Relativistic Equations Of Motion Research Articles

On the maximal and minimal distances of flying-around satellite formation

PurposeThe purpose of this paper is to present the obtained analytic solutions of maximal and minimal inter-satellite distances for flying-around satellite formation.Design/methodology/approachThe relative motion equation is used to express the inter-satellite distance as the function of the orbital elements of two participating satellites for the flying-around satellite formation. Then by taking the derivative of the distance function with respect to the true anomaly, some possible extreme value points are obtained. According to the detailed analysis, the maximal and minimal distance solutions are found. By a reverse process, the expected initial differential orbital elements that generate the required extreme inter-satellite distances are also obtained.FindingsThe maximal and minimal distances of the flying-around formation can be analytically written as the functions of three initial orbital elements differences, i.e. the differential orbital inclination, the differential eccentricity and the differential right ascension. For the given maximal and minimal distances, there are lots of solutions of the initial differential orbital elements, which can produce the expected relative motions.Research limitations/implicationsThe solutions of the maximal and minimal inter-satellite distances are only accurate for the circular or near circular reference orbit. For the elliptic reference orbit, there is a need to develop new methods to find the analytic solutions.Practical implicationsThe results here can be applied to design the factual flying-around formation with dimension requirements in mission analysis stage.Originality/valueBy using the solutions presented in this paper, the engineers can design the expected flying-around formation with required maximal and minimal inter-satellite distances in a very easy way.

Sliding mode predictive guidance for terminal rendezvous in eccentric orbits

This paper presents a robust guidance algorithm for a chaser spacecraft to rendezvous with a target spacecraft in Earth orbit. The basis of the proposed guidance method is finding an appropriate set of states as close as possible to the current states that would lead the spacecraft to the target in the desired mission time. In order to provide the prediction of states, the relative dynamics equations of motion are solved analytically for the chaser spacecraft rendezvous considering constant acceleration. Although the equations are solved for rendezvous with circular orbit target, it is shown by several simulations that the proposed guidance algorithm is applicable in perturbed elliptical orbits rendezvous as well. The sliding mode method as a robust nonlinear method is utilized as the steering law. The robust steering law tracks the desired states computed by the predictive guidance method. The Lyapunov stability method proves the asymptotic stability of the integrated guidance and steering laws. Because the proposed closed-loop guidance is simple and computationally easy, it is suitable for implementation in real-time applications. Some numerical simulations are conducted to show the performance of the proposed guidance method in different conditions. It is illustrated that compared with other steering laws, the fuel consumption is reduced utilizing the proposed guidance approach. The results reveal that the sliding mode guarantees the tracking of the required states and minimum final errors even in the presence of uncertainties and disturbances.

A New Glideslope Guidance Algorithm for Minimum-Fuel Fixed-Time Elliptic Rendezvous Using Semidefinite Programming

This paper presents a new minimum-fuel glideslope guidance algorithm for approaching a target evolving on an elliptic orbit. In addition to the usual rectilinear profile to follow as in Hablani’s seminal paper, two new features are requested for the new algorithm. The first one imposes bounds on the guidance error inherent to chemical propulsion glideslope guidance, such that the chaser’s trajectory does not escape from an admissible domain. The second one minimizes the consumption during rendezvous. Indeed, unlike the classical glideslope algorithm for which there is no direct control on the fuel consumption, additional degrees of freedom and relevant decision variables may be identified. By combining a useful parametrization of the Tschauner-Hempel relative equations of motion and results from polynomial optimization, a semidefinite formulation of the constraints on the maximal guidance error is obtained. For a fixed-time glideslope rendezvous with a pre-assigned number of maneuvers, a fuel-optimal solution with a bounded guidance error is obtained by solving a semidefinite programming problem. Two numerical examples illustrate the usefulness of the method compared to the classical ones when the approach corridor has to verify stringent geometrical restrictions such as line-of-sight constraints.

Application of the Nonlinear Tschauner-Hempel Equations to Satellite Relative Position Estimation and Control

In this paper we develop the nonlinear motion equations in terms of the true anomaly varying Tschauner–Hempel equations relative to a notional orbiting particle in a Keplerian orbit, relatively close to an orbiting primary satellite to estimate the position of a spacecraft. A second orbiting body in Earth orbit relatively close to the first is similarly modelled. The dynamic relative motion models of the satellite and the second orbiting body, both of which are modelled in terms of independent relative motion equations, include several perturbing effects, such as the asymmetry of the Earth gravitational field resulting in the Earth's oblateness effect and the third body accelerations due to the Moon and the Sun. Linear control laws are synthesised for the primary satellite using the transition matrix so it can rendezvous with the second orbiting body. The control laws are then implemented using the state estimates obtained earlier to validate the feedback controller. Thus, we demonstrate the application of a Linear Quadratic Nonlinear Gaussian (LQNG) design methodology to the satellite rendezvous control design problem and validate it.

Relativistic radiative transfer and relativistic spherical flows: Relation between the mass-loss rate and terminal speed

Abstract Radiatively driven relativistic spherical flows are numerically investigated under the fully special relativistic treatment and relativistic radiative transfer. We first solve the relativistic radiative transfer equation for spherically symmetric outflows iteratively, using a trial velocity distribution, and obtain specific intensities as well as moment quantities, and the Eddington factor. Using the obtained comoving flux, we next solve the relativistic equation of motion, and obtain the refined velocity distribution, the mass-loss rate being determined as an eigenvalue. Until both the intensity and velocity distributions converge, we repeat these double iteration processes. We found that the flows are quickly accelerated near to the central lumious core to reach the terminal speed. The Eddington factor has a complicated behavior, depending on the optical depth and flow speed. We further found that a relation between the flow terminal speed βout normalized by the speed of light and the mass-loss rate $\dot{m}$ normalized by the critical one is roughly approximated as $\dot{m} \propto \tau _* \beta _{\rm out}^{-5/2}$, where τ* is a typical optical depth of the flow, whereas we can analytically derive the relation of $\dot{m} \propto \tau _* \beta _{\rm out}^{-2}$ using a back-of-the-envelope calculation in the nonrelativistic case.

Prospects for asteroid mass determination from close encounters between asteroids: ESA's Gaia space mission and beyond

AbstractWe present a catalog of potential candidates for asteroid mass determination based on mutual close encounters of numbered asteroids with massive perturbers (D>20 km). Using a novel geometric approach tuned to optimize observability, we predict optimal epochs for mass determination observations. In contrast to previous studies that often used simplified dynamical models, we have numerically propagated the trajectories of all numbered asteroids over the time interval from 2013 to 2023 using relativistic equations of motion including planetary perturbations, J2 of the Sun, the 16 major asteroid perturbers and the perturbations due to non-sphericities of the planets. We compiled a catalog of close encounters between asteroids where the observable perturbation of the sky plane trajectory is greater than 0.5 mas so that astrometric measurements of the perturbed asteroids in the Gaia data can be leveraged. The catalog v1.0 is available at ftp://dosya.akdeniz.edu.tr/ivantsov.

Analysis of the lateral dynamics of a vehicle and driver model running straight ahead

In this paper, we show that even an extremely simple nonlinear vehicle and driver model can show complex behaviors, like multi-stability and sensible dependence on the initial condition. The mechanical model of the car has two degrees of freedom, and the related equations of motion contain the nonlinear characteristics of the tires. The driver model is described by a single (nonlinear) equation, characterized by three parameters that describe how the driver steers the vehicle. Namely such parameters are the gain (steering angle per lateral deviation from desired path), the preview distance, and the reaction time delay. Bifurcation analysis is adopted to characterize straight ahead motion at different speeds, considering separately the two cases of understeering or oversteering cars. In the first case, we show that at suitable speeds the model can have three different attracting oscillating trajectories on which the system can work and that are reached due to different disturbances. In the second case, we confirm that instability arises if the forward speed is too high. The final results of the paper, bifurcation diagrams, can be used for many considerations critical both from the theoretical and from the practical viewpoints.

New State Transition Matrices for Spacecraft Relative Motion in Perturbed Orbits

This paper presents new state transition matrices that model the relative motion of two spacecraft in arbitrarily eccentric orbits perturbed by and differential drag for three state definitions based on relative orbital elements. Both density-model-specific and density-model-free formulations of the effects of differential drag are included. The state transition matrices are derived by first performing a Taylor expansion on the equations of relative motion and subsequently computing an exact closed-form solution of the resulting linear differential equations. The resulting state transition matrices are used to generalize the geometric interpretation of the effects of and differential drag on relative motion in near-circular orbits provided in previous works to arbitrarily eccentric orbits. Additionally, this paper harmonizes current literature by demonstrating that a number of state transition matrices derived by previous authors using various techniques can be found by subjecting the models presented in this paper to more restrictive assumptions. Finally, the presented state transition matrices are validated through comparison with a high-fidelity numerical orbit propagator. It is found that these models are able to match or exceed the accuracy of comparable models in the literature over a broad range of orbit scenarios.

Long-term effects of main-body’s obliquity on satellite formation perturbed by third-body gravity in elliptical and inclined orbit

A new non-simplified model of formation flying is derived in the presence of an oblate main-body and third-body perturbation. In the proposed model, considering the perturbation of the third-body in an inclined orbit, the effect of obliquity (axial tilt) of the main-body is becoming important and has been propounded in the absolute motion of a reference satellite and the relative motion of a follower satellite. From a new point of view, J2 perturbed relative motion equations and considering a disturbing body in an elliptic inclined three dimensional orbit, are derived using Lagrangian mechanics based on accurate introduced perturbed reference satellite motion. To validate the accuracy of the model presented in this study, an auxiliary model was constructed as the Main-body Center based Relative Motion (MCRM) model. Finally, the importance of the main-body’s obliquity is demonstrated by several examples related to the Earth-Moon system in relative motion and lunar satellite formation keeping. The main-body’s obliquity has a remarkable effect on formation keeping in the examined in-track and projected circular orbit (PCO) formations.

Dynamic boundary layer based neural network quasi-sliding mode control for soft touching down on asteroid

Pinpoint landing is a critical step in some asteroid exploring missions. This paper is concerned with the descent trajectory control for soft touching down on a small irregularly-shaped asteroid. A dynamic boundary layer based neural network quasi-sliding mode control law is proposed to track a desired descending path. The asteroid’s gravitational acceleration acting on the spacecraft is described by the polyhedron method. Considering the presence of input constraint and unmodeled acceleration, the dynamic equation of relative motion is presented first. The desired descending path is planned using cubic polynomial method, and a collision detection algorithm is designed. To perform trajectory tracking, a neural network sliding mode control law is given first, where the sliding mode control is used to ensure the convergence of system states. Two radial basis function neural networks (RBFNNs) are respectively used as an approximator for the unmodeled term and a compensator for the difference between the actual control input with magnitude constraint and nominal control. To improve the chattering induced by the traditional sliding mode control and guarantee the reachability of the system, a specific saturation function with dynamic boundary layer is proposed to replace the sign function in the preceding control law. Through the Lyapunov approach, the reachability condition of the control system is given. The improved control law can guarantee the system state move within a gradually shrinking quasi-sliding mode band. Numerical simulation results demonstrate the effectiveness of the proposed control strategy.

Optimal Continuous-Thrust Rephasing Maneuver in Circular Orbit

The minimum-time constant-thrust circular orbit rephasing problem is studied using a curvilinear relative motion description for dynamics. The resulting optimal control problem in the thrust orientation is formulated using the indirect method and investigated both analytically and numerically. By linearizing the relative motion equations, a key nondimensional parameter is identified, which characterizes the duration of the maneuver and its qualitative structure when changing from a short-maneuver to a long-maneuver time regime, passing through a transition zone. Approximate analytical solutions are obtained for the short- and long-maneuver regimes, providing an accurate estimation of the minimum maneuver time and a clear understanding of the evolution of the optimal thrust profile, which approaches two opposite bang–bang structures in the limit cases. The nonlinear problem is investigated numerically and shown to be fully characterized by two nondimensional parameters instead of one. Finally, a comparison with different two-impulse maneuvers is conducted. The results can be used for constructing good initial guesses in more complex optimization problems and to support preliminary mission design.

Time and Covariance Threshold Triggered Optimal Uncooperative Rendezvous Using Angles-Only Navigation

A time and covariance threshold triggered optimal maneuver planning method is proposed for orbital rendezvous using angles-only navigation (AON). In the context of Yamanaka-Ankersen orbital relative motion equations, the square root unscented Kalman filter (SRUKF) AON algorithm is developed to compute the relative state estimations from a low-volume/mass, power saving, and low-cost optical/infrared camera’s observations. Multi-impulsive Hill guidance law is employed in closed-loop linear covariance analysis model, based on which the quantitative relative position robustness and relative velocity robustness index are defined. By balancing fuel consumption, relative position robustness, and relative velocity robustness, we developed a time and covariance threshold triggered two-level optimal maneuver planning method, showing how these results correlate to past methods and missions and how they could potentially influence future ones. Numerical simulation proved that it is feasible to control the spacecraft with a two-line element- (TLE-) level uncertain, 34.6% of range, initial relative state to a 100 m v-bar relative station keeping point, at where the trajectory dispersion reduces to 3.5% of range, under a 30% data gap per revolution on account of the eclipse. Comparing with the traditional time triggered maneuver planning method, the final relative position accuracy is improved by one order and the relative trajectory robustness and collision probability are obviously improved and reduced, respectively.

EXCHANGE SPATIAL-ENERGY INTERACTIONS

The notion of spatial-energy parameter (P-parameter) is introduced based on the modified Lagrangian equation for relative motion of two interacting material points, and is a complex characteristic of important atomic values responsible for interatomic interactions and having the direct connection with electron density inside an atom. Wave properties of P-parameter are found, its wave equation having a formal analogy with the equation of Ψ-function is given. With the help of P-parameter technique numerous calculations of exchange structural interactions are done, the applicability of the model for the evaluation of the intensity of fundamental interactions is demonstrated, initial theses of quark screw model are given.

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.

Discussion on Fundamental Problems of Physics Hidden in Cosmology

Discussion on Fundamental Problems of Physics Hidden in Cosmology

Angles-Only Initial Relative Orbit Determination Performance Analysis using Cylindrical Coordinates

The solution of the initial relative orbit determination problem using angles-only measurements is important for orbital proximity operations, satellite inspection and servicing, and the identification of unknown space objects in similar orbits. In this paper, a preliminary relative orbit determination performance analysis is conducted utilizing the linearized relative orbital equations of motion in cylindrical coordinates. The relative orbital equations of motion in cylindrical coordinates are rigorously derived in several forms included the exact nonlinear two-body differential equations of motion, the linear-time-varying differential equations of motion for an elliptical orbit chief, and the linear-time-invariant differential equations of motion for a circular orbit chief. Using the nonlinear angles-only measurement equation in cylindrical coordinates, evidence of full-relative-state observability is found, contrary to the range observability problem exhibited in Cartesian coordinates. Based on these results, a geometric approach to assess initial relative orbit determination performance is formulated. To facilitate a better understanding of the problem, the focus is on the 2-dimensional initial orbit determination problem. The results clearly show the dependence of the relative orbit determination performance on the geometry of the relative motion and on the time-interval between observations. Analysis is conducted for leader-follower orbits and flyby orbits where the deputy passes directly above or below the chief.

ON ENERGY PRINCIPLES OF STRUCTURAL INTERACTIONS

The notion of spatial-energy parameter (P-parameter) is introduced based on the modified Lagrangian equation for relative motion of two interacting material points, which is a complex characteristic of important atomic values. Wave properties of P-parameter are found, its wave equation having a formal analogy with the equation of Ψ-function is given. In the systems in which the interactions proceed along the potential gradient (positive performance) the resulting potential energy is found based on the principle of adding reciprocals of the corresponding energies of subsystems. Some correlations of P-parameter values with Lagrangian and Hamiltonian functions are obtained .

Reexamining cluster radioactivity in trans-lead nuclei with consideration of specific density distributions in daughter nuclei and clusters

We further investigate the cluster emission from heavy nuclei beyond the lead region in the framework of the preformed cluster model. The refined cluster-core potential is constructed by the double-folding integral of the density distributions of the daughter nucleus and the emitted cluster, where the radius or the diffuseness parameter in the Fermi density distribution formula is determined according to the available experimental data on the charge radii and the neutron skin thickness. The Schr\odinger equation of the cluster-daughter relative motion is then solved within the outgoing Coulomb wave-function boundary conditions to obtain the decay width. It is found that the present decay width of cluster emitters is clearly enhanced as compared to that in the previous case, which involved the fixed parametrization for the density distributions of daughter nuclei and clusters. Among the whole procedure, the nuclear deformation of clusters is also introduced into the calculations, and the degree of its influence on the final decay half-life is checked to some extent. Moreover, the effect from the bubble density distribution of clusters on the final decay width is carefully discussed by using the central depressed distribution.

Operational Methodology for Large-Scale Deployment of Nanosatellites into Low Earth Orbit

The on-orbit deployment of large numbers of nanosatellites is cast in terms of the relative motion equations, augmented with differential drag. Using this framework, a methodology is developed for minimizing the likelihood of recontact or interference among the deployed nanosatellites. This approach is applied to a representative test case derived from the Operationally Responsive Space Office’s Launch Enabler Mission, which delivered 28 small payloads to orbit in 2013. By strategically varying the deployment order, timing, and launch vehicle attitude, the net separation among each of the nanosatellites is demonstrated for the first three weeks with minimal close approaches. These results are validated using high-fidelity simulations.

Logical inference approach to relativistic quantum mechanics: Derivation of the Klein–Gordon equation

The logical inference approach to quantum theory, proposed earlier De Raedt et al. (2014), is considered in a relativistic setting. It is shown that the Klein–Gordon equation for a massive, charged, and spinless particle derives from the combination of the requirements that the space–time data collected by probing the particle is obtained from the most robust experiment and that on average, the classical relativistic equation of motion of a particle holds.