Hyperbolic Trajectories Research Articles

Smooth and continuous interplanetary trajectory design of spacecraft using iterative patched-conic method

This paper is concerned with the transfer trajectory design among multiple planets using multiple flybys. The proposed trajectory design method consists of a semi-analytical tuning strategy and iterative patched-conic method. The tuning strategy refers to a method realizing the smooth connection between a hyperbolic trajectory near a planet and a trajectory obtained by Lambert's problem between the planet and another one. The proposed method iteratively solves Lambert's problem and the tuning strategy so that the arrival or departure velocity obtained by Lambert's problem coincides with the velocity at the corresponding point in the sphere of influence (SOI), assigned by the tuning strategy. Consequently, a smooth and continuous trajectory with multiple flybys can be obtained, where the SOI of each flyby planet and the flyby time are explicitly taken into account. Moreover, analytical conditions with respect to the orbital elements of the arrival and departure hyperbolas are provided to minimize velocity mismatch at the periapsis of a flyby planet. Through numerical simulation using the Cassini example, it is shown that the resultant trajectory by the proposed method is close to the optimal trajectory obtained by the n-body problem. Thus, a suitable approximate trajectory can be effectively generated without solving the n-body problem.

Simulated annealing velocity analysis: Automating the picking process

We have developed an automated method for velocity picking that allows us to estimate appropriate velocity functions for the normal moveout correction of common-depth-point (CDP) gathers, valid for either hyperbolic or nonhyperbolic trajectories. In the hyperbolic velocity analysis case, the process involves the simultaneous search (picking) of a certain number of time-velocity pairs in which the semblance, or any other coherence measure, is high. In the nonhyperbolic velocity analysis case, a third parameter, usually associated with the layering and/or the anisotropy, is added to the searching process. Our technique relies on a simple but effective search of a piecewise linear curve defined by a certain number of nodes in a 2D or 3D space that follows the semblance maxima. The search is carried out efficiently using a constrained very fast simulated annealing algorithm. The constraints consist of static and dynamic bounding restrictions, which are viewed as a means to incorporate prior information about the picking process. This allows us to avoid those maxima that correspond to multiples, spurious events, and other meaningless events. Results using synthetic and field data indicate that our technique permits automatically obtaining accurate and consistent velocity picks that lead to flattened events, in agreement with the manual picks. As an algorithm, the method is very flexible for accommodating additional constraints (e.g., preselected events) and depends on a limited number of parameters. These parameters are easily tuned according to data requirements, available prior information, and the user’s needs. The computational costs are relatively low, ranging from a fraction of a second to, at most, 1–2 s per CDP gather, using a standard PC with a single processor.

A Modified Double Gyre with Ground Truth Hyperbolic Trajectories for Flow Visualization

AbstractThe model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.

Trajectory Design of Perseus: A CubeSat Mission Concept to Phobos

The Martian satellites Phobos and Deimos hold many unanswered questions that may provide clues to the origin of Mars. These moons are low Δv stopover sites to Mars. Some human missions to Mars typically identify Phobos and Deimos as staging bases for Mars surface exploration. Astronauts could base initial operations there in lieu of repeated voyages to and from the planet surface, to refuel transiting spacecraft, to teleoperate robotics and other critical machinery, and to develop habitable infrastructure ahead of human landings. Despite their strategic and scientific significance, there has been no successful dedicated mission to either moon. For this reason, we propose Perseus, a geological imaging CubeSat mission to Phobos. Perseus, a 27U, 54kg CubeSat will return thermal and visible images at resolutions better than currently available over most of Phobos’ surface. This includes visible images at 5m/pixel and thermal images at 25m/pixel of Phobos’ surface. The Perseus mission is nominally intended to be a co-orbital mission, where the spacecraft will encounter Phobos on its Martian orbit. However, a hyperbolic rendezvous mission concept, to image Phobos on a hyperbolic flyby, is also considered to reduce the risks associated with orbit capture and to reduce mission costs. This paper presents the preliminary feasibility, science objectives, and technological development challenges of achieving these science goals. We then formulate two rendezvous concepts as a series of three nonlinear optimization problems that span the design tree of mission concepts. The tree’s root node is the heliocentric cruise problem, which identifies the near-optimal launch and arrival windows for the Perseus spacecraft. The leaf nodes of the design tree are the two rendezvous concepts that identify near-optimal co-orbital and hyperbolic trajectories for Phobos’ reconnaissance. The design problems are solved using evolutionary algorithms, and the performance of the selected mission concepts is then examined. The results indicate that a co-orbital encounter allows about one encounter per day with about 6 min per encounter. The hyperbolic encounter, on the other hand, allows a single encounter where the spacecraft will spend about 2 min in the imaging region with respect to Phobos. The spacecraft will obtain higher resolution images of Phobos on this feasible region than have ever been seen for most of the surface. These detailed images will help identify candidate landing sites and provide critical data to derisk future surface missions to Phobos.

Spontaneous-focusing and self-healing of Airy-like beams

Modulated by the spatial phase, Airy-like beams that possess the faster and better self-healing as well as the greater acceleration along the desired arbitrary trajectory are introduced. Airy-like beams have the strong spontaneous-focusing ability and the peak value of the focal plane is 5.4 times that of the initial plane when traveling along the quadratic function trajectory. Besides, the larger quadratic coefficient and the smaller constant term of the quadratic function can enhance the self-healing and the spontaneous-focusing of the beams. When the beams transmit along the snakelike trajectory, the original main lobe separates a new main lobe, and the energy will concentrate on the new main lobe. Airy-like beams, along a hyperbolic-secant trajectory, can remain non-diffraction over long distances. However, when traveling along a hyperbolic trajectory, the evolution process is very similar to that of Airy-Gaussian vortex beams. These excellent properties can be applied to the special beams’ modulation and optimization. Furthermore, it may provide a way for constructing beams along the special trajectory with the desired spontaneous-focusing and self-healing.

Trajectory Estimation for Particles Observed in the Vicinity of (101955) Bennu

AbstractWe analyze the trajectories of 313 particles seen in the near‐Bennu environment between December 2018 and September 2019. Of these, 65% follow suborbital trajectories, 20% undergo more than one orbital revolution around the asteroid, and 15% directly escape on hyperbolic trajectories. The median lifetime of these particles is ∼6 hr. The trajectories are sensitive to Bennu's gravitational field, which allows us to reliably estimate the spherical harmonic coefficients through degree 8 and to resolve nonuniform mass distribution through degree 3. The particles are perturbed by solar radiation pressure, enabling effective area‐to‐mass ratios to be estimated. By assuming that particles are oblate ellipsoids of revolution, and incorporating photometric measurements, we find a median axis ratio of 0.27 and diameters for equivalent‐volume spheres ranging from 0.22–6.1 cm, with median 0.74 cm. Our size distribution agrees well with that predicted for fragmentation due to diurnal thermal cycling. Detailed models of known accelerations do not produce a match to the observed trajectories, so we also estimate empirical accelerations. These accelerations appear to be related to mismodeling of radiation pressure, but we cannot rule out contributions from mass loss. Most ejections take place at local solar times in the afternoon and evening (12:00–24:00), although they occur at any time of day. We independently identify ten ejection events, some of which have previously been reported. We document a case where a particle ricocheted off the surface, revealing a coefficient of restitution 0.57±0.01 and demonstrating that some apparent ejections are not related to surface processes.

Step-size effect in the time-transformed leapfrog integrator on elliptic and hyperbolic orbits

ABSTRACT A drift-kick-drift (DKD) type leapfrog symplectic integrator applied for a time-transformed separable Hamiltonian (or time-transformed symplectic integrator; TSI) has been known to conserve the Kepler orbit exactly. We find that for an elliptic orbit, such feature appears for an arbitrary step size. But it is not the case for a hyperbolic orbit: When the half step size is larger than the conjugate momenta of the mean anomaly, a phase transition happens and the new position jumps to the non-physical counterpart of the hyperbolic trajectory. Once it happens, the energy conservation is broken. Instead, the kinetic energy minus the potential energy becomes a new conserved quantity. We provide a mathematical explanation for such phenomenon. Our result provides a deeper understanding of the TSI method, and a useful constraint of the step size when the TSI method is used to solve the hyperbolic encounters. This is particular important when an (Bulirsch–Stoer) extrapolation integrator is used together, which requires the convergence of integration errors.

Extraction of Distinguished Hyperbolic Trajectories for 2D Time‐Dependent Vector Field Topology

AbstractThis paper does two main contributions to 2D time‐dependent vector field topology. First, we present a technique for robust, accurate, and efficient extraction of distinguished hyperbolic trajectories (DHT), the generative structures of 2D time‐dependent vector field topology. It is based on refinement of initial candidate curves. In contrast to previous approaches, it is robust because the refinement converges for reasonably close initial candidates, it is accurate due to its adaptive scheme, and it is efficient due to its high convergence speed. Second, we provide a detailed evaluation and discussion of previous approaches for the extraction of DHTs and time‐dependent vector field topology in general. We demonstrate the utility of our approach using analytical flows, as well as data from computational fluid dynamics.

Finite-time circulation changes from topological rearrangement of distinguished curves and non-advective fluxes

A vorticity budget diagnoses the growth or decay of a vortex from advective transport, or non-advective fluxes such as those due to friction or vortex tilting. However, when a budget calculation is performed with respect to a fixed coordinate, errors may result from time-dependence of the flow, leading to disagreement between the vorticity tendency and the observed vorticity field. An adaptive Lagrangian coordinate resolves this problem, provided that the resulting Lagrangian structure does not become too complicated.In this study, a numerical simulation of Hurricane Nate (2011), the vorticity tendency is evaluated along distinguished material curves. There can be no net advective flux along a closed material curve, therefore, the total circulation tendency for a material region includes only the non-advective uxes acting along its boundary. A distinguished set of material curves (DMCs) associated with a distinguished hyperbolic trajectory (DHT) form a Lagrangian topology similar to that of a cat’s eye flow or “pouch” at each Eulerian snapshot. The time-dependence of velocities allows additional regions called lobes, which are formed by the intersections of DMCs, to exchange fluid across the vortex boundary by redefining the boundary.Because the vortex boundary changes, we refer to this redefinition of material boundary as “topological rearrangement”. The method is useful for unsteady cat’s-eye flows and more complex interactions of multiple waves, vortices and background shear. All advective changes of the vortex circulation are identified by exchanges of the lobes, and all non-advective uxes act between the vortex and either the lobes or environmental flow. The Lagrangian topology and combination of advective and non-advective uxes relative to the topology is used to describe the evolution of the circulation of Nate during its time of formation.

Enhanced CORDIC Based Rotator Design for Sinusoidal Transforms

Transforms play an important role in conversion of information from one domain to the other. To be more specific transforms like Discrete Fourier transform (DFT) and Discrete Cosine transform (DCT) helps us to migrate from one time domain to frequency domain based on the basis function selected. The basis function of the every sinusoidal transform carries out a circular rotation to convert information from one domain to the other. There are applications related to communication which requires this rotation into the hyperbolic trajectory as well. Multiplierless algorithm like CORDIC improves the latency of the transforms by eliminating the number of multipliers in the basis function. In this paper we have designed and implemented enhanced version of CORDIC based Rotator design. The Enhanced version is simulated for order 1 to order 36 to emphasize on the results of the proposed algorithm. Results shows that the enhanced CORDIC rotator design surpasses the Mean square error after the order 18 compared to standard CORDIC. Unified CORDIC also can be implemented using the said algorithm to implement different three trajectories.

Hyperbolic Meteoroids Impacting the Moon

Abstract Since the discovery of the Moon’s asymmetric ejecta cloud, the origin of its sunward-canted density enhancement has not been well understood. We propose impact ejecta from meteoroids on hyperbolic trajectories (β-meteoroids) that hit the Moon’s sunward side could explain this unresolved asymmetry. β-meteoroids are submicron in size, comparable to or smaller than the regolith particles they hit, and can impact the Moon at very high speeds ∼100 km s−1. Therefore, their impact regime may differ from the significantly larger and slower sporadic meteoroids responsible for generating the bulk of the lunar impact ejecta cloud. We compare lunar impact ejecta production to β-meteoroid fluxes observed by multiple spacecraft. If β-meteoroids are able to liberate similar sized submicron particles, orbital dust detector measurements from the Lunar Dust Experiment on board the Lunar Atmosphere and Dust Environment Explorer spacecraft only need to detect one ejecta grain out of every 106 β-meteoroid impacts to the lunar surface to explain the sunward asymmetry with this additional population. This finding suggests β-meteoroids may also contribute to the evolution of other airless surfaces in the inner solar system, and by extension, at exozodiacal systems.

Analysis of Transport and Mixing Phenomenon to Invariant Manifolds Using LCS and KAM Theory Approach in Unsteady Dynamical Systems

The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent structures (LCSs). For the time-dependent perturbation problems, we analyze in detail the development of transport barriers that play an important role in the transport process. Firstly, the analytical study of KAM theory is adopted to explain the transport and mixing phenomenon in measured and simulated airfoil flow. Then, the flow topology of unsteady flow behind an airfoil is investigated for the low Reynolds number problem. Simulations are carried out based on a particular Finite-Time Lyapunov Exponent (FTLE) technique for the detection of invariant manifolds of the hyperbolic trajectories. Besides, the Characteristic Base Split (CBS) scheme combined with a dual time stepping technique is utilized to simulate such transient flow problems. Thus, in the course of the current research, the role of the velocity phase plot during vortex formation is explored that is highly periodic and resulted in the formation of a stable pattern of manifolds and invariant tori. Hence, the proposed study encouraged the new picture of the vortex shedding and flow separation process. As a conclusion, our results give a better understanding of invariant tori control transport phenomena that will lead to a new heuristic for unsteady flows.

A vortex pair in ground effect, dynamics and optimal control

The dynamics and control of a vortex pair in ground effect are investigated in a planar, incompressible and laminar setting. The evolution of the vortices obtained numerically shows vortex rebound as a consequence of the separation of the boundary layer induced at the wall by the vortices. An optimal control approach is developed and employed for vortex Reynolds numbers of 200 and 1000 in order to identify the optimal Dirichlet boundary condition at the wall to counteract this rebound and allow for an increased lateral displacement of the vortex, similarly to the inviscid evolution of the flow, which features hyperbolic trajectories. The work is primarily a conceptual approach to deal with aircraft separation distances in airport airspace by moving the vortices laterally, away from the runway but may also apply to the control of coherent structures in wall bounded turbulence. The most efficient control is able to double the lateral position and yields mostly vertical in and outflow at the wall. An optimal horizon time is found, equal to 5 characteristic time units of the vortex system, beyond which control is not able to further displace the vortices. The control is shown to delay the separation of the boundary layer at the origin of vortex rebound by applying suction ahead of the vortex, and to generate a vorticity flux at the wall, leading to a pusher vortex of sign opposite to that of the primary vortex, that attenuates the effect of the no-slip boundary condition at the wall by pushing the vortex outward.

Estimating normal moveout velocity using the recurrent neural network

Machine learning (ML) has recently gained immense popularity because of its successful application in complex problems. It develops an abstract relation between the input and output. We have evaluated the application of ML to the most basic seismic processing of normal moveout (NMO) correction. The arrival times of reflection events in a common midpoint (CMP) gather follow a hyperbolic trajectory; thus, they require a correction term to flatten the CMP gather before stacking. This correction term depends on an rms velocity, also referred to as the NMO velocity. In general, NMO velocity is estimated using the semblance measures and picking the peaks in the velocity panel. This process requires a lot of human intervention and computation time. We have developed a novel method using one of the tools based on an ML- approach and applied to the NMO velocity estimation problem. We use the recurrent neural network (RNN) to estimate the NMO velocity directly from the seismic data. The input to the network is a seismic gather and corresponding precalculated NMO velocity (as prelabeled data set) to flatten the gather. We first train the network to develop a relationship between the input gathers (before NMO correction) and the corresponding NMO velocities for a few CMPs as a supervised learning process. Adam optimization algorithm is used to train the RNN. The output from the network is then compared against the correct NMO velocity. The error between the two velocities is then used to update the weight of the neurons and to minimize the mean-squared error between the two velocities. After the network is trained, it can be used to calculate the NMO velocity for the rest of the seismic gathers. We evaluate our method on a noisy data set from Poland. We used only 10% of the CMPs to train the network, and then we used the trained network to predict NMO velocity for the remaining CMP locations. The stack section obtained by using RNN-generated NMO velocities is nearly identical to that obtained by the conventional semblance method.

X-ray interference fringes from a weakly bent plane-parallel crystal with negative strain gradient.

Under the anomalous transmission condition in the Bragg mode, X-ray interference fringes were observed between two beams with different hyperbolic trajectories in a very weakly bent plane-parallel perfect crystal with negative strain gradient. The origin of the fringes was analysed based on the dynamical theory of diffraction for a distorted crystal. In the reflected beam from the entrance surface, the interference fringes were observed between once- and twice-reflected beams from the back surface. In the transmitted beam from the back surface, the interference fringes were observed between the direct beam and once-reflected beam from the entrance surface. In the emitted beam from the lateral surface, the interference fringes were observed between the beams after different numbers of reflections in the crystal. The multiply reflected beams were formed by a combined result of long propagation length along the beam direction with large divergence of the refracted beams when the strain gradient was negative. The period of these interference fringes was sensitive to very weak strain, of the order of 10-7.

Aircraft planar trajectories in crosswind navigation: some hypergeometric solutions

We study the planar motion of a self-propelled object against a withstanding flow, such as wind or sea current, whose speed is capable of perturbing the trajectory of the object, heading it off its path. Seven cases are dealt with, corresponding to fairly general wind types and all leading to nonlinear ordinary differential equations for the object motion. The first two cases concern motions described in Cartesian coordinates. They are followed by motions under a cyclonic wind, analysed in a polar reference frame; the wind is modelled as a logarithmic or hyperbolic spiral and closed-form solutions are obtained by means of the Gauss hypergeometric function and the Lerch transcendent. Further wind instances, treated in a Cartesian or in a polar reference system, require some special functions, such as the Lambert function, while finding the motion time law leads to a trinomial equation, in the most interesting situation of a mobile object that is faster than the wind. In the last case of a horizontal hyperbolic wind, time law and trajectory are computed in a polar reference frame by means of the Lambert function.

Non-gravitational acceleration in the trajectory of 1I/2017 U1 ('Oumuamua).

'Oumuamua (1I/2017U1) is the first known object of interstellar origin to have entered the Solar System on an unbound and hyperbolic trajectory with respect to the Sun1. Various physical observations collected during its visit to the Solar System showed that it has an unusually elongated shape and a tumbling rotation state1-4 and that the physical properties of its surface resemble those of cometary nuclei5,6, even though it showed no evidence of cometary activity1,5,7. The motion of all celestial bodies is governed mostly by gravity, but the trajectories of comets can also be affected by non-gravitational forces due to cometary outgassing8. Because non-gravitational accelerations are at least three to four orders of magnitude weaker than gravitational acceleration, the detection of any deviation from a purely gravity-driven trajectory requires high-quality astrometry over a long arc. As a result, non-gravitational effects have been measured on only a limited subset of the small-body population9. Here we report the detection, at 30σ significance, of non-gravitational acceleration in the motion of 'Oumuamua. We analyse imaging data from extensive observations by ground-based and orbiting facilities. This analysis rules out systematic biases and shows that all astrometric data can be described once a non-gravitational component representing a heliocentric radial acceleration proportional to r-2 or r-1 (where r is the heliocentric distance) is included in the model. After ruling out solar-radiation pressure, drag- and friction-like forces, interaction with solar wind for a highly magnetized object, and geometric effects originating from 'Oumuamua potentially being composed of several spatially separated bodies or having a pronounced offset between its photocentre and centre of mass, we find comet-like outgassing to be a physically viable explanation, provided that 'Oumuamua has thermal properties similar to comets.

Gulf Stream Transport and Mixing Processes via Coherent Structure Dynamics

AbstractThe Gulf Stream has been characterized as either a barrier or blender to fluid transfer, a duality relevant to gyre‐scale climate adjustment. However, previous characterization depended on relatively sparse, Lagrangian in situ observations. The finite‐time Lyapunov exponent (FTLE) is calculated from satellite altimetry to identify Lagrangian coherent structures (LCS) in the Gulf Stream region. These LCS provide dense sampling of flow and capture distinct regions associated with mixing. Independent observations of ocean color contain similar flow‐dependent structures, providing verification of the method and highlighting transport and mixing processes that influence sea surface temperature and chlorophyll, among other water properties. Diagnosed LCS support the existing Bower kinematic model of the Gulf Stream, but also highlight novel behavior of comparable importance. These include vortex pinch‐off and formation of spiral eddies, clearly identified by LCS and which may be explained by considering changes to flow topology and the dynamics of shear‐flow instability at both small and large Rossby number. Such processes, seen though LCS, may further enable validation of climate models. The spatial distribution of these intermittent processes is characterized in terms of the criticality of jet dynamics with respect to Rossby wave propagation, and whether the jet is in an unstable or wave‐maker regime. The generation and connectivity of hyperbolic trajectories in the flow appears to play an important role in governing large‐scale transport and mixing across the Gulf Stream.

Allowance for more realistic trajectories of plasma electrons in the Stark broadening of hydrogenlike spectral lines

In the Conventional Theory (CT) of the electron broadening of hydrogenlike spectral lines in plasmas, the perturbing electrons are considered moving along hyperbolic trajectories in the Coulomb field of the effective charge Z − 1 (in atomic units), where Z is the nuclear charge of the radiating ion. Thus, the CT assumed that the motion of the perturbing electron can be described in frames of a two-body problem, one particle being the perturbing electron and the other ‘particle’ being the charge Z − 1. However, in reality it is a three-body problem, involving the perturbing electron, the nucleus, and the bound electron, so that trajectories of the perturbing electrons should be more complicated. In the present paper we take this into account via the standard analytical method of separating rapid and slow subsystems by using the fact that the characteristic frequency of the motion of the bound electron around the nucleus is much higher than the characteristic frequency of the motion of the perturbing electron around the radiating ion. By applying this method we obtain more accurate analytical results for the electron broadening operator than in the CT. We show by examples of the electron broadening of the Lyman lines of He II that the allowance for this effect increases with the electron density Ne, becomes significant already at Ne ∼ 1017 cm−3 and very significant at higher densities. We also briefly discuss a paper by Baryshnikov and Lisitsa (Sov. Phys. JETP 53 (1981) 471) where the fundamental symmetry of the effective potential for this problem was used in a different way.

Hyperbolic accelerating beams and their relation with Hermite-Gaussian beams.

We derive the initial distributions of phase and complex amplitude of accelerating beams with arbitrary predesigned hyperbolic trajectories using the caustic-design method and explore the relation between these beams and Hermite-Gaussian beams. The results show the hyperbolic accelerating beams are a larger class of beams than Hermite-Gaussian beams. When the bending parameter is an integer, the hyperbolic accelerating beams have a similar initial complex amplitude distribution and almost the same propagating characteristics as Hermite-Gaussian beams. Through the analysis of the ray-based method, we also derive an approximate expression for the initial complex amplitude of Hermite-Gaussian beams after introducing an amplitude distribution function. Although the proposed approximate expressions of complex amplitude are more complex than the usually used Hermite-Gaussian function, they explicitly indicate the information on local amplitude, wave vector, and internal ray structure (including caustics) of these beams and thus provide us clearer geometrical insights into these beams.