Circular Restricted Three-body Model Research Articles

The effect of zonal harmonics on dynamical structures in the circular restricted three-body problem near the secondary body

The circular restricted three-body model is widely used for astrodynamical studies in systems where two major bodies are present. However, this model relies on many simplifications, such as point-mass gravity and planar, circular orbits of the bodies, and limiting its accuracy. In an effort to achieve higher-fidelity results while maintaining the autonomous simplicity of the classic model, we employ zonal harmonic perturbations since they are symmetric about the z-axis, thus bearing no time-dependent terms. In this study, we focus on how these perturbations affect the dynamic environment near the secondary body in real systems. Concise, easily implementable equations for gravitational potential, particle motion, and modified Jacobi constant in the perturbed model are presented. These perturbations cause a change in the normalized mean motion, and two different formulations are addressed for assigning this new value. The shifting of collinear equilibrium points in many real systems due to J_2 of each body is reported, and we study how families of common periodic orbits—Lyapunov, vertical, and southern halo—shift and distort when J_2, J_4, and J_6 of the primary and J_2 of the secondary body are accounted for in the Jupiter–Europa and Saturn–Enceladus systems. It is found that these families of periodic orbits change shape, position, and energy, which can lead to dramatically different dynamical behavior in some cases. The primary focus is on moons of the outer planets, many of which have very small odd zonal harmonic terms, or no measured value at all, so while the developed equations are meant for any and all zonal harmonic terms, only even terms are considered in the simulations. Early utilization of this refined CR3BP model in mission design will result in a more smooth transition to full ephemeris model.

Solution set calculation of the Sun-perturbed optimal two-impulse trans-lunar orbits using continuation theory

The solution set of the Sun-perturbed optimal two-impulse trans-lunar orbit is helpful for overall optimization of the lunar exploration mission. A model for computing the two-impulse trans-lunar orbit, which strictly satisfies the boundary constraints, is established. The solution set is computed first with a circular restricted three-body model using a generalized local gradient optimization algorithm and the strategy of design variable initial continuation. By taking the solution set of a circular restricted three-body model as the initial values of the design variables, the Sun-perturbed solution set is calculated based on the dynamic model continuation theory and traversal search methodology. A comparative analysis shows that the fuel cost may be reduced to some extent by considering the Sun’s perturbation and choosing an appropriate transfer window. Moreover, there are several optimal two-impulse trans-lunar methods for supporting a lunar mission to select a scenario with a certain ground measurement and to control the time cost. A fitted linear dependence relationship between the Sun’s befitting phase and the trans-lunar duration could thus provide a reference to select a low-fuel-cost trans-lunar injection window in an engineering project.

Low thrust Earth–Moon transfer trajectories via lunar capture set

A cislunar cargo spacecraft with low-thrust propulsion traveling between the Earth and the Moon is essential for sustainable, long-term manned lunar exploration. In low-thrust Earth–Moon transfer (LTEMT), lunar capture is the primary prerequisite for spacecraft subject to the circular restricted three-body model. Therefore, this study identifies sufficient conditions for lunar capture, which are determined by the Jacobi integral and Hill’s region. This paper proposes a guidance scheme that includes thrust direction, thrust efficiency, and a five-stage flight control sequence based on the variation of the Jacobi integral. The LTEMT problem is then converted to an initial value problem of a differential equation with three parameters. Lunar capture set theories (LCSTs), which are convenient for identifying lunar capture sets, are presented and proved according to the continuous properties of the ordinary differential equation. Finally, the solutions of the LTEMT trajectories departing from a geosynchronous orbit with an altitude of approximately 35,827 km are discussed for different thrust accelerations and cut-off values of the thrust efficiency. The robustness is analyzed assuming that navigation and switching time errors are present to demonstrate the adaptability of this method. The results reveal that the proposed guidance scheme and LCSTs can provide technical support for future cislunar cargo missions.

Inclined asymmetric librations in exterior resonances

Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.

Novel Solar-Sail Mission Concepts for High-Latitude Earth and Lunar Observation

Solar-sail periodic orbits in the Earth–moon circular restricted three-body problem are proposed for continuous observation of the polar regions of the Earth and the moon. The existence of families of solar-sail periodic orbits in the Earth–moon system has previously been demonstrated by the authors and is expanded by introducing additional orbit families. Orbits for near-term solar-sail technology originate by maintaining the solar sail at a constant attitude with respect to the sun such that mission operations are greatly simplified. The results of this investigation include a constellation of two solar-sail -vertical Lyapunov orbits that achieves continuous observation of both the lunar South Pole and the center of the Aitken Basin at a minimum elevation of 15 deg. At Earth, a set of two, clover-shaped orbits can provide continuous coverage of one of the Earth’s poles at a minimum elevation of 20 deg. Results generated in the Earth–moon circular restricted three-body model are easily transitioned to one that includes eccentricity effects and demonstrates that the orbits are feasible in realistic regimes.

Multi-objective transfer to libration-point orbits via the mixed low-thrust and invariant-manifold approach

The multi-objective optimization of transfer trajectories from an orbit near Earth to a periodic libration-point orbit in the Sun–Earth system using the mixed low-thrust and invariant-manifold approach is investigated in this paper. A two-objective optimization model is proposed based on the mixed low-thrust and invariant-manifold approach. The circular restricted three-body model (CRTBP) is utilized to represent the motion of a spacecraft in the gravitational field of the Sun and Earth. The transfer trajectory is broken down into several segments; both low-thrust propulsion and stable manifolds are utilized based on the CRTBP in different segments. The fuel cost, which is generated only by the low-thrust trajectory for transferring the spacecraft from an orbit near Earth to a stable manifold, is minimized. The total flight time, which includes the time during which the spacecraft is controlled by the low-thrust trajectory and the time during which the spacecraft is moving on the stable manifold, is also minimized. Using the nondominated sorting genetic algorithm for the resulting multi-objective optimization problem, highly promising Pareto-optimal solutions for the transfer of the spacecraft are found. Via numerical simulations, it is shown that tradeoffs between time of flight and fuel cost can be quickly evaluated using this approach. Furthermore, for the same time of flight, transfer trajectories based on the mixed-transfer method can save a larger amount of fuel than the low-thrust method alone.

Leveraging Resonant-Orbit Manifolds to Design Transfers Between Libration-Point Orbits

Resonant orbits have been widely employed in mission design for planetary flyby trajectories (Jupiter Europa Orbiter) and, more recently, as a source of long-term stability (Interstellar Boundary Explorer). Yet, resonant orbits have not been explored extensively as transfer mechanisms between nonresonant orbits in multibody systems. To highlight the benefit of employing resonant orbits for transfers and given the increased interest in employing libration-point orbits for a variety of purposes, planar and three-dimensional transfers from a low Earth orbit to the vicinity of the Earth–moon libration points via resonant arcs are constructed. Solutions are efficiently generated in the circular restricted three-body model, and transitioned to a higher-fidelity model for validation. Direct optimization techniques are applied to further reduce the propellant requirements, and a system translation process is defined to allow for the quick translation of Earth–moon transfers to other three-body systems.

Optimal Free-Return Trajectories to Near-Earth Asteroids

A method is presented to calculate optimal trajectories to near-Earth asteroids that incorporate free-return trajectory segments. These trajectories are useful because they return to Earth with no fuel penalty, which is useful for risk mitigation on manned missions. The algorithm minimizes total velocity change over an entire nominal mission, encompassing a single impulsive maneuver each for Earth departure, leaving the free-return trajectory, rendezvous with and departure from the asteroid, culminating with unpowered reentry into Earth’s atmosphere. Two example cases are presented in the circular restricted three-body model: a mission to 2000 SG344 from 23 August 2028 to 9 October 2029 with a cost of , and a mission to 2006 BZ147 from 9 February 2033 to 21 December 2033 with a cost of . The results presented here showcase the usage of the algorithm, which may be applied to free-return trajectories to a wide variety of near-Earth objects.

Optimal Lunar Orbit Insertion from a Variable Symmetric Free-Return Trajectory

A method for the optimization of a multiple-impulse lunar orbit insertion sequence from a symmetric free-return trajectory is presented. Both the inclination and altitude at lunar periapsis along the free return are variable. The algorithm systematically optimizes the orbit insertion sequence while simultaneously generating a free-return outbound trajectory. With a direct multiple-shooting method, optimization is performed in the circular restricted three-body model with a constrained sequential quadratic programming algorithm. The necessary gradients are derived with linear perturbation theory, and a closed-form expression is derived for computing gradients along a general n-impulse trajectory. Examples are presented for two classes of symmetric free returns, and a range of lunar orbit orientations is examined to assess the impact on velocity impulse requirements.

Computation and Optimization of Lunar Orbit Insertion from a Fixed Free Return

In support of future human missions to the Moon, the present research combines free return trajectories with lunar orbit insertion (LOI) sequences to provide both crew safety and selenographic versatility. Because future missions necessitate global lunar access, the required spacecraft plane change at the Moon may be large enough that a multi-maneuver sequence offers velocity impulse cost savings. Approximating the Earth-Moon system with the circular restricted three-body model, this paper develops a targeting and optimization procedure to transfer a spacecraft from a lunar free return trajectory to a closed lunar orbit through a multi-maneuver LOI sequence. Thus, a safe Earth-return for crew is guaranteed while not compromising the ability to achieve any lunar orbit.