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。     

Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G

The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV _Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change.     

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.     

Three-body problem, its Lagrangian points and how to exploit them using an alternative transfer to L4 and L5

An alternative transfer strategy to send spacecraft to stable orbits around the Lagrangian equilibrium points L4 and L5 based in trajectories derived from the periodic orbits around L1 is presented in this work. The trajectories derived, called Trajectories G, are described and studied in terms of the initial generation requirements and their energy variations relative to the Earth through the passage by the lunar sphere of influence. Missions for insertion of spacecraft in elliptic orbits around L4 and L5 are analysed considering the restricted three-body problem Earth–Moon-particle and the results are discussed starting from the thrust, time of flight and energy variation relative to the Earth.     

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.     

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.     

Controlled Solar Sail Transfers into Near-Sun Regions Combined with Planetary Gravity-Assist Flybys

Results of numerical simulations of 'local-optimal' (or 'instantaneously optimal') trajectories of a space probe with a flat solar sail which moves from the circular Earth orbit to near-Sun regions are presented. We examine planar (ecliptic) solar sail transfer with gravity-assist flybys of Earth, Venus and Mercury. Several complex control modes of the sail tilt orientation angle for near-Sun orbits and for some 'falling onto the Sun' trajectories are investigated. The numerical simulations are used to examine the flight duration of some sail missions and to investigate the evolution of osculating elliptical orbits.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

Comparison of low-energy lunar transfer trajectories to invariant manifolds

In this study, transfer trajectories from the Earth to the Moon that encounter the Moon at various flight path angles are examined, and lunar approach trajectories are compared to the invariant manifolds of selected unstable orbits in the circular restricted three-body problem. Previous work focused on lunar impact and landing trajectories encountering the Moon normal to the surface, and this research extends the problem with different flight path angles in three dimensions. The lunar landing geometry for a range of Jacobi constants is computed, and approaches to the Moon via invariant manifolds from unstable orbits are analyzed for different energy levels.     

Optimal transfers between unstable periodic orbits using invariant manifolds

This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.     

Study of lunar gravity assist orbits in the restricted four-body problem

In this paper, the lunar gravity assist (LGA) orbits starting from the Earth are investigated in the Sun–Earth–Moon–spacecraft restricted four-body problem (RFBP). First of all, the sphere of influence of the Earth–Moon system (SOIEM) is derived. Numerical calculation displays that inside the SOIEM, the effect of the Sun on the LGA orbits is quite small, but outside the SOIEM, the Sun perturbation can remarkably influence the trend of the LGA orbit. To analyze the effect of the Sun, the RFBP outside the SOIEM is approximately replaced by a planar circular restricted three-body problem, where, in the latter case, the Sun and the Earth–Moon barycenter act as primaries. The stable manifolds associated with the libration point orbit and their Poincaré sections on the SOIEM are applied to investigating the LGA orbit. According to our research, the patched LGA orbits on the Poincaré sections can efficiently distinguish the transit LGA orbits from the non-transit LGA orbits under the RFBP. The former orbits can pass through the region around libration point away from the SOIEM, but the latter orbits will bounce back to the SOIEM. Besides, the stable transit probability is defined and analyzed. According to the variant requirement of the space mission, the results obtained can help us select the LGA orbit and the launch window.     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

Delta-v requirements for earth co-orbital rendezvous missions

Earth co-orbital asteroids present advantages as potential targets for future asteroid rendezvous missions. Their prolonged proximity to Earth facilitates communication, while their Earth-like orbits mean a steady flux of solar power and no significant periodic heating and cooling of the spacecraft throughout the course of the mission. Theoretical studies show that low-inclination co-orbital orbits are more stable than high-inclination orbits. As inclination is the most significant indicator of low delta-v rendezvous orbits, there is the potential for a large population of easily accessible asteroids, with favorable engineering requirements. This study first looks at phase-independent rendezvous orbits to a large number of objects, then looks in more detail at the phase-dependent orbits to the most favorable objects. While rendezvous orbits to co-orbital objects do not have a low delta-v necessarily, some objects present energy requirements significantly less than previous rendezvous missions. Currently we find no ideal co-orbital asteroids for rendezvous missions, although theoretical Earth Trojans present very low-energy requirements for rendezvous.     

Design and analysis of Weak Stability Boundary trajectories to Moon

An algorithm is developed to find Weak Stability Boundary transfer trajectories to Moon in high fidelity force model using forward propagation. The trajectory starts from an Earth Parking Orbit (circular or elliptical). The algorithm varies the control parameters at Earth Parking Orbit and on the way to Moon to arrive at a ballistic capture trajectory at Moon. Forward propagation helps to satisfy launch vehicle’s maximum payload constraints. Using this algorithm, a number of test cases are evaluated and detailed analysis of capture orbits is presented.     

On the stability of high inclination Trojans

Numerical integrations have been performed for orbits of Venus', Earth's and Mars' Trojan-asteroid test particles in a self consistent model of the solar system to study the stability of inclined Trojan orbits. In the case of Mars low inclination orbits tend to be unstable while the contrary seems to apply to Venus and Earth, although the stability of some very high inclination orbits may not be excluded on the basis of these computations.     

Discovery of an asteroid and quasi‐satellite in an Earth‐like horseshoe orbit

Abstract— The newly discovered asteroid 2002 AA₂₉ moves in a very Earth‐like orbit that relative to Earth has a unique horseshoe shape and allows transitions to a quasi‐satellite state. This is the first body known to be in a simple heliocentric horseshoe orbit, moving along its parent planet's orbit. It is similarly also the first true co‐orbital object of Earth, since other asteroids in 1:1 resonance with Earth have orbits very dissimilar from that of our planet. When a quasi‐satellite, it remains within 0.2 AU of the Earth for several decades. 2002 AA₂₉ is the first asteroid known to exhibit this behavior. 2002 AA₂₉ introduces an important new class of objects offering potential targets for space missions and clues to asteroid orbit transfer evolution.     

On the construction of low-energy cislunar and translunar transfers based on the libration points

There exist cislunar and translunar libration points near the Moon, which are referred to as the LL ₁ and LL ₂ points, respectively. They can generate the different types of low-energy trajectories transferring from Earth to Moon. The time-dependent analytic model including the gravitational forces from the Sun, Earth, and Moon is employed to investigate the energy-minimal and practical transfer trajectories. However, different from the circular restricted three-body problem, the equivalent gravitational equilibria are defined according to the geometry of the instantaneous Hill boundary due to the gravitational perturbation from the Sun. The relationship between the altitudes of periapsis and eccentricities is achieved from the Poincaré mapping for all the captured lunar trajectories, which presents the statistical feature of the fuel cost and captured orbital elements rather than generating a specified Moon-captured segment. The minimum energy required by the captured trajectory on a lunar circular orbit is deduced in the spatial bi-circular model. The idea is presented that the asymptotical behaviors of invariant manifolds approaching to/traveling from the libration points or halo orbits are destroyed by the solar perturbation. In fact, the energy-minimal cislunar transfer trajectory is acquired by transiting the LL ₁ point, while the energy-minimal translunar transfer trajectory is obtained by transiting the LL ₂ point. Finally, the transfer opportunities for the practical trajectories that have escaped from the Earth and have been captured by the Moon are yielded by the transiting halo orbits near the LL ₁ and LL ₂ points, which can be used to generate the whole of the trajectories.     

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.