Relative Orbital Motion Research Articles

Dynamics and capability of along orbit debris self-collecting with rotating constant magnets

Orbital debris with equivalent size of 1∼10 cm is hard to be detected and its number is huge, which is the most harmful to on-orbit spacecraft while there is still no effective removal means. Inspired by vacuum cleaner, applying the conductive property of orbital debris and the physical law that variable magnetic flux would induce eddy current on the surface of a conductive material, this paper put forward a novel approach of ‘along orbit debris self-collecting with rotating constant magnetic dipole array’. Based on dynamic modeling, self-collecting feasibility and control capability analysis, evolution law of relative orbital motion with induced magnetic force derivation, and on-orbit debris magnetic collecting numerical simulation, this novel approach has been verified and the corresponding requirements of magnetic collector are put forward. In addition, this paper also gave how to use several key parameters to adjust the region of along orbit debris self-collecting, and pointed out several problems needing to be researched in-depth for actual application.

DETERMINATION OF THE CONDITIONS FOR THE SAFE APPROACH OF AN INSPECTOR SATELLITE WITH A SPACECRAFT IN EARTH ORBIT

The article is devoted to the study of the movement of the inspector satellite in the vicinity of the observed spacecraft. The inspector satellite is developed in order to monitor the spacecraft, carry out timely maintenance in order to eliminate malfunctions for its further functioning, refueling, towing, etc. One of the most important stages of the satellite is the approach to the spacecraft, which is planned by modeling using the equations of relative orbital motion, which generation is presented in this article. The rendezvous procedure is carried out by two-pulse maneuver. The article also describes conducted numerical experiments modeling the approach under different initial conditions, in which conditions for collision and safe approach of the satellite with the spacecraft, the most convenient observation positions near the spacecraft and the time when it can be observed are determined.

Lyapunov functions and regions of attraction for spherically constrained relative orbital motion

Lyapunov functions and regions of attraction for spherically constrained relative orbital motion

Applicability of Clohessy-wiltshire Model For Optimal Low-thrust Rendezvous

This paper investigates the applicability of the continuous thrust Clohessy-Wiltshire model for the optimal low-thrust rendezvous in the orbit. While simplifying the non-linear equations of relative orbital motion, certain assumptions were taken in account by Clohessy and Wiltshire to linearize the system of equations. When the same model is used for optimization of the trajectory of a relative transfer in the orbit using low-thrust, because of the assumptions in the dynamic model there are errors in the result which in turn depends upon the initial conditions for the optimal control problem. A study has been done to investigate the possible trend of error in the control input by varying the initial conditions for the relative transfer and comparing the results with those obtained using non-linear equations. Further, the trend of error in control input has also been studied by increasing the altitude of the target spacecraft orbit. Optimization of the relative motion trajectory has been performed to account for minimum propellant expenditure.

A Geometrical, Reachable Set Approach for Constrained Pursuit–Evasion Games with Multiple Pursuers and Evaders

This paper presents a solution strategy for deterministic time-optimal pursuit–evasion games with linear state constraints, convex control constraints, and linear dynamics that is consistent with linearized relative orbital motion models such as the Clohessy–Wiltshire equations and relative orbital elements. The strategy first generates polytopic inner approximations of the players’ reachable sets by solving a sequence of convex programs. A bisection method then computes the optimal termination time, which is the least time at which a set containment condition is satisfied. The pursuit–evasion games considered are games with (1) a single pursuer and single evader, (2) multiple pursuers and a single evader, and (3) a single pursuer and multiple evaders. Compared to variational methods, this reachable set strategy leads to a tractable formulation even when there are state and control constraints. The efficacy of the strategy is demonstrated in three numerical simulations for a constellation of satellites in close proximity in low earth orbit.

Rendezvous and Docking of Co-Operative Target with Dual-Axis Gimbaled Electric Thruster

This paper presents a simulation study on the viability of a dual axis gimbaled electric thruster for the relative orbital position control. Relative attitude of the spacecraft is controlled using reaction wheels (RWs). This study considers proportional derivative (PD) algorithm for orbital position control and nonlinear dynamic inversion for relative attitude control. The relative attitude dynamics is derived assuming that the target spacecraft is cooperative and is inactive, therefore, its relative attitude changes only due to its orbital motion around the central body. The dual-gimbal dynamics is approximated for control design due to coupling between gimbal rates. Numerical simulation results are presented to show the efficacy of the control algorithm as well the viability of the dual-axis gimbaled thruster for relative orbital motion control.

A Full-Body Relative Orbital Motion of Spacecraft Using Dual Tensor Algebra and Dual Quaternions

This paper proposes a new non-linear differential equation for the six degrees of freedom (6-DOF) relative rigid bodies motion. A representation theorem is provided for the 6-DOF differential equation of motion in the arbitrary non-inertial reference frame. The problem of the 6-DOF relative motion of two spacecraft in the specific case of Keplerian confocal orbits is proposed. The result is an analytical method without secular terms and singularities. Tensors dual algebra and dual quaternions play a fundamental role, with the solution representation being the relative problem. Furthermore, the representation theorems for the rotation and translation parts of the 6-DOF relative orbital motion problems are obtained.

The Binary and the Disk: The Beauty is Found within NGC3132 with JWST

The planetary nebula (PN) NGC 3132 is a striking example of the dramatic but poorly understood mass-loss phenomena that (1–8) M ⊙ stars undergo during their death throes as they evolve into white dwarfs (WDs). From an analysis of JWST multiwavelength (0.9–18 μm) imaging of NGC 3132, we report the discovery of an extended dust cloud around the WD central star (CS) of NGC 3132, seen most prominently in the 18 μm image, with a surface-brightness-limited radial extent of ≳2″. We show that the A2V star located 1.″7 to CS’s northeast (and 0.75 kpc from Earth) is gravitationally bound to the latter, as evidenced by the detection of relative orbital angular motion of 0.°24 ± 0.°045 between these stars over ∼20 yr. Using aperture photometry of the CS extracted from the JWST images, together with published optical photometry and an archival UV spectrum, we have constructed the spectral energy distribution (SED) of the CS and its extended emission over the UV to mid-IR (0.091–18 μm) range. We find that fitting the SED of the CS and the radial intensity distributions at 7.7, 12.8, and 18 μm with thermal emission from dust requires a cloud that extends to a radius of ≳1785 au, with a dust mass of ∼1.3 × 10−2 M ⊕ and grains that are 70% silicate and 30% amorphous carbon. We propose plausible origins of the dust cloud and an evolutionary scenario in which a system of three stars—the CS, a close low-mass companion, and a more distant A2V star—forms a stable hierarchical triple system on the main sequence but becomes dynamically unstable later, resulting in the spectacular mass ejections that form the current, multipolar PN.

Relative Orbital Motion Control of Spacecraft Based on Multi-Objective Optimization

In this paper, we consider the relative motion of a follower spacecraft orbiting the Earth in a near circular orbit with respect to a leader spacecraft. The relative orbital motion-control problem of the follower spacecraft is studied here. In order to allow the follower spacecraft to succeed in its target orbital motion maneuver, we proposed a multi-objective optimization method to solve the relative orbital motion-control problem. Firstly, a Hill–Clohessy–Wiltshire equation was used to describe the continuous relative orbital motion-control system. Then, the system was discretized into a discrete system using numerical methods. Next, a multi-objective optimization model of the relative orbital motion-control problem was formulated. In the model, two objectives, i.e., the orbital motion error and the energy consumption, were minimized simultaneously. Furthermore, the ε-constraint method was used to solve the multi-objective optimization problem and the Pareto front, which demonstrates that the trade-off between the two objectives can be achieved. Finally, numerical experiments were carried out to validate the effectiveness of the proposed multi-objective optimization approach.

Convex Optimization-Based Techniques for Trajectory Design and Control of Nonlinear Systems with Polytopic Range

This paper presents new techniques for the trajectory design and control of nonlinear dynamical systems. The technique uses a convex polytope to bound the range of the nonlinear function and associates with each vertex an auxiliary linear system. Provided controls associated with the linear systems can be generated to satisfy an ordering constraint, the nonlinear control is computable by the interpolation of controls obtained by convex optimization. This theoretical result leads to two numerical approaches for solving the nonlinear constrained problem: one requires solving a single convex optimization problem and the other requires solving a sequence of convex optimization problems. The approaches are applied to two practical problems in aerospace engineering: a constrained relative orbital motion problem and an attitude control problem. The solve times for both problems and approaches are on the order of seconds. It is concluded that these techniques are rigorous and of practical use in solving nonlinear trajectory design and control problems.

Strong Observability of LTV Systems With Feedthrough and On-Orbit Reconnaissance and Evasion Applications

This letter considers linear time-varying control systems with feedthrough. A necessary and sufficient condition for strong observability (also known as observability with unknown inputs) is stated in terms of the observability matrix and newly redefined invertibility matrix. This is followed by an observer for pointwise reconstruction of the state from the output and its time derivatives. A variable-time weakly unobservable subspace is then introduced such that the rank test can be recast in terms of this subspace and the kernel of a certain matrix. The subspace characterization leads to results on the existence and construction of unobservable state and control functions. The theoretical results are illustrated in an on-orbit reconnaissance application. Within the context of a time-varying relative orbital motion model, it is shown that strong observability is satisfiable with any constant or bounded feedthrough matrix. In the absence of strong observability, control functions to evade observation are constructed.

Equilibria, periodicity, and chaotic behavior in spherically constrained relative orbital motion

Equilibria, periodicity, and chaotic behavior in spherically constrained relative orbital motion

A Mass–Magnitude Relation for Low-mass Stars Based on Dynamical Measurements of Thousands of Binary Star Systems

Stellar mass is a fundamental parameter that is key to our understanding of stellar formation and evolution, as well as the characterization of nearby exoplanet companions. Historically, stellar masses have been derived from long-term observations of visual or spectroscopic binary star systems. While advances in high-resolution imaging have enabled observations of systems with shorter orbital periods, measurements of stellar masses remain challenging, and relatively few have been precisely measured. We present a new statistical approach to measuring masses for populations of stars. Using Gaia astrometry, we analyze the relative orbital motion of >3800 wide binary systems comprising low-mass stars to establish a mass–magnitude relation in the Gaia G RP band spanning the absolute magnitude range 14.5 > > 4.0, corresponding to a mass range of 0.08 M ⊙ ≲ M ≲ 1.0 M ⊙. This relation is directly applicable to >30 million stars in the Gaia catalog. Based on comparison to existing mass–magnitude relations calibrated for K s magnitudes from the Two Micron All Sky Survey, we estimate that the internal precision of our mass estimates is ∼10%. We use this relation to estimate masses for a volume-limited sample of ∼18,200 stars within 50 pc of the Sun and the present-day field mass function for stars with M ≲ 1.0 M ⊙, which we find peaks at 0.16 M ⊙. We investigate a volume-limited sample of wide binary systems with early-K dwarf primaries, complete for binary mass ratios q > 0.2, and measure the distribution of q at separations >100 au. We find that our distribution of q is not uniform, rather decreasing toward q = 1.0.

Relative Orbital Motion of a Charged Object Near a Spaceborne Radially Directed Rotating Magnetic Dipole

Relative Orbital Motion of a Charged Object Near a Spaceborne Radially Directed Rotating Magnetic Dipole

Role of Topocentric Parallax in Near-Earth Object Initial Orbit Determination

Near-Earth Object (NEO) initial orbit determination typically uses astrometric measurements during a close approach over a time window much shorter than the orbital period of the NEO. The initial orbit is only weakly determined with dominant uncertainties in the distance of the NEO from the Earth. Topocentric astrometric measurements allow us to estimate NEO distances using observed nonlinear motions of the NEOs relative to observers, which come from the relative orbital motion of the NEOs to the Earth plus the topocentric parallax (parallax) from the diversity of observatory locations relative to the Earth center. We calculate the ratio of the contributions to the nonlinear motion from the relative orbital motion and the parallax to be approximately (TΔ/(day au))2, where T is the arc length measured in days and Δ is the distance of close approach. The dominant nonlinear motion for ranging the NEO comes from the relative orbital motion of the NEO to the Earth center, due to mainly the differential solar gravitational acceleration, when TΔ ≳ 1 day au and the parallax when TΔ ≲ 1 day au. This is confirmed by simulation data and supported by observational data of real NEOs. In the regime TΔ ≲ 1 day au, the orbit determination uncertainties are inversely proportional to the amplitude of the parallax. Introducing diversities of hour angles and observatory latitudes (especially alternating between extreme values) into scheduled follow-up observations can improve the parallax amplitude, thus the orbit accuracy. Most of the newly discovered NEOs are in this regime, we recommend optimizing parallax by properly scheduling observations when the NEO is very close to the Earth and using synthetic tracking to improve astrometry accuracy for initial orbit determination.

Misaligned Circumstellar Disks and Orbital Motion of the Young Binary XZ Tau

Abstract We report our analyses of the multi-epoch (2015–2017) Atacama Large Millimeter/submillimeter Array (ALMA) archival data of the Class II binary system XZ Tau at Bands 3, 4, and 6. The millimeter dust-continuum images show compact, unresolved (r ≲ 15 au) circumstellar disks (CSDs) around the individual binary stars, XZ Tau A and B, with a projected separation of ∼39 au. The 12CO (2–1) emission associated with those CSDs traces the Keplerian rotations, whose rotational axes are misaligned with each other (P.A. ∼ −5° for XZ Tau A and ∼130° for XZ Tau B). The similar systemic velocities of the two CSDs (V LSR ∼ 6.0 km s−1) suggest that the orbital plane of the binary stars is close to the plane of the sky. From the multi-epoch ALMA data, we have also identified the relative orbital motion of the binary. Along with the previous NIR data, we found that the elliptical orbit (e = 0.742 − 0.034 + 0.025 , a = 0 .″ 172 − 0 .″ 003 + 0 .″ 002 , and ω = − 54 .° 2 − 4 .° 7 + 2 .° 0 ) is preferable to the circular orbit. Our results suggest that the two CSDs and the orbital plane of the XZ Tau system are all misaligned with each other, and possible mechanisms to produce such a configuration are discussed. Our analyses of the multi-epoch ALMA archival data demonstrate the feasibility of time-domain science with ALMA.

Discovery and modelling of broad-scale plasma lensing in black-widow pulsar J2051 − 0827

ABSTRACT We report on an unusually bright observation of PSR J2051 − 0827 recorded during a regular monitoring campaign of black-widow pulsar systems with the Effelsberg 100-m telescope. Through fortunate coincidence, a particularly bright scintillation maximum is simultaneous with the eclipse by the companion, enabling precise measurements of variations in the flux density, dispersion measure (DM), and scattering strength throughout the eclipse. The flux density is highly variable throughout the eclipse, with a peak 1.7 times the average away from the eclipse, and yet does not significantly decrease on average. We recover the flux density variations from the measured DM variations using geometric optics, with a relative velocity as the only free parameter. We measure an effective velocity of 470 ± 10 km s−1, consistent with the relative orbital motion of the companion, suggesting that the outflow velocity of the lensing material is low, or is directly along the line of sight. The $2{{\ \rm per\ cent}}$ uncertainty on the effective velocity is a formal error; systematics related to our current model are likely to dominate, and we detail several extensions to the model to be considered in a full treatment of lensing. This is a demonstration of the causal link between DM and lensing; the flux density variations can be predicted directly through the derivatives of DM. Going forward, this approach can be applied to investigate the dynamics of other eclipsing systems, and to investigate the physical nature of scintillation and lensing in the ionized interstellar medium.

Mathematical modeling and simulation of nonlinear spacecraft rendezvous and formation flying problems via averaging method

The relative orbital motion problem of a deputy spacecraft with respect to a chief spacecraft is, generally, described using a set of differential equations governing the motion of the spacecraft relative to each other instead of describing their motion, separately, relative to the Earth. In this paper, new time-varying, time periodic cubic approximation model of spacecraft relative motion is developed from the original nonlinear relative equations of motion with the chief in elliptical orbit. Afterwards, averaging method is applied to the cubic model to obtain asymptotic approximations and periodic solutions. The formulation of the averaging solutions using averaging method affords us the opportunity to have a better insight of the relative motion dynamics. From the numerical simulations, the averaging model is in close agreement with the nonlinear model. This can be attributed to the inclusion of cubic nonlinear terms. The model is amenable to the long-term prediction of the behavior of the relative motion and useful for spacecraft formation flying analysis.

Two-epoch Orbit Estimation for Wide Binaries Resolved in Hipparcos and Gaia

Abstract The Hipparcos catalog and its Double and Multiple System Annex (DMSA) lists 4099 components with individual proper motions and coordinates on the epoch 1991.25. Many of these long-period binary stars are also present in Gaia Data Release 2 (DR2). Using the available relative positions and proper motions separated by 24.25 yr, the equations of relative orbital motion can be solved for the two epoch eccentric anomalies, orbital period, and eccentricity. This method employs elimination of the linear Thiele–Innes unknowns and nonlinear optimization of the remaining condition equations. The quality of these solutions is compromised by the insufficient condition and modest precision of the Hipparcos astrometric data, as revealed by Monte Carlo simulations with artificially perturbed data points. The presence of multiple systems and optical pairs can also perturb the results. Limited experiments with artificial data indicate that useful estimates can be obtained with a 25 yr epoch difference for wide binaries with orbital periods up to ∼500 yr. The prospects of this method dramatically improve with the proposed next-generation space astrometry missions such as Gaia-NIR and Theia, especially when additional conditions are included from astrometric or spectroscopic measurements. An ancillary catalog of cross-identification and astrometric information for 1295 double-star pairs cross-matched in Gaia DR2 and Hipparcos is also published.

Orbital Path and Spacecraft Attitude Correction for the MODIS Lunar Spatial Characterization

For the Moderate Resolution Imaging Spectroradiometer (MODIS) on the Terra and Aqua platforms, regularly scheduled lunar observations using spacecraft roll maneuvers have been used extensively for sensor characterization. While the primary purpose of these observations is for radiometric calibration of the reflective solar bands, they have also been leveraged for a number of other sensor performance assessments, such as the band-to-band spatial registration (BBR) and detector-to-detector spatial registration (DDR). The spatial registration calculations are complicated by the fact that the Moon does not move in a straight path across the sensor field of view (FOV). This path is determined by the relative orbital motion between the spacecraft and the Moon and the instrument attitude error that results from the roll maneuver. In this article, we develop a correction for the MODIS lunar spatial characterization measurements by calculating the predicted path of the Moon across the sensor FOV using spacecraft and lunar ephemeris data to model the relative orbital motion between the spacecraft and the Moon along with spacecraft attitude error data acquired during the roll maneuver. The difference between the measured and predicted positions of the Moon in the MODIS FOV can be used to calculate the BBR and DDR results. Since the predicted path across the FOV will be the same for each band, the BBR results will be minimally affected. However, we will show that the along-scan spread in the DDR can be significantly reduced, which results in a much greater consistency throughout the full mission for both Aqua and Terra MODIS.