as z-tubes (or as a sub-family of z-tubes). Since the latter are parented by x2 orbits, this result supports the contribution of the bar (in which x2 orbits are dominant in the inner region) in the formation of the NSD. Moreover, most of the chaotic orbits are found to be contaminants from the bar or bulge which would confirm the predicted contamination from the most recent NSD models. Conclusions. Based on a detailed orbital analysis, we were able to classify orbits into various families, most of which are parented by x2-type orbits, which are dominant in the inner part of the bar.

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Two-parametric families of orbits produced by 3D potentials inside a material concentration: an application to galaxy models

Two-parametric families of orbits produced by 3D potentials inside a material concentration: an application to galaxy models

Dynamical Structures under Nonrestricted Hierarchical Planetary Systems with Different Mass Ratios

Secular dynamics have been extensively studied in both the inner and outer restricted hierarchical three-body systems. In the inner restricted problem, the quadrupole-order resonance (i.e., the well-known Kozai resonance) causes large coupled oscillations of eccentricity and inclination when the maximum inclination is higher than 39.2°, and the octupole-order resonance leads to the behavior of orbital flips. In the outer restricted problem, the behavior of orbital flips is due to the quadrupole-order resonance. Secular dynamics under the inner and outer restricted systems are distinctly different. The mass ratio of inner and outer bodies could change the ratio of circular orbital angular momenta β, which significantly influences dynamical structures of the system. But this influence is still unclear. In this paper, we focus on nonrestricted hierarchical planetary systems where β > 1 and investigate the secular dynamics by changing mass ratios. Dynamical structures are systematically explored from four aspects: periodic orbits, secular resonances, orbital flips, and chaos detection. We find that (a) it tends to lead to more bifurcations in the host family of prograde periodic orbits associated with Kozai resonance with smaller β; (b) with the decrease of β, fewer orbits inside the octupole-order resonance can realize flip; (c) for given initial conditions, the forbidden region appears in the retrograde region and becomes larger as β decreases, meaning that the mutual inclination cannot reach a very high value if β is small; and (d) chaotic orbits are distributed in the low-eccentricity, high-inclination region when β > 1.

Particle motion around traversable wormholes: Possibility of closed timelike geodesics

The present work investigates the general wormhole solution in Einstein gravity with an exponential shape function around an ultrastatic and a finite redshift geometry. The geodesic motion around the wormholes is studied in which the deflection angle of the orbiting photon sphere is found to be negative after a certain region, indicating the presence of repulsive effect of gravity in both the ultrastatic and finite redshift wormholes. Various unbounded and bounded timelike trajectories are presented on the wormhole embedding diagrams, in which some of the bound orbits involve intersection points that may lead to causality violating geodesics. Another class of closed timelike geodesics are obtained in the unstable circular trajectory that appeared at the wormhole throat. Finally, the trajectories are classified in terms of the family of CTG orbits.

Applications of knot theory to the detection of heteroclinic connections between quasi-periodic orbits

Heteroclinic connections represent unique opportunities for spacecraft to transfer between isoenergetic libration point orbits for zero deterministic ΔV expenditure. However, methods of detecting them can be limited, typically relying on human-in-the-loop or computationally intensive processes. In this paper we present a rapid and fully systematic method of detecting heteroclinic connections between quasi-periodic invariant tori by exploiting topological invariants found in knot theory. The approach is applied to the Earth–Moon, Sun–Earth, and Jupiter–Ganymede circular restricted three-body problems to demonstrate the robustness of this method in detecting heteroclinic connections between various quasi-periodic orbit families in restricted astrodynamical problems.

Investigation of near-rectilinear halo orbit search and rescue using staging [formula omitted]/[formula omitted] Lyapunov and distant retrograde orbit families

Cislunar space is a region of growing interest with nations investing resources to cultivate long presence habitations on the lunar surface. With this increased attention and expansion of missions, both crewed and uncrewed, the likelihood of a mishap or a spacecraft becoming impaired and unable to continue its mission will also increase. The present research adds to the field of cislunar mission operations and trajectory analysis by investigating search and rescue (SAR) operations via rendezvous and proximity operations (RPO) with an impaired notional spacecraft located in a Near-Rectlinear Halo Orbit (NRHO). This research compares the response times of rescuer spacecraft located in sample distant retrograde orbits (DROs) and L1/L2 Lyapunov orbits for the timely far rendezvous with the impaired spacecraft located in the NRHO. This will simulate a variety of far rendezvous whereby the impaired spacecraft’s location within the NRHO and the rescuer spacecraft in the L1/L2 Lyapunov and DRO orbit families are varied. A series of minimum time optimal control problems are posed using the circular restricted three-body problem (CR3BP) dynamics, and pseudospectral methods are used to find solutions given an example maximum ΔV constraint of 3 km/s. The results reinforces our intuition that rendezvous time of flight (TOF) between orbits within the L1, L2, and DRO families and the targeted NRHO correlate with proximity to the NRHO, with the shortest far rendezvous times in each family found to be approximately 6 hours, 4.5 hours, and 10 hours respectively. The results further show that a constellation of two rescue spacecraft could be positioned within the three orbit families to achieve far rendezvous with the chosen NRHO in under one day.

Age of Geminids derived from the statistics of meteoroid orbits

ABSTRACT Statistical analysis of samples of the orbits of celestial bodies is complicated by the fact that the Keplerian orbit is a multidimensional object, the coordinate representation of which non-linearly depends on the choice of orbital elements. In this work, using the construction of the Fréchet mean, concepts of mean orbit and dispersion of the orbit family are introduced, consistent with the distance function on the orbit set. The introduced statistical characteristics serve as analogues of sample mean and variance of a one-dimensional random variable. Exact formulas for calculating the elements of mean orbits and dispersion quantities with respect to two metrics on the orbit space are derived. For a large sample of meteoroid orbits from the Geminid stream, numerical simulations of orbit evolution over 20 000 yr in the past were conducted. By analysing the dependency of statistical characteristics on time, estimates for the age of the stream and the gas outflow velocity are obtained under the assumption of the birth of the Geminids due to the rapid destruction of the cometary nucleus. The estimate of the age of the stream lies in the interval from 1200 to 2400 yr, and the speed of gas outflow at perihelion should have been more than 1.2 km s−1.

Periodicity and lifetime of orbits around elongated asteroids

This paper presents dynamics and control of orbits around an asteroid that is modelled as a uniformly rotating homogeneous rectangular parallelepiped (HRP). Aside from the non-uniform gravitational field of the asteroid, we include the solar radiation pressure (SRP) as a perturbing force. A simple model based on the polyhedral method is employed for calculating the gravity around the HRP. First, circular periodic orbits are identified both in the presence and absence of solar radiation pressure. Then, families of general non-circular periodic orbits in the presence of SRP are determined and subsequently analysed for stability. Next, for a general orbit in the equatorial plane, its stability against a catastrophic impact of an orbiting spacecraft onto the asteroid’s surface is checked in the presence of SRP. Orbits, where the solar radiation pressure is less/more dominant over the central body’s gravity, are identified with the aim of finding the best orbits for parking a spacecraft. Then, for long-period orbits, SRP is averaged and the time averaged evolution of the orbital elements is obtained. For a special set of initial condition, a closed form solution for the evolution of the orbital eccentricity is obtained. Using this solution, a lower limit for the life-time of a spacecraft that is left uncontrolled is derived in a closed form. Finally, active control strategies are developed for orbit-keeping ofspacecraft for any choice of orbital parameters. A linear-quadratic regulator (LQR) is designed to track a given Keplerian orbit with minimum fuel expenditure. Fuel expenditure on various orbits are calculated and compared to determine the orbits of cheapest operation. The work presented is applicable to small body (asteroids and comets) exploration missions which require online cancellation of the perturbing forces with minimal knowledge of the accurate shape, size, and composition of the small body prior to the mission.