Arbitrary Orbit Research Articles

Quantum instability and Ehrenfest time for an inverted harmonic oscillator

We use out-of-time order correlators (OTOCs) to investigate the quantum instability and Ehrenfest time for an inverted harmonic oscillator (IHO). For initial states located in the stable manifolds of the IHO we find that the corresponding OTOC exhibits identical evolutionary characteristics to the saddle point before the Ehrenfest time. For initial states located in the unstable manifolds, the OTOCs still grow exponentially but the time to maintain exponential growth is related to the center position of its wave packet in phase space. Moreover, we use the Husimi Q function to visualize the quantum wave packets during exponential growth of the OTOCs. Our results show that quantum instability exists at arbitrary orbits in the IHO system, and the Ehrenfest time in the IHO system depends not only on the photon number of the initial system but also on the central positions of the initial states in phase space.

Fully automatic online geometric calibration for non-circular cone-beam CT orbits using fiducials with unknown placement.

Cone-beam CT (CBCT) with non-circular scanning orbits can improve image quality for 3D intraoperative image guidance. However, geometric calibration of such scans can be challenging. Existing methods typically require a prior image, specialized phantoms, presumed repeatable orbits, or long computation time. We propose a novel fully automatic online geometric calibration algorithm that does not require prior knowledge of fiducial configuration. The algorithm is fast, accurate, and can accommodate arbitrary scanning orbits and fiducial configurations. The algorithm uses an automatic initialization process to eliminate human intervention in fiducial localization and an iterative refinement process to ensure robustness and accuracy. We provide a detailed explanation and implementation of the proposed algorithm. Physical experiments on a lab test bench and a clinical robotic C-arm scanner were conducted to evaluate spatial resolution performance and robustness under realisticconstraints. Qualitative and quantitative results from the physical experiments demonstrate high accuracy, efficiency, and robustness of the proposed method. The spatial resolution performance matched that of our existing benchmark method, which used a 3D-2D registration-based geometric calibrationalgorithm. We have demonstrated an automatic online geometric calibration method that delivers high spatial resolution and robustness performance. This methodology enables arbitrary scan trajectories and should facilitate translation of such acquisition methods in a clinical setting.

Unveiling the Merger Structure of Black Hole Binaries in Generic Planar Orbits.

The precise modeling of binary black hole coalescences in generic planar orbits is a crucial step to disentangle dynamical and isolated binary formation channels through gravitational-wave observations. The merger regime of such coalescences exhibits a significantly higher complexity compared to the quasicircular case, and cannot be readily described through standard parametrizations in terms of eccentricity and anomaly. In the spirit of the effective one body formalism, we build on the study of the test-mass limit, and introduce a new modeling strategy to describe the general-relativistic dynamics of two-body systems in generic orbits. This is achieved through gauge-invariant combinations of the binary energy and angular momentum, such as a dynamical "impact parameter" at merger. These variables reveal simple "quasi-universal" structures of the pivotal merger parameters, allowing us to build an accurate analytical representation of generic (bounded and dynamically bounded) orbital configurations. We demonstrate the validity of these analytical relations using 311 numerical simulations of bounded noncircular binaries with progenitors from the RIT and SXS catalogs, together with a custom dataset of dynamical captures generated using the Einstein Toolkit, and test-mass data in bound orbits. Our modeling strategy lays the foundations of accurate and complete waveform models for systems in arbitrary orbits, bolstering observational explorations of dynamical formation scenarios and the discovery of new classes of gravitational wave sources.

Performance-Designated Optimal Circumnavigation Control for UAVs along Arbitrary Smooth Orbits

Performance-Designated Optimal Circumnavigation Control for UAVs along Arbitrary Smooth Orbits

Optimal Encirclement Control Along Arbitrary Orbits for Vehicles Considering Safety Constraints: A Safe Learning Approach

Optimal Encirclement Control Along Arbitrary Orbits for Vehicles Considering Safety Constraints: A Safe Learning Approach

Bringing Order to Disorder: A method for stabilising a chaotic system around an arbitrary unstable periodic orbit

This paper introduces a two-step procedure for stabilising a chaotic system around an arbitrary periodic orbit in a Poincaré map. The periodic orbit may be a period-n orbit traversing the Poincaré plane an integer n times before completing the periodic cycle. The order of traversing can also be chosen arbitrarily. Chaotic systems lack predictability due to positive Lyapunov exponents. For this reason, predictive control methods tend to perform poorly because the predictable horizon may be short. Current methods for stabilising chaotic systems work by controlling existing unstable periodic orbits. This paper proposes a method for stabilising the system on an arbitrary point in the Poincaré section. The method in this paper consists of two steps. In the first step, we pose an optimisation problem that computes an input signal that introduces a desired unstable or stable periodic orbit. In the second step, we use existing methods, such as the method using delay coordinate embedding developed by Ott, Grebogi and Yorke in 1990, to stabilise the system around the newly introduced periodic orbit. As an example, we demonstrate the method on the resistively and capacitively shunted driven Josephson junction.

Research for Optimal Programs for Controlling the Relative Motion of a Spacecraft with Limited Thrust

The problem of optimal control of the relative motion of a spacecraft with a finite thrust engine in arbitrary circumcircular orbits using the Pontryagin’s maximum principle is considered. Motion is studied in an orbital cylindrical coordinate system, using variables written in the form of secular and periodic components of relative motion in the orbital plane. The main attention is paid to the analysis of the optimal control structure with a free and transversal orientation of the thrust vector in the presence of passive areas on the trajectory. As a criterion for choosing the optimal control, the motor operating time of the corrective motors was considered. Characteristic control structures for various areas of initial driving conditions have been determined, and estimates of the marginal costs of motor time have been obtained.

Uniform bifurcation of comet-type periodic orbits in the restricted (n + 1)-body problem with non-Newtonian homogeneous potential

We treat the restricted (n+1)-body problem with a non-Newtonian homogeneous potential where the n primaries move on an arbitrary 2π-periodic orbit. We prove that the satellite equation has infinitely many periodic solutions that emerge from the infinity, asymptotically homothetic to the circular solutions of a central force problem. These solutions are obtained as critical solutions of a family of time-dependent perturbed Lagrangian systems, bifurcating uniformly from a compact set of periodic solutions of the unperturbed Lagrangian system.

Practical workflow for arbitrary non-circular orbits for CT with clinical robotic C-arms.

Non-circular orbits in cone-beam CT (CBCT) imaging are increasingly being studied for potential benefits in field-of-view, dose reduction, improved image quality, minimal interference in guided procedures, metal artifact reduction, and more. While modern imaging systems such as robotic C-arms are enabling more freedom in potential orbit designs, practical implementation on such clinical systems remains challenging due to obstacles in critical stages of the workflow, including orbit realization, geometric calibration, and reconstruction. In this work, we build upon previous successes in clinical implementation and address key challenges in the geometric calibration stage with a novel calibration method. The resulting workflow eliminates the need for prior patient scans or dedicated calibration phantoms, and can be conducted in clinically relevant processing times.

Fast CBCT Reconstruction using Convolutional Neural Networks for Arbitrary Robotic C-arm Orbits.

Cone-beam CT (CBCT) with non-circular acquisition orbits has the potential to improve image quality, increase the field-of view, and facilitate minimal interference within an interventional imaging setting. Because time is of the essence in interventional imaging scenarios, rapid reconstruction methods are advantageous. Model-Based Iterative Reconstruction (MBIR) techniques implicitly handle arbitrary geometries; however, the computational burden for these approaches is particularly high. The aim of this work is to extend a previously proposed framework for fast reconstruction of non-circular CBCT trajectories. The pipeline combines a deconvolution operation on the backprojected measurements using an approximate, shift-invariant system response prior to processing with a Convolutional Neural Network (CNN). We trained and evaluated the CNN for this approach using 1800 randomized arbitrary orbits. Noisy projection data were formed from 1000 procedurally generated tetrahedral phantoms as well as anthropomorphic data in the form of 800 CT and CBCT images from the Lung Image Database Consortium Image Collection (LIDC). Using this proposed reconstruction pipeline, computation time was reduced by 90% as compared to MBIR with only minor differences in performance. Quantitative comparisons of nRMSE, FSIM and SSIM are reported. Performance was consistent for projection data simulated with acquisition orbits the network has not previously been trained on. These results suggest the potential for fast processing of arbitrary CBCT trajectory data with reconstruction times that are clinically relevant and applicable - facilitating the application of non-circular orbits in CT image-guided interventions and intraoperative imaging.

An Extension of the Three Gap Theorem to Interval Exchange Transformations

Abstract The three gap theorem asserts that for any real $\alpha $ and any positive integer $N$, the fractional parts of the sequence $0, \alpha , 2\alpha , \cdots , (N-1)\alpha $ have at most three distinct gap lengths. In this note, we extend the three gap theorem to arbitrary orbits of interval exchange transformations, and in the process, we generalize the three gap theorem and a result by Boshernitzan.

Orbital Rendezvous via a General Nonsingular Method

This paper develops a general method to execute orbital rendezvous in continuation of previous work, where a spacecraft’s orbit was represented using the angular momentum and eccentricity vectors of the orbit, along with the orbital energy. A discrete feedback control method using this orbit representation that can be used to autonomously maneuver to arbitrary target orbits is designed. Full rendezvous is accomplished in two distinct phases: orbit matching and a phasing maneuver, which match the five-degree-of-freedom orbit and along-track position, respectively. Once within a short distance of the target position, the spacecraft could enter a close-proximity mode if higher precision is required, such as for docking. The method is verified in simulations, and the usage is compared to two-impulse transfer and rendezvous and continuous low-thrust trajectories. Effects of perturbations are also examined quantitatively. The developed control method is shown to give similar expenditure to unbounded impulsive maneuvers while being applicable to arbitrary orbits, as it avoids singularities or linearizing the dynamics. It is usable by small spacecraft using infrequent, impulsive actuation, which may improve the feasibility and robustness of small spacecraft missions.

Time-domain metric reconstruction for hyperbolic scattering

Self-force methods can be applied in calculations of the scatter angle in two-body hyperbolic encounters, working order by order in the mass ratio (assumed small) but with no recourse to a weak-field approximation. This, in turn, can inform ongoing efforts to construct an accurate model of the general-relativistic binary dynamics via an effective-one-body description and other semi-analytical approaches. Existing self-force methods are to a large extent specialised to bound, inspiral orbits. Here we develop a technique for (numerical) self-force calculations that can efficiently tackle scatter orbits. The method is based on a time-domain reconstruction of the metric perturbation from a scalar-like Hertz potential that satisfies the Teukolsky equation, an idea pursued so far only for bound orbits. The crucial ingredient in this formulation are certain jump conditions that (each multipole mode of) the Hertz potential must satisfy along the orbit, in a 1+1-dimensional multipole reduction of the problem. We obtain a closed-form expression for these jumps, for an arbitrary geodesic orbit in Schwarzschild spacetime, and present a full numerical implementation for a scatter orbit. In this paper we focus on method development, and go only as far as calculating the Hertz potential; a calculation of the self-force and its physical effects on the scatter orbit will be the subject of forthcoming work.

Memories of past close encounters in extreme trans-Neptunian space: Finding unseen planets using pure random searches

Context. The paths followed by the known extreme trans-Neptunian objects (ETNOs) effectively avoid direct gravitational perturbations from the four giant planets, yet their orbital eccentricities are in the range between 0.69−0.97. Solar system dynamics studies show that such high values of the eccentricity can be produced via close encounters or secular perturbations. In both cases, the presence of yet-to-be-discovered trans-Plutonian planets is required. Recent observational evidence cannot exclude the existence, at 600 AU from the Sun, of a planet of five Earth masses. Aims. If the high eccentricities of the known ETNOs are the result of relatively recent close encounters with putative planets, the mutual nodal distances of sizeable groups of ETNOs with their assumed perturber may still be small enough to be identifiable geometrically. In order to confirm or reject this possibility, we used Monte Carlo random search techniques. Methods. Two arbitrary orbits may lead to close encounters when their mutual nodal distance is sufficiently small. We generated billions of random planetary orbits with parameters within the relevant ranges and computed the mutual nodal distances with a set of randomly generated orbits with parameters consistent with those of the known ETNOs and their uncertainties. We monitored which planetary orbits had the maximum number of potential close encounters with synthetic ETNOs and we studied the resulting distributions. Results. We provide narrow ranges for the orbital parameters of putative planets that may have experienced orbit-changing encounters with known ETNOs. Some sections of the available orbital parameter space are strongly disfavored by our analysis. Conclusions. Our calculations suggest that more than one perturber is required if scattering is the main source of orbital modification for the known ETNOs. Perturbers might not be located farther than 600 AU and they have to follow moderately eccentric and inclined orbits to be capable of experiencing close encounters with multiple known ETNOs.

Dynamical evolution of massive perturbers in realistic multicomponent galaxy models I: implementation and validation

ABSTRACT Galaxies are self-gravitating structures composed by several components encompassing spherical, axial, and triaxial symmetry. Although real systems feature heterogeneous components whose properties are intimately connected, semi-analytical approaches often exploit the linearity of the Poisson’s equation to represent the potential and mass distribution of a multicomponent galaxy as the sum of the individual components. In this work, we expand the semi-analytical framework developed in Bonetti et al. (2020) by including both a detailed implementation of the gravitational potential of exponential disc (modelled with a sech2 and an exponential vertical profile) and an accurate prescription for the dynamical friction experienced by massive perturbers (MP) in composite galaxy models featuring rotating disc structures. Such improvements allow us to evolve arbitrary orbits either within or outside the galactic disc plane. We validate the results obtained by our numerical model against public semi-analytical codes as well as full N-body simulations, finding that our model is in excellent agreement to the codes it is compared with. The ability to reproduce the relevant physical processes responsible for the evolution of MP orbits and its computational efficiency make our framework perfectly suited for large parameter-space exploration studies.

Analysis of limitations on recovery of gravity field based on satellite gravity gradient data

Although satellite gravity gradient data plays a great role in determining short-wavelength part of static gravity field model, accuracy of the long-wavelength part of gravity field model recovered by them are poor, which leads to only a few applications in time-variable gravity investigation. The reason is that some factors limit the accuracy of the gravity field recovered using gradient data, including accuracy of the gravity gradient observations, measurement bandwidth (MBW) of gradiometer, satellite inclination, etc. This paper aims at analyzing the influence of these limitations on gravity field recovery and discusses the possibility of time-variable gravity field detection by using gravity gradient observations. Firstly, for arbitrary satellite orbit inclination, we give the frequency distributions of all the components of gravity gradients (i.e. Txx,Tyy,Tzz,Txy,TxzandTyz, ). The results show that the maximum frequency of each component of the gravity gradients is the same, i.e. l/Ts (l is degree of the gravity field model, Ts is the orbital periods), and it is not influenced by the inclination of the satellite orbits. Secondly, the paper gives a theory proof to explain why only the low orders of the coefficients are influenced by polar gaps. Big polar gaps are experimented by a numerical test with inclination of 45°. Finally, considering that the measurement bandwidth can be expanded and accuracy of gradient observations can be improved by superconducting gravity gradiometer (SGG) compared to gradiometer used in Gravity field and steady-state Ocean Circulation Explorer (GOCE), the possibility of detecting time-variable gravity using gravity gradient observations is discussed. The results show that the SGG creates errors in MBW with magnitude of 0.014 mE, which is smaller than the magnitude of the time-variable gravity gradient signals (i.e., 0.02 mE) derived from Gravity Recovery and Climate Experiment (GRACE) gravity field models. This indicates the potential of SGG in time-variable gravity detection.

Rotordynamic Performance of the Interlocking Labyrinth Seal With a Tilting Rotor

Abstract Numerical analysis model of an interlocking labyrinth seal (ILS) is established for studying the effect of tilting rotor on its rotordynamic characteristics. The dynamic characteristic identification method based on infinitesimal theory is applied to solve the dynamic force coefficient of the seal with arbitrary elliptical orbits and eccentric positions under field conditions. The paper investigated the dynamic characteristics of the interlocking labyrinth seal with various misalignment angles (θ = 0, 0.1 deg, 0.2 deg, 0.3 deg, 0.4 deg, 0.5 deg, 0.6 deg), different pressure ratios (Pin = 6.9 bar, PR = 0.5, 0.8), locations of misalignment center (Loc = 0, L/2, L). Results show that the tilting rotor could minimize the leakage flow rate of the ILS. When the misalignment angle θ = 0.6 deg, the mass flow rate can be reduced about 2.5%. The effect of each cavity in the ILS on the stability of the system is different. The cavity with the inlet close to the rotor and the outlet away from the rotor helps to improve the system stability due to its locally antirotational flow. The effective damping of the entire ILS increases as the misalignment angle increases. The system shows the best stability when the misalignment center is close to the seal inlet. The tilting rotor has a positive effect on the stability of the ILS only except for high whirling frequency (>100 Hz) under Loc = L.

Bounded relative motions near Keplerian orbits in spherical coordinates

The problem of relative motion for arbitrary Keplerian orbits is modeled in a curvilinear coordinate system. A set of spherical coordinates rather than the classical Cartesian coordinates is used. The techniques such as linearization and normalization are performed to simplify the relative equations of motion as a linear form that has a similar structure as the Tschauner-Hempel (T-H) equations but expressed in spherical coordinates. The analytical solution in spherical coordinates is obtained by using the same technique as shown in the classical T-H equations. Based on this solution, bounded relative motions and the boundedness conditions are recognized for elliptical, parabolic and hyperbolic orbits, respectively. The geometry of bounded relative motions near parabolic and hyperbolic orbits are revealed, which has not been reported in literatures yet. It shows a strong connection with the “leader-follower” formation near elliptical orbits. Numerical simulation demonstrates the improved accuracy of the newly-developed solution and shows the existence of bounded relative trajectories for unbounded reference orbits.

Two-impulse, bi-parabolic, and bi-elliptical transfers between coplanar elliptical orbits

Necessary conditions for minimum characteristic velocity are applied to the problem of optimal impulsive transfer between arbitrary coplanar elliptical orbits. Additionally some existence and sufficient conditions for optimal transfer are presented. Equations for calculating optimal bi-parabolic transfers and optimal two-impulse transfers between orbits reduce to the problem of solving three nonlinear equations in three unknowns. Minimizing impulses are found to be applied tangentially if and only if the initial elliptical orbits have directionally aligned apses. In these cases minimizing impulses can be found in closed form.Four fundamental theorems are presented to unify these closed-form solutions. All of these closed-form minimizing transfers are either generalized Hohmann or bi-parabolic. Bi-elliptic transfers are never optimal but can approach arbitrarily close to a bi-parabolic transfer. An example is presented for perigees aligned in the same direction and another for aligned perigees in opposition.

Rapid Evaluation of Low-Thrust Transfers from Elliptical Orbits to Geostationary Orbit

Low-thrust orbit transfers to geostationary equatorial orbit (GEO) will likely use a combination of chemical and electrical propulsion stages to strike a balance between transit time, minimal power degradation, and delivered payload mass. Therefore, mission designers need a method for rapidly evaluating low-thrust transfers to GEO. Relying on full trajectory optimization programs for preliminary studies is not a desirable option due to the time associated with optimizing multiple trajectories. This paper develops an algorithm that rapidly determines the for a low-thrust transfer from an arbitrary elliptical orbit to GEO. Transfer time to GEO is accurately computed by incorporating thrust interruption (due to Earth-shadow effects) and power degradation (due to passage through the Van Allen radiation belts). The method employs curve fits of optimal transfers and simple polynomial expressions for the orbital element histories. Hence, the technique is fast and does not require numerical integration of the powered equations of motion or a numerical search. The method is demonstrated by presenting several transfers to GEO and comparing the performance metrics with the associated optimal transfers.