Hierarchical global search for fuel-optimal impulsive transfers between lunar libration orbits

The attention to the periodic orbit in the Earth-Moon restricted three-body system continues to grow due to its special environment and locations. This research investigates the feasibility of constructing fuel-optimal single and multiple impulse transfers between unstable periodic orbits at L₁ and L₂ points. Invariant manifolds, which could provide the appropriate initial trajectories for optimization, are analyzed deeply to enable previously unknown orbit options and potentially to reduce mission cost. A global search strategy based on comparing the orbital state of the unstable and stable manifolds, incorporated with low-thrust techniques, is performed to seek a suitable matching point for maneuver application. Then the sequential quadratic programming (SQP) is adopted to further optimize the velocity increment and obtain the single/multiple impulse optimal transfers. The associated constraint gradients are derived to achieve higher accuracy and rapidity of the algorithm. To highlight the effectivity of the transfer scheme, three-dimensional low-energy transfers between different types and spatial regions of performing single and multiple impulses are explored. The total Delta-V required varies between a few meters per second and tens of meters per second, and the related flight time is about several weeks, mainly depending on the energy of periodic orbits and the invariant manifold structure. The results obtained in this paper can provide a useful reference for the selection of escape and capture site along the manifolds, maneuver magnitude and transfer time.     

从日地系统L2出发借力月球飞越近地小行星

对于停留在日地系统L2的“嫦娥2号”探测器,其后续飞行方案有多个选项,例如主动撞月或重返月球轨道、返回地球轨道或再入大气、飞往地月系统L1/L2或日地系统L1、进入深空飞越近地小行星(最终,“嫦娥2号”于2012年12月13日成功地实现了对Toutatis小行星的近距离飞越)。探讨上述的飞行方案需要对飞行轨道进行初步设计,总的速度脉冲限制在100 m/s以内并且需要考虑探测器同时受到太阳、地球、月球的引力作用。本研究设计了探测器从日地系统L2出发借力月球实现Toutatis小行星飞越的飞行方案,与直接飞越方案相比,借力月球可以进一步节省探测器的燃料消耗,其等效速度脉冲设计值为58.47 m/s。     

Configuration space and stability analysis of solar sail near-vertical earth-trailing orbits

Regions outside the reach of traditional propulsion systems or the ones that require significant propellant, may be reached by harnessing the solar radiation pressure and leveraging coupled dynamics to maneuver a sail-based spacecraft. Earth-trailing orbits have recently been investigated for getting a unique perspective of the Sun while maintaining the spacecraft in close proximity to Earth. Vertical orbits trailing the Earth exhibit the additional capability to view the Sun from above and below the ecliptic plane. In this work, families of sail-based orbits are explored for varying Earth-trailing angles and Z amplitudes in the Sun-Earth circular restricted three-body problem. Optimization is carried out to ensure that the non-traditional vertical orbits exhibit a constant pitch angle control history, as well as symmetry across the X-Y plane. The stability of the resulting orbit families is assessed using an extension of Flouquet theory to Differential Algebraic Equations. Results indicate that sail-based Earth-trailing vertical orbits can be more stable than traditional sub-L₁ sail-based vertical orbits.     

Heliotropic dust rings for Earth climate engineering

This paper examines the concept of a Sun-pointing elliptical Earth ring comprised of dust grains to offset global warming. A new family of non-Keplerian periodic orbits, under the effects of solar radiation pressure and the Earth’s J₂ oblateness perturbation, is used to increase the lifetime of the passive cloud of particles and, thus, increase the efficiency of this geoengineering strategy. An analytical model is used to predict the orbit evolution of the dust ring due to solar-radiation pressure and the J₂ effect. The attenuation of the solar radiation can then be calculated from the ring model. In comparison to circular orbits, eccentric orbits yield a more stable environment for small grain sizes and therefore achieve higher efficiencies when the orbit decay of the material is considered. Moreover, the novel orbital dynamics experienced by high area-to-mass ratio objects, influenced by solar radiation pressure and the J₂ effect, ensure the ring will maintain a permanent heliotropic shape, with dust spending the largest portion of time on the Sun facing side of the orbit. It is envisaged that small dust grains can be released from a circular generator orbit with an initial impulse to enter an eccentric orbit with Sun-facing apogee. Finally, a lowest estimate of 1 × 10¹² kg of material is computed as the total mass required to offset the effects of global warming.     

Families of halo orbits in the elliptic restricted three-body problem for a solar sail with reflectivity control devices

Solar sail halo orbits designed in the Sun-Earth circular restricted three-body problem (CR3BP) provide inefficient reference orbits for station-keeping since the disturbance due to the eccentricity of the Earth’s orbit has to be compensated for. This paper presents a strategy to compute families of halo orbits around the collinear artificial equilibrium points in the Sun-Earth elliptic restricted three-body problem (ER3BP) for a solar sail with reflectivity control devices (RCDs). In this non-autonomous model, periodic halo orbits only exist when their periods are equal to integer multiples of one year. Here multi-revolution halo orbits with periods equal to integer multiples of one year are constructed in the CR3BP and then used as seeds to numerically continue the halo orbits in the ER3BP. The linear stability of the orbits is analyzed which shows that the in-plane motion is unstable while the out-of-plane motion is neutrally stable and a bifurcation is identified. Finally, station-keeping is performed which shows that a reference orbit designed in the ER3BP is significantly more efficient than that designed in the CR3BP, while the addition of RCDs improve station-keeping performance and robustness to uncertainty in the sail lightness number.     

基于准流形方法日地系L3点转移轨道设计

针对深空探测中轨道转移时间长且能量消耗较大的问题,提出基于准流形实现从地球停泊轨道到日地系L3点转移轨道的设计方法。在日地限制性三体问题模型下,在L1点或L2点Halo轨道上施加扰动推力,构造准流形,利用其非线性三体动力学特性,通过霍曼转移轨道与近地轨道进行拼接,使航天器进入准流形后能够无动力滑行到L3点附近区域。在准流形与L3点周期轨道交点,施加速度脉冲,使航天器进入相应周期轨道,从而完成轨道转移。仿真结果表明,利用该方法所得结果与基于不变流形的转移轨道相比,能将速度增量从4398m/s减少为4014m/s,并将转移时间从9年以上缩短到7.3年以内,有效地提高了航天器的工作效率。     

On the station keeping of a solar sail in the elliptic Sun–Earth system

In this work we focus on the dynamics of a solar sail in the Sun–Earth Elliptic Restricted Three-Body Problem with solar radiation pressure. The considered situation is the motion of a sail close to the L₁ point, but displacing the equilibrium point with the sail so that it is possible to have continuous communication with the Earth. In previous works we derived a station keeping strategy for this situation but using the Circular RTBP as a model.     

Navigating BepiColombo during the weak-stability capture at Mercury

BepiColombo is scheduled for launch in August 2013 and to arrive after a nearly six-year long transfer at Mercury in June 2019. The trajectory has a number of challenging elements: a launch with Soyuz/Fregat into a geostationary transfer orbit, followed by a lunar flyby, long low-thrust arcs and five more planetary flybys (one at the Earth, two at Venus and two at Mercury). At arrival the low thrust arcs reduce the approach velocity so much that BepiColombo passes by the Sun–Mercury Lagrange points L₁ and L₂ and gets weakly captured in a highly eccentric orbit around Mercury in case the orbit insertion manoeuvre would fail.This paper describes the navigation strategy during the final phase. Five trajectory correction manouevres during the last 65 days requiring up to 20 m/s (3σ) are proposed. With this strategy it is possible to navigate BepiColombo safely through the weak-stability boundary of Mercury and to reach the target periherm with a precision of 11 km.     

Space debris mitigation strategies and practices in geosynchronous transfer orbits

Missions to geosynchronous orbits remain one of the most important elements of space launch traffic, accounting for 40% of all missions to Earth orbit and beyond during the four-year period 2000–2003. The vast majority of these missions leave one or more objects in geosynchronous transfer orbits (GTOs), contributing on a short-term or long-term basis to the space debris population. National and international space debris mitigation guidelines seek to curtail the accumulation of debris in orbits which penetrate the regions of low Earth orbit and of geosynchronous orbit. The orbital lifetime of objects in GTO can be greatly influenced by the initial values of perigee, inclination, and right ascension of the orbital plane, leading to orbital lifetimes of from less than one month to more than 100 years. An examination of the characteristic GTOs employed by launch vehicles from around the world has been conducted. The consequences of using perigees above 300 km and super-synchronous apogees, typically above 40,000 km, have been identified. In addition, the differences in orbital behavior of launch vehicle stages and mission-related debris in GTOs have been investigated. Greater coordination and cooperation between space launch service providers and spacecraft designers and owners could significantly improve overall compliance with guidelines to mitigate the accumulation of debris in Earth orbit.     

The distributions of positions of Minimal Orbit Intersection Distances among Near Earth Asteroids

This paper presents the distributions of the positions of the Minimal Orbit Intersection Distances (MOID) among three subgroups of the Near Earth Asteroids (NEAs). This includes 683 Atens, 4185 Apollos and 3538 Amors which makes over 15 millions combinations of the pairs of orbits. The results which are obtained in this analysis show very interesting distributions of positions of the MOIDs and circumstances of close approaches of the asteroids and emphasize influence of different orbital elements on these distributions.     

Resonant quasi-periodic near-rectilinear Halo orbits in the Elliptic-Circular Earth-Moon-Sun Problem

Motivated by the near-future re-exploration of the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L₂ southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.     

High energy charged particle experiment – A payload proposal to KuaFu-B mission

High Energy Charged Particle Experiment (HECPE) is to measure the fluxes of MeV electrons and tens of MeV protons. The two satellites of KuaFu-B are in the same polar orbit with apogee 7.0R_E, perigee 1.8R_E. They can sweep large L values and pass through the inner and outer radiation belts. The high energy electrons and protons in the radiation belts are principal sources for failures of satellites and spacecrafts in the Earth orbits. The enhancements of the high energy electrons and protons, so-called energetic particle events, are important phenomena of the Space Weather. The energy ranges monitored by HECPE are 0.3–0.5 MeV, 0.5–1.0 MeV, 1.0–2.0 MeV, and E > 2.0 MeV for electrons, 5–10 MeV, 10–20 MeV, 20–40 MeV, and 40–80 MeV for protons.     

Dynamics of a spacecraft and normalization around Lagrangian points in the Neptune–Triton system

The problem of a spacecraft orbiting the Neptune–Triton system is presented. The new ingredients in this restricted three body problem are the Neptune oblateness and the high inclined and retrograde motion of Triton. First we present some interesting simulations showing the role played by the oblateness on a Neptune’s satellite, disturbed by Triton. We also give an extensive numerical exploration in the case when the spacecraft orbits Triton, considering Sun, Neptune and its planetary oblateness as disturbers. In the plane a × I (a = semi-major axis, I = inclination), we give a plot of the stable regions where the massless body can survive for thousand of years. Retrograde and direct orbits were considered and as usual, the region of stability is much more significant for the case of direct orbit of the spacecraft (Triton’s orbit is retrograde). Next we explore the dynamics in a vicinity of the Lagrangian points. The Birkhoff normalization is constructed around L₂, followed by its reduction to the center manifold. In this reduced dynamics, a convenient Poincaré section shows the interplay of the Lyapunov and halo periodic orbits, Lissajous and quasi-halo tori as well as the stable and unstable manifolds of the planar Lyapunov orbit. To show the effect of the oblateness, the planar Lyapunov family emanating from the Lagrangian points and three-dimensional halo orbits are obtained by the numerical continuation method.     

Experimental measurement of particle deceleration and survival in multiple thin foil targets

Experimental results of deceleration of iron particles by means of a stack of thin Mylar foils are presented. The mass range of the iron particles is 3 × 10⁻¹² to 1 × 10⁻¹⁰8 the velocity ranges from 2.5 km/s to 5 km/s. Decelerations of up to ca. 70% were measured using 2 and 3 foils. The measured deceleration can be expressed over the whole velocity range studied as a function of the particle mass. Comparison with previous studies of thin foil penetration /1,2/ supports the approach to a scaling in terms of momentum conservation. We discuss limits for intact dust particle capture in space, and conclude that such techniques for deceleration and capture for comet flyby missions are effective. This is instanced already in Solar Max data where capture is effected at higher velocities and identifiable fragments of incident particles have been analysed.     

Intracellular molecular distributions in spacecraft experiments in orbit around Earth

It is possible that the nucleolous inside the cell plays the role of a “gravity receptor”. Furthermore, cells up to 10 μm in diameter can demonstrate some effect due to the redistribution of mitochondria or nucleolous. Effects of gravity should be present in various cell systems where larger objects such as the ribosomes move from cell to cell. In this paper we study the effects of gravity on cells. In particular, we examine the resulting intracellular molecular distribution due to Brownian motion and the ordered distribution of molecules under the action of gravity, where n₀ is the number per unit volume at certain level, and n is the number per unit volume above that level. This is an experiment that takes place at a certain orbital altitude in a spacecraft in orbit around Earth, where the acceleration due to the central field is corrected for the oblateness and also the rotation of the Earth. We found that equatorial circular and elliptical orbits have the highest n/n₀ ratios. This experiment takes place in circular and elliptical orbits, with eccentricities e = 0, 0.1 and involves a bacterial cell at an orbital altitude of 300 km. We found that n/n₀ = 1.00299 and 1.0037 respectively, which is still a 0.6–0.7 % higher than n/n₀ = 0.0996685 calculated on the surface of the Earth. Examining mitochondria in similar orbital experiments we found that equatorial orbits result to higher n/n₀ ratios. In particular, we found that n/n₀ = 8.38119, where an elliptical orbit of eccentricity e = 0.1 results to n/n₀ = 13.8525. Both are high above 100%, signifying the importance of Brownian motion over gravity. Our results are of interest to biomedical applications. Molecular concentrations are important for various processes such as the embryogenesis, positional homeostasis and its relation to cell energy expenditure, cell torque, cell deformation, and more. These results indicate that statistical molecular distributions play an important role for the recognition of a particular environment by the cell, in biological space experiment to come.     

Optimal station-keeping near Earth–Moon collinear libration points using continuous and impulsive maneuvers

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–Moon Lagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points. The most critical perturbation effect near the L₁ and L₂ Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and the Sun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restricted five-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control or impulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuous control has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multiple shooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimized MS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multiple impulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory should close itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almost zero station-keeping maneuver.     

Phasing with near rectilinear Halo orbits: Design and comparison

In the next future, space agencies are planning to return to the Moon. The objective is to assemble an orbiting space station, called Gateway, on a Near Rectilinear Halo Orbit around the Moon as a base for future Moon and deep space missions. Within this framework, multiple side missions will be planned to sustain the Gateway (Artemis mission). The proposed work is thought in framework of the preliminary design of future cargo missions, in particular on the design of an efficient phasing trajectory, under the Circular Restricted Three body problem hypotheses, to bring a cargo vehicle from the end of the Earth-Moon transfer to the beginning of the proximity operations such as rendezvous and docking with the space station. The work aims covering the lack of literature in phasing trajectories with the NRHO by proposing three different strategies to connect the Earth-Moon transfer trajectory with the proximity operations. The three strategies are classified based on the choice of the parking orbits or the choice of the manifolds. Two strategies use butterfly and Halo orbits to park the vehicle before transferring to the target orbit. The third strategy, instead, uses manifolds to allow a direct phasing. In the paper, the three innovative strategies are designed and compare in a specific scenario.     

Low-thrust transfer to Backflip orbits

The aim of the work is to design a low-thrust transfer from a Low Earth Orbit to a “useful” periodic orbit in the Earth–Moon Circular Restricted Three Body Model (CR3BP). A useful periodic orbit is here intended as one that moves both in the Earth–Moon plane and out of this plane without any requirements of propellant mass. This is achieved by exploiting a particular class of periodic orbits named Backflip orbits, enabled by the CR3BP. The unique characteristics of this class of periodic solutions allow the design of an almost planar transfer from a geocentric orbit and the use of the Backflip intrinsic characteristics to explore the geospace out of the Earth–Moon plane. The main advantage of this approach is that periodic plane changes can be obtained by performing an almost planar transfer. In order to save propellant mass, so as to increase the scientific payload of the mission, a low-powered transfer is considered. This foresees a thrusting phase to gain energy from a departing circular geocentric orbit and a second thrusting phase to match the state of the target Backflip orbit, separated by an intermediate ballistic phase. This results in a combined application of a low-thrust manoeuvre and of a periodical solution in the CR3BP to realize a new class of missions to explore the Earth–Moon neighbourhoods in a quite inexpensive way. In addition, a low-thrust transit between two different Backflip orbits is analyzed and considered as a possible extension of the proposed mission. Thus, also a Backflip-to-Backflip transfer is addressed where a low-powered probe is able to experience periodic excursions above and below the Earth–Moon plane only performing almost planar and very short transfers.     

Longitudinal distribution of thermospheric densities and ionization rates in the upper atmosphere of Mars at mid-latitude using MGS ACC data

The Accelerometer Experiment (ACC) onboard Mars Global Surveyor (MGS) measured 1600 density profiles in the upper atmosphere of Mars during aerobraking. These measurements reveal large-scale and small-scale structure in the thermosphere of Mars. Here, the measurements of mass density for 115 orbits (#P0670–P0789) from November 1 to 30, 1998, under spring equinox and medium solar activity conditions (average F_10.7 ∼ 137) during phase 2 of the aerobraking in the thermosphere of Mars at different altitudes and longitudes are presented for northern mid-latitude (17–42°N) in the dayside atmosphere using ACC onboard MGS. From these mass densities, the neutral densities of different gases are derived from their mixing ratios. Using these neutral densities, the longitudinal distribution of photoionization rates and photoelectron impact ionization rates are calculated at wavelength range 1–102.57 nm due to EUV and soft X-ray radiation under photochemical controlled region using Analytical Yield Spectrum approach (AYS). These conditions are appropriate for MGS Phase 2 aerobraking period from which the accelerometer data is used. Under the photochemical equilibrium condition, the electron density near the peak varies as the square root of the total peak ionization rate. Using this fact, an attempt is being made to estimate the mean primary and secondary peak electron density by averaging the longitudinal variations of total peak ionization rates in the northern mid-latitude (17–42°N) ionosphere of Mars, as there is no radio science measurement at this latitude region by MGS.     

Impact analysis of the transponder time delay on radio-tracking observables

Accurate tracking of probes is one of the key points of space exploration. Range and Doppler techniques are the most commonly used. In this paper we analyze the impact of the transponder delay, $i.e$. the processing time between reception and re-emission of a two-way tracking link at the satellite, on tracking observables and on spacecraft orbits. We show that this term, only partially accounted for in the standard formulation of computed space observables, can actually be relevant for future missions with high nominal tracking accuracies or for the re-processing of old missions. We present several applications of our formulation to Earth flybys, the NASA GRAIL and the ESA BepiColombo missions.