Halo Orbit Research Articles

Solar sail transfer strategy for long-duration multi-objective mission in cislunar space utilizing solar radiation and reflectivity control device

Abstract This study delves into the complexities of optimizing solar sail trajectories for long-duration multi-objective missions in cislunar space, focusing on orbital transfers between halo orbits and distant retrograde orbits. It presents an innovative transfer strategy that integrates solar radiation propulsion and reflectivity control devices to effectively adjust thrust force magnitude and direction, thereby improving maneuver capabilities. The optimization method is founded on the coupled dynamics of a solar sail, aiming to minimize the transfer time of flight using the Hermite-Simpson collocation method while ensuring optimal performance. Through thorough analysis and comparison, the research strives to determine the most efficient trajectory for solar sail orbital transfers in cislunar space.

A note on the computation of multi-revolution NRHO under the ephemeris model

Near Rectilinear Halo Orbits (NRHOs) are vital to manned lunar and deep space exploration, which nowadays are of great interest for different space agencies and in particular with regard to the future space station. However, the required computation of multi-revolution NRHO under the ephemeris model is difficult, especially for the NRHOs with low periapsis relative to the secondary body. This paper explores this issue from the perspective of multiple shooting, first analyzing the influence of the state transition matrices by means of their condition number and then, focusing on a good selection of trajectory segments with suitable patch points. The methodology considerably improves the convergence and the computation under the ephemeris model. Numerical simulations show that at least 30 revolutions can be achieved for NRHOs with perilune radius of less than 12,000 km around L1 or period less than 8.8 days around L2; meanwhile, the number of segments used can be as low as 2 for each single revolution. As for the position of patch points, the first and last points of each revolution should be apart from the Moon, and the distance can be reduced only when the number of segments increases. The proposed method requires no dedicated optimization algorithm or commercial software to produce the multi-revolution NRHOs.

Semi-analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

A unified semi-analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits, quasihalo orbits, Axial orbits, and their invariant manifolds, as well as transit and non-transit orbits. Based on classical in-plane and out-of-plane frequency resonance mechanisms, the Lindstedt–Poincaré method could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. In this paper, by introducing a coupling coefficient η and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt–Poincaré method. Analyzing the bifurcation equation obtained from different coupling forms reveals bifurcation conditions for all kinds of orbits near collinear libration points. When η = 0, the series solution describes non-bifurcated orbits, while when η ≠ 0, the solution describes bifurcated orbits, including quasihalo orbits, Axial orbits, and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified semi-analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.

Autonomous Lunar rendezvous trajectory planning and control using nonlinear MPC and Pontryagin’s principle

This paper explores the application of Nonlinear Model Predictive Control (NMPC) techniques, based on the Pontryagin Minimum Principle, for a minimum-propellant autonomous rendezvous maneuver in non-Keplerian Lunar orbits. The relative motion between the chaser and the target is described by the nonlinear dynamics of the circular restricted three body-problem, posing unique challenges due to the complex and unstable dynamics of near-rectilinear halo orbits. Key aspects of the proposed NMPC include trajectory optimization, maneuver planning, and real-time control, leveraging on its ability to satisfy complex mission requirements while ensuring safe and efficient spacecraft operations and in the presence of input and nonlinear/non-convex state constraints. The proposed formulation allows the design of a minimum-propellant controller, whose optimal control signal results to be bang–bang in time. A case study based on the Artemis III mission – where the docking of the Orion spacecraft to the Gateway station is planned – is illustrated in order to demonstrate the efficiency of the proposed approach, showcasing its potential for enhancing target tracking accuracy, while reducing propellant consumption.

Bifurcation Mechanism of Quasi-Halo Orbit from Lissajous Orbit

This paper presents a general analytical method to describe the center manifolds of collinear libration points in the restricted three-body problem (RTBP). It is well known that these center manifolds include Lissajous orbits, halo orbits, and quasi-halo orbits. Previous studies have traditionally treated these orbits separately by iteratively constructing high-order series solutions using the Lindstedt–Poincaré method. Instead of relying on resonance between their frequencies, this study identifies that halo and quasi-halo orbits arise due to intricate coupling interactions between in-plane and out-of-plane motions. To characterize this coupling effect, a novel concept, coupling coefficient η, is introduced in the RTBP, incorporating the coupling term ηΔx in the z-direction dynamics equation, where Δ represents a formal power series concerning the amplitudes. Subsequently, a uniform series solution for these orbits is constructed up to a specified order using the Lindstedt–Poincaré method. For any given paired in-plane and out-of-plane amplitudes, the coupling coefficient η is determined by the bifurcation equation Δ=0. When η=0, the proposed solution describes Lissajous orbits around libration points. As η transitions from zero to nonzero values, the solution describes quasi-halo orbits, which bifurcate from Lissajous orbits. Particularly, halo orbits bifurcate from planar Lyapunov orbits if the out-of-plane amplitude is zero. The proposed method provides a unified framework for understanding these intricate orbital behaviors in the RTBP.

Aerogel-based collection of ejecta material from asteroids from libration point orbits: Dynamics and capture design

Scientific interest in asteroids and their physical characteristics is growing. These bodies provide insights into the primordial solar system and represent a valuable source of metals, silicates, and water. Several missions over the past few years have aimed to improve and better identify the main properties of these poorly known celestial bodies. However, these missions relied on touchdown(s) on the target asteroid to gather samples, which is complicated owing to the difficulty of accurately reaching and rendezvousing with the body. This study aims to assess the feasibility of an in-orbit asteroid sample collection mission. Such a strategy could prevent complex operations related to landing and touchdown maneuvers and avoid the dead times present in a mission requiring several landings. The presented collection scenario, which focuses on the asteroid Ryugu, proposes gathering samples using a spacecraft injected into a halo orbit around the second libration point, L2. For this purpose, the orbits in the neck region of the zero velocity curves are analyzed. A novel methodology to characterize bouncing behavior is introduced. An interpolation-based approach was used to recover the appropriate restitution coefficients for each collision occurring at a specific impact angle. This was applied to both the rigid body model and the point mass approximation studied for two different sites on the asteroid. Furthermore, the study enlarged the region of interest from only L2 to its neighboring zones to return a more global and realistic point of view. Considering the solar radiation pressure and asteroid aspherical potential, particles of different sizes ejected from different longitudes and with different ejection angles were classified according to their trajectories to finally build a database. Based on this analysis, an aerogel-based collection strategy inspired by that used in the Stardust-NExT (NASA) mission was investigated to assess its possible applicability to the analyzed scenario.

Extended perilune rendezvous method for low Delta-V transition to NRHO for cargo transportation mission to gateway

The construction of Gateway in lunar orbit is planned as a post-ISS mission by NASA, and JAXA is considering a cargo supply mission to provide service. The Near-Rectilinear Halo Orbit (NRHO) is a candidate for a Gateway orbit, and several methods for the transfer trajectory are being researched. The In-Direct Transfer (IDT) has a short transition period of seven days, but the ΔV application is inefficient. The Weak Stability Boundary (WSB) transfer maintains a small ΔV, but the transition takes over 100 days. However, a trajectory intermediate between IDT and WSB has not yet been actively studied. This paper proposes the Extended Perilune Rendezvous Method (EPRM) of intermediate transfer to NRHO. This method reduces ΔV by orbiting in an elliptical lunar orbit for over two weeks. EPRM has several transition methods corresponding to ΔV and transition periods. This paper systematically reveals the transition mechanisms. Moreover, an effective TCM, considering several error types, is presented for an actual mission. Finally, the rendezvous accuracy with Gateway is discussed.

The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating–pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we implement a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.

A population of neutron star candidates in wide orbits from Gaia astrometry

We report discovery and spectroscopic follow-up of 21 astrometric binaries containing solar-type stars and dark companions with masses near 1.4 M⊙. The simplest interpretation is that the companions are dormant neutron stars (NSs), though ultramassive white dwarfs (WDs) and tight WD+WD binaries cannot be fully excluded. We selected targets from Gaia DR3 astrometric binary solutions in which the luminous star is on the main sequence and the dynamically-implied mass of the unseen companion is (a) more than 1.25M⊙ and (b) too high to be any non-degenerate star or close binary. We obtained multi-epoch radial velocities (RVs) over a period of 670 days, spanning a majority of the orbits’ dynamic range in RV. The RVs broadly validate the astrometric solutions and significantly tighten constraints on companion masses. Several systems have companion masses that are unambiguously above the Chandrasekhar limit, while the rest have masses between 1.25 and 1.4 M⊙. The orbits are significantly more eccentric at fixed period than those of typical WD + MS binaries, perhaps due to natal kicks. Metal-poor stars are overrepresented in the sample: 3 out of 21 objects (14%) have [Fe/H] ∼−1.5 and are on halo orbits, compared to ∼0.5% of the parent Gaia binary sample. The metal-poor stars are all strongly enhanced in lithium. The formation history of these objects is puzzling: it is unclear both how the binaries escaped a merger or dramatic orbital shrinkage when the NS progenitors were red supergiants, and how they remained bound when the NSs formed. Gaia has now discovered 3 black holes (BHs) in astrometric binaries with masses above 9 M⊙, and 21 NSs with masses near 1.4M⊙. The lack of intermediate-mass objects in this sample is striking, supporting the existence of a BH/NS mass bimodality over 4 orders of magnitude in orbital period.

Cislunar [formula omitted] and [formula omitted] axial orbits and their applications

Cislunar L4 or L5 axial orbit families are continuous asymmetrical periodic orbits from the figure-8 orbit around L4 or L5 Lagrange point of the Earth–Moon system to the halo-family around the Moon. They provide special locations of lunar relay satellites for the lunar south pole due to their offset locations. In this paper, we investigate the orbital characters and applications of cislunar L4 and L5 axial orbits. A construction method of cislunar L4 and L5 axial orbit families is proposed according to their topological feature in the torus space. Then, lunar relay performances, including coverage and eclipse, are analyzed for different cislunar L4 and L5 axial orbits. Numerical computation indicates that 3:1 resonant L4 and L5 axial orbits are the optimal options in terms of the stability, coverage and eclipse avoidance. Lunar relay performances of a two-satellite constellation deployed in these orbits are compared with that in a near rectilinear halo orbit (NRHO). Numerical results show that the constellation of axial orbits can provide longer contact duration to the lunar equator and southern hemisphere than that of the NRHO. Finally, a station-keeping strategy and a transfer design of cislunar L4 and L5 axial orbits are developed, and their fuel costs are studied.

Design of a Cislunar space navigation constellation based on special long-period orbits

This paper proposes a method for designing a navigation constellation in the Cislunar space. Firstly, the dynamic characteristics of the Cislunar space were analyzed. Northern Halo orbits and southern Halo orbits around L1 and L2 libration point are designed. Then, based on the invariant manifold theory, the homoclinic low-energy transfer trajectories of L1 Halo orbits and the low-energy heteroclinic transfer trajectories between L1 and L2 Halo orbits are provided respectively. Afterthat, two types of special long-period orbits are attained for the constellation design. Based on the geometric dilution of precision and the minimum number of visible satellites, design requirements for two parts are given. Next, two navigation constellation configurations are proposed. Lastly, simulations are conducted on the performance of two constellation configurations with different number of satellites. Compared to constellations in one orbit, constellations that configured in two orbits require fewer satellites to meet the design requirements. In different lunar transfer orbits, the performance of the dual orbit constellation was validated.

Optimization, guidance, and control of low-thrust transfers from the Lunar Gateway to low lunar orbit

The Lunar Gateway will represent a primary space system useful for the Artemis program, Earth–Moon transportation, and deep space exploration. It is expected to serve as a staging location and logistic outpost on the way to the lunar surface. This study focuses on low-thrust transfer dynamics, from the Near-Rectilinear Halo Orbit traveled by Gateway to a specified Low-altitude Lunar Orbit (LLO). More specifically, this research addresses two closely-related problems: (i) determination of the minimum-time low-thrust trajectory and (ii) design, implementation, and testing of a guidance and control architecture, for a space vehicle that travels from Gateway to LLO. Orbit dynamics is described in terms of modified equinoctial elements, with the inclusion of all the relevant perturbations, in the context of a high-fidelity multibody ephemeris model. The minimum-time trajectory from Gateway to a specified lunar orbit is detected through an indirect heuristic approach, which uses the analytical conditions arising in optimal control theory in conjunction with a heuristic technique. However, future missions will pursue a growing level of autonomy, and this circumstance implies the mandatory design and implementation of an efficient feedback guidance scheme, capable of compensating for nonnominal flight conditions. This research proposes nonlinear orbit control as a viable and effective option for autonomous explicit guidance of low-thrust transfers from Gateway to LLO. This approach allows defining a feedback law that enjoys quasi-global stability properties without requiring any offline reference trajectory. The overall spacecraft dynamics is modeled and simulated, including attitude control and actuation. The latter is demanded to an array of reaction wheels, arranged in a pyramidal configuration. Guidance, attitude control, and actuation are implemented in an iterative scheme. Monte Carlo simulations demonstrate that the guidance and control architecture at hand is effective in nonnominal flight conditions, i.e. with random starting point from Gateway as well as in case of temporary unavailability of the propulsion system. The numerical results also point out that only a modest propellant penalty is associated with the use of feedback guidance and control in comparison to the minimum-time optimal trajectory.

Thermal-structural and prestressed modal analyses for a solar sail with nonlinear shape memory alloy spring

The objective of this research is to forecast the impact of alternating non-uniform temperature distributions on the deformation and internal stress within the thin film of a solar photon sail, as well as to assess how stress variations affect the sail’s characteristics. To maintain the internal stress amplitude within the film during extreme temperature fluctuations, the solar sail employs an innovative pre-tensioning technique utilizing shape memory alloy (SMA) springs. A thermal-structural model is formulated specifically for a hexagonal solar sail. Subsequently, thermal-structural analyses are conducted to quantify deformations and stresses within the sail during its operation on the Earth-Moon L1 halo orbit. Incorporating the prestress factor, the out-of-plane deflection of the film under solar pressure is computed. Furthermore, prestressed modal analyses are performed to determine mode shapes and natural frequencies, taking into account both the prestress and the solar pressure acting on the film. Finally, numerical results are used to compare the performance of the solar sail in hot and cold environments.

Rapid Approximation of Low-Thrust Spacecraft Reachable Sets within Complex Two-Body and Cislunar Dynamics

The reachable set of controlled dynamical systems is the set of all reachable states from an initial condition over a certain time horizon, subject to operational constraints and exogenous disturbances. In astrodynamics, rapid approximation of reachable sets is invaluable for trajectory planning, collision avoidance, and ensuring safe and optimal performance in complex dynamics. Leveraging the connection between minimum-time trajectories and the boundary of reachable sets, we propose a sampling-based method for rapid and efficient approximation of reachable sets for finite- and low-thrust spacecraft. The proposed method combines a minimum-time multi-stage indirect formulation with the celebrated primer vector theory. Reachable sets are generated under two-body and circular restricted three-body (CR3B) dynamics. For the two-body dynamics, reachable sets are generated for (1) the heliocentric phase of a benchmark Earth-to-Mars problem, (2) two scenarios with uncertainties in the initial position and velocity of the spacecraft at the time of departure from Earth, and (3) a scenario with a bounded single impulse at the time of departure from Earth. For the CR3B dynamics, several cislunar applications are considered, including L1 Halo orbit, L2 Halo orbit, and Lunar Gateway 9:2 NRHO. The results indicate that low-thrust spacecraft reachable sets coincide with invariant manifolds existing in multi-body dynamical environments. The proposed method serves as a valuable tool for qualitatively analyzing the evolution of reachable sets under complex dynamics, which would otherwise be either incoherent with existing grid-based reachability approaches or computationally intractable with a complete Hamilton–Jacobi–Bellman method.

Natural and Forced Spacecraft Loitering in a Near Rectilinear Halo Orbit

Natural and Forced Spacecraft Loitering in a Near Rectilinear Halo Orbit

Investigation of near-rectilinear halo orbit search and rescue using staging [formula omitted]/[formula omitted] Lyapunov and distant retrograde orbit families

Cislunar space is a region of growing interest with nations investing resources to cultivate long presence habitations on the lunar surface. With this increased attention and expansion of missions, both crewed and uncrewed, the likelihood of a mishap or a spacecraft becoming impaired and unable to continue its mission will also increase. The present research adds to the field of cislunar mission operations and trajectory analysis by investigating search and rescue (SAR) operations via rendezvous and proximity operations (RPO) with an impaired notional spacecraft located in a Near-Rectlinear Halo Orbit (NRHO). This research compares the response times of rescuer spacecraft located in sample distant retrograde orbits (DROs) and L1/L2 Lyapunov orbits for the timely far rendezvous with the impaired spacecraft located in the NRHO. This will simulate a variety of far rendezvous whereby the impaired spacecraft’s location within the NRHO and the rescuer spacecraft in the L1/L2 Lyapunov and DRO orbit families are varied. A series of minimum time optimal control problems are posed using the circular restricted three-body problem (CR3BP) dynamics, and pseudospectral methods are used to find solutions given an example maximum ΔV constraint of 3 km/s. The results reinforces our intuition that rendezvous time of flight (TOF) between orbits within the L1, L2, and DRO families and the targeted NRHO correlate with proximity to the NRHO, with the shortest far rendezvous times in each family found to be approximately 6 hours, 4.5 hours, and 10 hours respectively. The results further show that a constellation of two rescue spacecraft could be positioned within the three orbit families to achieve far rendezvous with the chosen NRHO in under one day.

Sun-Earth system Lagrange point spacecraft orbit computation by numerical method and deep learning technique

In this article, the Lagrange points locations are computed in the Rotating Lagrangian Point (RLP) as well as in J2000 inertial reference frames. The well-known Sun-Earth system is considered for the circular restricted three-body problem (CRTBP). In the synodic frame, the Sun-Earth Lagrangian L1 point halo orbit is computed via Differential Correction and Lindstedt-Poincaré (LP) methods. The periodic halo orbits are expressed with respect to barycenter, heliocenter and geocenter, respectively. The deep learning-based model is further explored for the orbit accuracy obtained by the numerical method considering the third, fourth and fifth order LP method.

Study of Nuclear Structure and Elastic Electron Scattering Form Factors for Neutron Halo Nuclei (22N ,23O and 24F) Using Full Correlation Functions with Two Size Parameters

The study of neutron-rich nuclei's form factors, root-mean-square radius (rms), and nuclear density distributions is the focus of this work for the nuclei (22N,23 O and 24F). With the use of a strong short-range effect and a strong tensor force, the nucleons distribution function of the two oscillating harmonic particles in a two-frequency shell model operates with two different parameters: the first (bcore), for the inner (core) orbits, and the second (bvalence) for the outer (halo) orbits. This work demonstrated the existence of neutron halo nuclei for the nuclei (22N, 23O and 24F) in the (2s1/2) shell. The computed density distribution of neutron, proton and matter for these nuclei displayed the long tail performance. Using the Borne approximation of the plane wave, the elastic form factor of the electron scattering from the alien nucleus was calculated, this form factor is independent of the neutrons that make up the halo, but rather it results from a difference in the proton density distribution of the last proton in the nuclei. Fortran 95 power station program was used to the neutron, proton and matter density, elastic electron scattering form factor, and rms radii. The calculated outcomes for these exotic nuclei were in good agreement with the experimental data.

The Moon as an effective propellant source: A comprehensive exergy analysis from extraction to depot

Establishing a permanent lunar base has gained increasing attention since it offers opportunities for international cooperation and the commercialization of space, forming the foundation and testing ground for a human existence independent from Earth. Essential to future missions beyond cislunar space is the exploration and in situ processing of the Moon's resources, especially the sustainable production of energetic resources and propellants. Utilizing in situ generated propellants can dramatically reduce transportation costs by removing the need to source propellants from Earth. Resources on the Moon are limited, and the extraction of available resources are energy-intensive processes demanding advanced techniques and technologies. Consequently, one of the biggest challenges lies in developing process architectures with a positive energy balance, for which comprehensive analyses are still missing. The focus currently lies on the extraction of water ice from lunar regolith and the production of hydrogen and oxygen through water electrolysis. However, alternative fuel and process options may reduce the energy cost while providing equivalent energetic revenue. In the scope of this research, the infrastructure and technologies required for extraction, refining, and storing are assumed to exist in cislunar space; therefore, only the operating cost is considered. Exergy analyses of in situ extraction methods are conducted to investigate whether the required energetic budget allows sustainable implementation. The analysis includes extraction methods and propellant options to reveal the extent to which alternatives to hydrogen are feasible. Exergy analyses determine thermodynamic losses of energy flows giving the ground for process optimization. The exergy destructed represents the margin of improvement within the process architecture and thus reflects the process's thermodynamic and economic value while allowing a more distinct examination of energy use. Assuming the availability of water and carbon dioxide ice in permanently shadowed regions, the analysis shows that choosing methane instead of hydrogen in combination with oxygen as propellants can reduce the required exergy input by up to a third. An example mission allows to directly compare the operating cost of the extraction processes for the different propellant options. The mission entails a spacecraft propelled by a liquid bipropellant engine utilizing the extracted propellant and transporting a payload of the same propellant to a depot located in lunar near-rectilinear halo orbit (NRHO). Although abundant in space, the results suggest that hydrogen may not be the only or even energetically cost-effective resource for developing cislunar and Martian space infrastructures. Likewise, sustainable extraction of propellants suitable for current and future propulsion systems will foster humanity's reach further into the solar system.

Variable-fidelity sensors and observer uncertainty using touring multi-body periodic orbits to conduct cislunar SSA: preliminary study

An accelerating interest in cislunar space and lunar orbit for civilian, commercial, and scientific missions requires a space situational awareness (SSA) architecture extending beyond geosynchronous orbit to promote space traffic management and safety. Space-based SSA in cislunar space is challenging due to difficulties associated with accurately estimating the position of the surveillance satellite, which is a foundational requirement for effectively performing the general SSA mission. Using multiple surveillance satellites with lower-fidelity sensors helps alleviate these concerns by aggregating multiple data sets with higher variance to achieve the same level or potentially improved accuracy as compared to fewer higher-quality sensors. A subset of Earth–Moon periodic orbits, herein identified as “touring” orbits, are used for an optical surveillance constellation with a target resident space object (RSO) in a L1 Halo orbit. Angles-only measurement data are processed utilizing an extended Kalman filter to estimate the position of the RSO. The analysis focuses on assessing the effectiveness of different numbers of surveillance satellites using touring cislunar periodic orbits for conducting the SSA mission relative to L1. Overall, this study finds that the use of an SSA constellation with low-fidelity sensors can match the performance achieved by a constellation featuring higher-fidelity sensors and reduced observer uncertainty for the observer orbits examined.