Unstable Orbits Research Articles

Heliotropic orbits at oblate asteroids: balancing solar radiation pressure and J2 perturbations

The combined effect of significant solar radiation pressure and $$J_2$$ perturbations on spacecraft orbits is investigated using both singly and doubly-averaged disturbing potentials with the Lagrange Planetary Equations. The resulting dynamics are applied to a spacecraft around an oblate asteroid. Several Sun-frozen families of orbits are identified using the singly-averaged potential, including two new families of orbits and a previously-discovered equatorial heliotropic orbit family. Families of both stable and unstable Sun-frozen orbits are mapped and characterized in the singly-averaged case. In addition, a heliotropic constraint is implemented to locate heliotropic orbits out of the equatorial plane using a constrained, doubly-averaged potential. Dynamic bounds for these 3D heliotropic orbits are shown to have an inclination limit of approximately 46 degrees for oblate bodies, and this limit is independent of the value of $$J_2$$ and radiation parameters. The resulting heliotropic and related periodic families of orbits are good candidates to consider for low-altitude science orbits around small oblate bodies with low or near-180 degree obliquity like Bennu, the target for the OSIRIS-REx mission.

Application of three-body stability to globular clusters – II. Observed velocity dispersions

The velocity dispersion profile in globular clusters (GCs) is explained here without having to rely on dark matter or a modification of Newtonian dynamics (MOND). The flattening of the velocity dispersion at large radii in certain Milky Way GCs, or lack thereof, is explained by recourse to the stability of the three-body problem in Newtonian dynamics. The previous paper in this series determined an analytical formula for the transition radius between stable and unstable orbits for a star in a GC. This stability boundary is used here to predict where the velocity dispersion profile is expected to flatten in GCs, given known orbital parameters of the GC-galaxy orbit. Published observational data for the velocity dispersion as a function of radius of 15 Milky Way GCs with approximately known orbital parameters are used here. We find that the stability boundary predicts flattening in the majority of clusters. While observational uncertainties in the orbital parameters prevent MOND from being ruled out entirely for some clusters, it is not the preferred model in any cluster. Based on the results of this study, we recommend further velocity dispersion observations and orbital determination for NGC 6171 and NGC 6341 as these are promising candidates for distinguishing Newtonian and MOND models. In particular, NGC 6171 may already be showing evidence of the chaotic diffusion of stars leading to flattening at the predicted stability boundary.

Algebraic orbits on period-3 window for the logistic map

Algebraic stable and unstable orbits are presented for the famous period-3 window of the logistic map $$x_{n+1}=rx_n(1-x_n)$$ . It is exhibited the general polynomial that gives rise to both stable and unstable period-3 orbits. These orbits are shown for three different fixed control parameter values of $$r$$ : at tangent bifurcation (birth), at super-stability and at ending pitchfork bifurcation (death) of the period-3 window. All orbits are exposed in two different ways: a sum of complex numbers $$x_i=a+bc+\overline{bc}$$ , as proposed by Gordon (Math Mag 69:118–120, 1996), and via Euler’s formula $$x_i=a+2|b|\cos (\theta )$$ . The algebraic expressions of $$a, b, c, |b|$$ and $$\theta $$ are given for each $$r$$ value for both stable and unstable orbits, as well as their numerical values and the Lyapunov exponent. It is shown that $$a$$ and $$|b|$$ are statistical quantities of the orbits.

Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. With regard to lunar missions, exterior and interior transfers, based on the transit through the regions where the collinear libration points L1 and L2 are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is focused on the definition and use of a special isomorphic mapping for low-energy mission analysis. A convenient set of cylindrical coordinates is employed to describe the spacecraft dynamics (i.e. position and velocity), in the context of the circular restricted three-body problem, used to model the spacecraft motion in the Earth–Moon system. This isomorphic mapping of trajectories allows the identification and intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Heteroclinic connections, i.e. the trajectories that belong to both the stable and the unstable manifolds of two distinct periodic orbits, can be easily detected by means of this representation. This paper illustrates the use of isomorphic mapping for finding (a) periodic orbits, (b) heteroclinic connections between trajectories emanating from two Lyapunov orbits, the first at L1, and the second at L2, and (c) heteroclinic connections between trajectories emanating from the Lyapunov orbit at L1 and from a particular unstable lunar orbit. Heteroclinic trajectories are asymptotic trajectories that travels at zero-propellant cost. In practical situations, a modest delta-v budget is required to perform transfers along the manifolds. This circumstance implies the possibility of performing complex missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining suitable, convenient end-of-life strategies for spacecraft orbiting the Earth. Seven distinct options are identified, and lead to placing the spacecraft into the final disposal orbit, which is either (a) a lunar capture orbit, (b) a lunar impact trajectory, (c) a stable lunar periodic orbit, or (d) an outer orbit, never approaching the Earth or the Moon. Two remarkable properties that relate the velocity variations with the spacecraft energy are employed for the purpose of identifying the optimal locations, magnitudes, and directions of the velocity impulses needed to perform the seven transfer trajectories. The overall performance of each end-of-life strategy is evaluated in terms of time of flight and propellant budget.

Application of three-body stability to globular clusters – I. The stability radius

The tidal radius is commonly determined analytically by equating the tidal field of the galaxy to the gravitational potential of the cluster. Stars crossing this radius can move from orbiting the cluster centre to independently orbiting the galaxy. In this paper, the stability radius of a globular cluster is estimated using a novel approach from the theoretical standpoint of the general three-body problem. This is achieved by an analytical formula for the transition radius between stable and unstable orbits in a globular cluster. A stability analysis, outlined by Mardling, is used here to predict the occurrence of unstable stellar orbits in the outermost region of a globular cluster in a distant orbit around a galaxy. It is found that the eccentricity of the cluster-galaxy orbit has a far more significant effect on the stability radius of globular clusters than previous theoretical results of the tidal radius have found. A simple analytical formula is given for determining the transition between stable and unstable orbits, which is analogous to the tidal radius for a globular cluster. The stability radius estimate is interior to tidal radius estimates and gives the innermost region from which stars can random walk to their eventual escape from the cluster. The time-scale for this random walk process is also estimated using numerical three-body scattering experiments.

Delayed feedback control and phase reduction of unstable quasi-periodic orbits.

The delayed feedback control (DFC) is applied to stabilize unstable quasi-periodic orbits (QPOs) in discrete-time systems. The feedback input is given by the difference between the current state and a time-delayed state in the DFC. However, there is an inevitable time-delay mismatch in QPOs. To evaluate the influence of the time-delay mismatch on the DFC, we propose a phase reduction method for QPOs and construct a phase response curve (PRC) from unstable QPOs directly. Using the PRC, we estimate the rotation number of QPO stabilized by the DFC. We show that the orbit of the DFC is consistent with the unstable QPO perturbed by a small state difference resulting from the time-delay mismatch, implying that the DFC can certainly stabilize the unstable QPO.

On the stability of the three classes of Newtonian three-body planar periodic orbits

Currently, the fifteen new periodic orbits of Newtonian three-body problem with equal mass were found by Šuvakov and Dmitra šinović [Phys Rev Lett, 2013, 110: 114301] using the gradient descent method with double precision. In this paper, these reported orbits are checked stringently by means of a reliable numerical approach (namely the “Clean Numerical Simulation”, CNS), which is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision with a procedure of solution verification. It is found that seven among these fifteen orbits greatly depart from the periodic ones within a long enough interval of time, and are thus most possibly unstable at least. It is suggested to carefully check whether or not these seven unstable orbits are the so-called “computational periodicity” mentioned by Lorenz in 2006. This work also illustrates the validity and great potential of the CNS for chaotic dynamic systems.

Extreme events in chaotic lasers with modulated parameter.

We study a theoretical model describing a laser with a modulated parameter, concentrating on the appearance of extreme events, also called optical rogue pulses. It is shown that two conditions are required for the appearance of such events in this type of nonlinear system: the existence of generalized multi-stability and the collisions of chaotic attractors with unstable orbits in external crisis, expanding the attractor to visit new regions in phase space.

Cylindrical isomorphic mapping applied to invariant manifold dynamics for Earth–Moon Missions

Several families of periodic orbits exist in the context of the circular restricted three-body problem. This work studies orbital motion of a spacecraft among these periodic orbits in the Earth–Moon system, using the planar circular restricted three-body problem model. A new cylindrical representation of the spacecraft phase space (i.e., position and velocity) is described, and allows representing periodic orbits and the related invariant manifolds. In the proximity of the libration points, the manifolds form a four-fold surface, if the cylindrical coordinates are employed. Orbits departing from the Earth and transiting toward the Moon correspond to the trajectories located inside this four-fold surface. The isomorphic mapping under consideration is also useful for describing the topology of the invariant manifolds, which exhibit a complex geometrical stretch-and-folding behavior as the associated trajectories reach increasing distances from the libration orbit. Moreover, the cylindrical representation reveals extremely useful for detecting periodic orbits around the primaries and the libration points, as well as the possible existence of heteroclinic connections. These are asymptotic trajectories that are ideally traveled at zero-propellant cost. This circumstance implies the possibility of performing concretely a variety of complex Earth–Moon missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining a suitable, convenient end-of-life strategy for spacecraft placed in any of the unstable orbits. The final disposal orbit is an externally confined trajectory, never approaching the Earth or the Moon, and can be entered by means of a single velocity impulse (of modest magnitude) along the right unstable manifold that emanates from the Lyapunov orbit at \(L_2\).

Dynamic Characteristics of Periodic Motions in a Restricted N-Body Problem

AbstractWe investigate the existence and follow the evolution and dynamic stability of periodic motions in a restricted version of the N-body problem, where the positions of the mass bodies reside persistently on a planar polygon ring and its geometrical centre. The results show the existence of both (linearly) stable and unstable periodic orbits around the body placed on the ring centre (inside the ring), around all the periphery bodies (outside the ring) and around one or more of the periphery bodies. The stable periodic trajectories form regions of gravitational stability, whereas some classes of the unstable orbits exhibit bifurcations to non-symmetric periodic orbits of the same period, and less often period-doubling bifurcations.

Approaching Moons from Resonance via Invariant Manifolds

In this work, the final approach typical of a trajectory traveling from the last resonance of an endgame scenario in a tour down to a moon is examined within the context of invariant manifolds. Previous analyses have usually solved this problem either by using numerical techniques or by computing a catalog of suitable trajectories. The invariant manifolds of a selected set of libration orbits and unstable resonant orbits are computed here to serve as guides for desirable approach trajectories. The analysis focuses on designing an approach phase that may be tied into the final resonance in the endgame sequence while also targeting desired conditions at the moon.

Steady-state electron trajectories in a free electron laser with ion-channel and axial magnetic field in the presence of self-fields

A theory of self-fields in a one-dimensional helical wiggler free electron laser with ion-channel guiding and axial magnetic field is presented. The steady-state orbits under the influence of self-field are derived and discussed. The Φ function that determines the rate of change of axial velocity with energy is derived. The numerical results show the effects of self-fields and the two electron-beams guiding devices (ion-channel and axial magnetic field) on the trajectories when used separately and simultaneously. The study shows that new unstable orbits, in the first part of the group I and II orbits, are found. A detailed stability analysis of orbits is presented.

Assessing the physical nature of near-Earth asteroids through their dynamical histories

We analyze a sample of 139 near-Earth asteroids (NEAs), defined as those that reach perihelion distances q<1.3au, and that also fulfill the conditions of approaching or crossing Jupiter’s orbit (aphelion distances Q>4.8au), having Tisserand parameters 2<T<3 and orbital periods P<20yr. In order to compare the dynamics, we also analyze a sample of 42 Jupiter family comets (JFCs) in near-Earth orbits, i.e. with q<1.3au. We integrated the orbits of these two samples for 104yr in the past and in the future. We find that the great majority of the NEAs move on stable orbits during the considered period, and that a large proportion of them are in one of the main mean motion resonances with Jupiter, in particular the 2:1. We find a strong coupling between the perihelion distance and the inclination in the motion of most NEAs, due to Kozai mechanism, that generates many sungrazers. On the other hand, most JFCs are found to move on very unstable orbits, showing large variations in their perihelion distances in the last few 102–103yr, which suggests a rather recent capture in their current near-Earth orbits. Even though most NEAs of our sample move in typical ‘asteroidal’ orbits, we detect a small group of NEAs whose orbits are highly unstable, resembling those of the JFCs. These are: 1997 SE5, 2000 DN1, 2001 XQ, 2002 GJ8, 2002 RN38, 2003 CC11, 2003 WY25, 2009 CR2, and 2011 OL51. These objects might be inactive comets, and indeed 2003 WY25 has been associated with comet Blanpain, and it is now designed as Comet 289P/Blanpain. Under the assumption that these objects are inactive comets, we can set an upper limit of ∼0.17 to the fraction of NEAs with Q>4.8au of cometary origin, but it could be even lower if the NEAs in unstable orbits listed before turn out to be bona fide asteroids from the main belt. This study strengthens the idea that NEAs and comets essentially are two distinct populations, and that periods of dormancy in comets must be rare. Most likely, active comets in near-Earth orbits go through a continuous erosion process in successive perihelion passages until disintegration into meteoritic dust and fragments of different sizes. In this scenario, 289P/Blanpain might be a near-devolatized fragment from a by now disintegrated parent comet.

Active Disturbance Rejection Station-Keeping Control of Unstable Orbits around Collinear Libration Points

An active disturbance rejection station-keeping control scheme is derived and analyzed for station-keeping missions of spacecraft along a class of unstable periodic orbits near collinear libration points of the Sun-Earth system. It is an error driven, rather than model-based control law, essentially accounting for the independence of model accuracy and linearization. An extended state observer is designed to estimate the states in real time by setting an extended state, that is, the sum of unmodeled dynamic and external disturbance. This total disturbance is compensated by a nonlinear state error feedback controller based on the extended state observer. A nonlinear tracking differentiator is designed to obtain the velocity of the spacecraft since only position signals are available. In addition, the system contradiction between rapid response and overshoot can be effectively solved via arranging the transient process in tracking differentiator. Simulation results illustrate that the proposed method is adequate for station-keeping of unstable Halo orbits in the presence of system uncertainties, initial injection errors, solar radiation pressure, and perturbations of the eccentric nature of the Earth's orbit. It is also shown that the closed-loop control system performance is improved significantly using our method comparing with the general LQR method.

Output-Feedback Control of a Chaotic MEMS Resonator for Oscillation Amplitude Enhancement

The present work addresses the problem of chaos control in an electrostatic MEMS resonator by using an output-feedback control scheme. One of the unstable orbits immersed in the chaotic attractor is stabilized in order to produce a sustained oscillation of the movable plate composing the microstructure. The orbit is carefully chosen so as to produce a high amplitude oscillation. This approach allows the enhancement of oscillation amplitude of the resonator at a reduced control effort, since the unstable orbit already exists in the system and it is not necessary to spend energy to create it. Realistic operational conditions of the MEMS are considered including parametric uncertainties in the model and constraints due to the difficulty in measuring the speed of the plates of the microstructure. A control law is constructed recursively by using the technique of backstepping. Finally, numerical simulations are carried out to confirm the validity of the developed control scheme and to demonstrate the effect of controlling orbits immersed in the chaotic attractor.

Analysis of hand motion differentiates expert and novice surgeons

BackgroundThe number of operations performed by a surgeon may be an indicator of surgical skill. The hand motions made by a surgeon also reflect skill and level of expertise. We hypothesized that the hand motions of expert and novice surgeons differ significantly, regardless of whether they are familiar with specific tasks during an operation. MethodsThis study compared 11 expert surgeons, each of whom had performed >100 laparoscopic procedures, and 27 young surgeons, each of whom had performed <15 laparoscopic procedures. Each examinee performed a specific skill assessment task, in which instrument motion was monitored using magnetic tracking system. We analyzed the paths of the centers of gravity of the tips of the needle holders and the relative paths of the tips using two mathematical methods of detrended fluctuation analysis and unstable periodic orbit analysis. ResultsDetrended fluctuation analysis showed that the exponent in the function describing the initial scaling exponent (α1) differed significantly for experts and novices, being close to 1.0 and 1.5, respectively (P < 0.01). This indicated that the expert group had a greater long-range coherence with an intrinsic sequence and smooth continuity among a series of motions. Likewise, unstable periodic orbit analysis showed that the second period of unstable orbit was significantly longer for experts in comparison with novices (P < 0.01). This demonstrates mathematically that the hands of experts are more stable when performing laparoscopic procedures. ConclusionsObjective evaluation of hand motion during a simulated laparoscopic procedure showed a significant difference between experts and novices.

Particle motion in the Schwarzschild-Quintessence space-time

The influence of free static spherically symmetric quintessence on particle motion in the Schwarzschild-quintessence space-time has been studied by numerical calculation. In the Schwarzschild space-time, the particle motion can be determined by an effective potential. However, this potential is dependent on the quintessence’s state parameter w q . We find that when the quintessence’s state parameter w q is in the range of $[-\frac{1}{3},0]$ , the massive particle’s motion is just like that in the Schwarzschild space-time. And when $-1\leqslant w_{q}<-\frac{1}{3}$ , a maximum unstable circular orbit exists for every L, and no matter how small L is, the scattering state exists, which leads to the accelerating expansion of our universe. The exists of the maximum orbit can even explain why galaxies is in a ball.

Resonant periodic orbits in the exoplanetary systems

The planetary dynamics of $4/3$, $3/2$, $5/2$, $3/1$ and $4/1$ mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to $60^\circ$ of mutual planetary inclination, but in most families, the stability does not exceed $20^\circ$-$30^\circ$, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the $4/3$ and $5/2$ resonance, respectively.

Stability Analysis and Control of A 3-D Autonomous Ai-Yuan-Zhi-Hao Hyperchaotic System

The stability analysis and control of a three-dimensional autonomous chaotic system that exhibits hyperchaotic properties is presented in this paper. The system's algebraic structure consistsof six variable control parameters, three nonlinear terms and two cross-products which triggers hypersensitivity in its dynamic state evolutions. Stability analysis and control via fuzzy modelling confirms the controllability of its unstable orbits in well-tuned regions of stability using Lyapunov principles.

Viewing black holes by waves

We study scattering of waves by black holes. Solving a massless scalar field with a point source in the Schwarzschild spacetime, waves scattered by the black hole is obtained numerically. We then reconstruct images of the black hole from scattered wave data for specified scattering angles. For the forward and the backward directions, obtained wave optical images of black holes show rings that correspond to the black hole glories associated with the existence of the unstable circular photon orbit in the Schwarzschild spacetime.