Distant quasi-periodic orbits around Mercury

In this paper, distant quasi-periodic orbits around Mercury are studied for future Mercury missions. All of these orbits have relatively large sizes, with their altitudes near or above the Mercury sphere of influence. The research is carried out in the framework of the elliptic restricted three-body problem (ER3BP) to account for the planet’s non-negligible orbital eccentricity. Retrograde and prograde quasi-periodic trajectories in the planar ER3BP are generalized from periodic orbits in the CR3BP by the homotopy algorithm, and the shape evolution of such quasi-periodic trajectories around Mercury is investigated. Numerical simulations are performed to evaluate the stability of these distant orbits in the long term. These two classes of orbits present different characteristics: retrograde orbits can maintain shape stability with a large size, although the trajectories in some regions may oscillate with larger amplitudes; for prograde orbits, the range of existence is much smaller, and their trajectories easily move away from the vicinity of Mercury when the orbits become larger. Distant orbits can be used to explore the space environment in the vicinity of Mercury, and some orbits can be taken as transfer orbits for low-cost Mercury return missions or other programs for their high maneuverability.     

Numerical study of the three-dimensional transit orbits in the circular restricted three-body problem

Transit orbits are defined as the trajectories that can pass through the neck region of the zero velocity surface in the circular restricted three-body problem (CR3BP). The low-energy transfers in the CR3BP or between two CR3BPs are always through the instrumentality of the transit orbits. In this paper, the distribution of the transit orbits in the six-dimensional phase space is explored by using numerical methods. The necessary and sufficient condition of transition is introduced, which defines the distribution of the transit orbits by using the manifolds of the vertical and horizontal Lyapunov orbits and the transit cones. The relationship between the manifolds of the libration point orbits and the boundary of the transit orbits is discovered. By using this relationship, a fast algorithm for detecting the boundary of the transit orbits is developed. Moreover, this boundary is parametrized by using Fourier series, which makes easy to use the conclusions of this paper in future trajectory optimization and mission design. All the analyses in this paper are based on the Sun?CEarth CR3BP, but the methods introduced here can be extended to any CR3BPs.     

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.     

Zonal harmonics of azimuthally averaged potential of rotating inhomogeneous ellipsoids

The distant retrograde orbits (DROs) can serve as the parking orbits for a long-term cis-lunar space station. This paper gives a comprehensive study on the transfer problem from DROs to Earth orbits, including low Earth orbits (LEOs), medium Earth orbits (MEOs), and geosynchronous orbits (GSOs), in the bicircular restricted four-body problem (BR4BP) via optimizations within a large solution space. The planar transfer problem is firstly solved by grid search and optimization techniques, and two types of transfer orbits, direct ones and low-energy ones, are both constructed. Then, the nonplanar transfer problem to Earth orbits with inclinations between 0 and 90 degrees are solved via sequential optimizations based on the planar transfers. The transfer characteristics in the cases of different destination orbit inclinations are discussed for both the direct and the low-energy transfer orbits. The important role of the lunar gravity in the low-energy transfers is also discussed, which can overcome the increase of transfer cost caused by the high inclination of Earth orbits. The distinct features of different transfer scenarios, including multiple revolutions around the Earth and Moon, the exterior phase, and the lunar flyby, are discovered. The energy of transfer orbits is exploited to discuss the effects of close lunar flybys. The results will be helpful for the transfer design in future manned or unmanned return missions, and can also provide valuable information for selecting proper parking DROs for cis-lunar space stations.

     

月球卫星轨道力学综述

月球探测器的运动通常可分为3个阶段，这3个阶段分别对应3种不同类型的轨道：近地停泊轨道、向月飞行的过渡轨道与环月飞行的月球卫星轨道。近地停泊轨道实为一种地球卫星轨道；过渡轨道则涉及不同的过渡方式(大推力或小推力等)；环月飞行的月球卫星轨道则与地球卫星轨道有很多不同之处，它决不是地球卫星轨道的简单克隆。针对这一点，全面阐述月球卫星的轨道力学问题，特别是环月飞行中的一些热点问题，如轨道摄动解的构造、近月点高度的下降及其涉及的卫星轨道寿命、各种特殊卫星(如太阳同步卫星和冻结轨道卫星等)的轨道特征、月球卫星定轨等。     

Orbits and manifolds near the equilibrium points around a rotating asteroid

We study the orbits and manifolds near the equilibrium points of a rotating asteroid. The linearised equations of motion relative to the equilibrium points in the gravitational field of a rotating asteroid, the characteristic equation and the stable conditions of the equilibrium points are derived and discussed. First, a new metric is presented to link the orbit and the geodesic of the smooth manifold. Then, using the eigenvalues of the characteristic equation, the equilibrium points are classified into 8 cases. A theorem is presented and proved to describe the structure of the submanifold as well as the stable and unstable behaviours of a massless test particle near the equilibrium points. The linearly stable, the non-resonant unstable, and the resonant equilibrium points are discussed. There are three families of periodic orbits and four families of quasi-periodic orbits near the linearly stable equilibrium point. For the non-resonant unstable equilibrium points, there are four relevant cases; for the periodic orbit and the quasi-periodic orbit, the structures of the submanifold and the subspace near the equilibrium points are studied for each case. For the resonant equilibrium points, the dimension of the resonant manifold is greater than 4, and we find at least one family of periodic orbits near the resonant equilibrium points. As an application of the theory developed here, we study relevant orbits for the asteroids 216 Kleopatra, 1620 Geographos, 4769 Castalia and 6489 Golevka.     

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.     

Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem

We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries.     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

A simple algorithm for orbit classification

We describe a simple algorithm for classifying orbits into orbit families. This algorithm works by finding patterns in the sign changes of the principal coordinates. Orbits in the logarithmic potential are studied as an application; we classify orbits into boxlet families and examine the influence of the core radius on the set of stable orbit families.     

Numerical study of the rectilinear isosceles restricted problem

We study the orbit of a particle in the plane of symmetry of two equal mass primaries in rectilinear keplerian motion. Using the surfaces of section we look for periodic orbits, examine their stability and search for quasi-periodic orbits and regions of escape. For large values of the angular momentumC, we verify the validity of the approximation of two fixed centers. However, we also find irregular families of orbits and resonance zones.For small values ofC, the approximation is no longer valid, but we find invariant curves whose interpretation might be interesting.     

A numerical investigation of the one-dimensional newtonian three-body problem

Numerical orbit integrations have been conducted to characterize the types of trajectories in the one-dimensional Newtonian three-body problem with equal masses and negative energy. Essentially three different types of motions were found to exist. They may be classified according to the duration of the bound three-body state. There are zero-lifetime predictable trajectories, finite lifetime apparently chaotic orbits, and infinite lifetime quasi-periodic motions. The quasi-periodic orbits are confined to the neighbourhood of Schubart's stable periodic orbit. For all other trajectories the final state is of the type binary + single particle in both directions of time. The boundaries of the different orbit-type regions seem to be sharp. We present statistical results for the binding energies and for the duration of the bound three-body state. Properties of individual orbits are also summarized in the form of various graphical maps in a two-dimensional grid of parameters defining the orbit. Supported by the Academy of Finland.     

New families of Sun-centered non-Keplerian orbits over cylinders and spheres

This paper introduces new families of Sun-centered non-Keplerian orbits (NKOs) that are constrained to a three-dimensional, cylindrical or spherical surface. As such, they are an extension to the well-known families of displaced NKOs that are confined to a two-dimensional plane. The cylindrical and spherical orbits are found by investigating the geometrically constrained spacecraft dynamics. By imposing further constraints on the orbit’s angular velocity and propulsive acceleration, the set of feasible orbits is defined. Additionally, the phase spaces of the orbits are explored and a numerical analysis is developed to find periodic orbits. The richness of the problem is further enhanced by considering both an inverse square acceleration law (mimicking solar electric propulsion) and a solar sail acceleration law to maintain the spacecraft on the three-dimensional surface. The wealth of orbits that these new families of NKOs generate allows for a range of novel space applications.     

Mars interplanetary trajectory design via Lagrangian points

With the increase in complexities of interplanetary missions, the main focus has shifted to reducing the total delta-V for the entire mission and hence increasing the payload capacity of the spacecraft. This paper develops a trajectory to Mars using the Lagrangian points of the Sun-Earth system and the Sun-Mars system. The whole trajectory can be broadly divided into three stages: (1) Trajectory from a near-Earth circular parking orbit to a halo orbit around Sun-Earth Lagrangian point L₂. (2) Trajectory from Sun-Earth L₂ halo orbit to Sun-Mars L₁ halo orbit. (3) Sun-Mars L₁ halo orbit to a circular orbit around Mars. The stable and unstable manifolds of the halo orbits are used for halo orbit insertion. The intermediate transfer arcs are designed using two-body Lambert’s problem. The total delta-V for the whole trajectory is computed and found to be lesser than that for the conventional trajectories. For a 480 km Earth parking orbit, the total delta-V is found to be 4.6203 km/s. Another advantage in the present approach is that delta-V does not depend upon the synodic period of Earth with respect to Mars.     

Natural and sail-displaced doubly-symmetric Lagrange point orbits for polar coverage

This paper proposes the use of doubly-symmetric, eight-shaped orbits in the circular restricted three-body problem for continuous coverage of the high-latitude regions of the Earth. These orbits, for a range of amplitudes, spend a large fraction of their period above either pole of the Earth. It is shown that they complement Sun-synchronous polar and highly eccentric Molniya orbits, and present a possible alternative to low thrust pole-sitter orbits. Both natural and solar-sail displaced orbits are considered. Continuation methods are described and used to generate families of these orbits. Starting from ballistic orbits, other families are created either by increasing the sail lightness number, varying the period or changing the sail attitude. Some representative orbits are then chosen to demonstrate the visibility of high-latitude regions throughout the year. A stability analysis is also performed, revealing that the orbits are unstable: it is found that for particular orbits, a solar sail can reduce their instability. A preliminary design of a linear quadratic regulator is presented as a solution to stabilize the system by using the solar sail only. Finally, invariant manifolds are exploited to identify orbits that present the opportunity of a ballistic transfer directly from low Earth orbit.     

A general method for the generation and extension of collinear libration point orbits

This paper is devoted to the study on applying numerical techniques to accurately compute and robustly extend the libration point orbits (LPOs). A new methodology is proposed exploiting the hyperbolic dynamics of the collinear libration points. Numerical tools are developed to facilitate the efficient computation process, which are applicable to realistic force models and inherently parallelizable. Extensive numerical explorations in the Earth–Moon system are carried out, revealing the delicate structures of nested island chains and bounded chaotic motions on the center manifold. Numerical results confirm that the proposed approach can handle the computations of various types of LPOs in a unified manner and is operational over a wide range of energy levels. LPOs obtained with this approach offer a broad range of future mission possibilities in an extended vicinity of the collinear libration points.     

从地面发射月球探测器的窗口选择

典型的月球探测器飞行轨道包括地球停泊轨道段、地月转移轨道段、月球卫星轨道段和着月轨道段。首先介绍了设计从地面发射月球探测器轨道典型的约束条件;然后,借助于二体假设,建立解析表达式,分析各种约束对窗口选择的影响,给出了各轨道段概略的飞行时间和粗窗口;最后,利用精确的探测器轨道动力学模型,计算精窗口,并给出了一则算例,所得结论可为月球探测器轨道发射、轨道设计提供依据。     

Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem

The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size.     

Families of periodic orbits in the general three-body problem for the Sun-Jupiter-Saturn mass-ratio and their stability

Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits.