A low-thrust transfer between the Earth–Moon and Sun–Earth systems based on invariant manifolds

A low-energy, low-thrust transfer between two halo orbits associated with two coupled three-body systems is studied in this paper. The transfer is composed of a ballistic departure, a ballistic insertion and a powered phase using low-thrust propulsion to connect these two trajectories. The ballistic departure and insertion are computed by constructing the unstable and stable invariant manifolds of the corresponding halo orbits, and a complete low-energy transfer based on the patched invariant manifolds is optimized using the particle swarm optimization (PSO) algorithm on the criterion of smallest velocity discontinuity and limited position discontinuity (less than 1 km). Then, the result is expropriated as the boundary conditions for the subsequent low-thrust trajectory design. The fuel-optimal problem is formulated using the calculus of variations and Pontryagin's Maximum Principle in a complete four-body dynamical environment. Then, a typical bang–bang control is derived and solved using the indirect method combined with a homotopic technique. The contributions of the present work mainly consist of two points. Firstly, the global search method proposed in this paper is simply handled using the PSO algorithm, a number of feasible solutions in a fairly wide range can be delivered without a priori or perfect knowledge of the transfers. Secondly, the indirect optimization method is used in the low-thrust trajectory design and the derivations of the first-order necessary conditions are simplified with a modified controlled, restricted four-body model.     

Manifold dynamics in the Earth–Moon system via isomorphic mapping with application to spacecraft end-of-life strategies

Recently, manifold dynamics has assumed an increasing relevance for analysis and design of low-energy missions, both in the Earth–Moon system and in alternative multibody environments. With regard to lunar missions, exterior and interior transfers, based on the transit through the regions where the collinear libration points L₁ and L₂ are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. This paper is focused on the definition and use of a special isomorphic mapping for low-energy mission analysis. A convenient set of cylindrical coordinates is employed to describe the spacecraft dynamics (i.e. position and velocity), in the context of the circular restricted three-body problem, used to model the spacecraft motion in the Earth–Moon system. This isomorphic mapping of trajectories allows the identification and intuitive representation of periodic orbits and of the related invariant manifolds, which correspond to tubes that emanate from the curve associated with the periodic orbit. Heteroclinic connections, i.e. the trajectories that belong to both the stable and the unstable manifolds of two distinct periodic orbits, can be easily detected by means of this representation. This paper illustrates the use of isomorphic mapping for finding (a) periodic orbits, (b) heteroclinic connections between trajectories emanating from two Lyapunov orbits, the first at L₁, and the second at L₂, and (c) heteroclinic connections between trajectories emanating from the Lyapunov orbit at L₁ and from a particular unstable lunar orbit. Heteroclinic trajectories are asymptotic trajectories that travels at zero-propellant cost. In practical situations, a modest delta-v budget is required to perform transfers along the manifolds. This circumstance implies the possibility of performing complex missions, by combining different types of trajectory arcs belonging to the manifolds. This work studies also the possible application of manifold dynamics to defining suitable, convenient end-of-life strategies for spacecraft orbiting the Earth. Seven distinct options are identified, and lead to placing the spacecraft into the final disposal orbit, which is either (a) a lunar capture orbit, (b) a lunar impact trajectory, (c) a stable lunar periodic orbit, or (d) an outer orbit, never approaching the Earth or the Moon. Two remarkable properties that relate the velocity variations with the spacecraft energy are employed for the purpose of identifying the optimal locations, magnitudes, and directions of the velocity impulses needed to perform the seven transfer trajectories. The overall performance of each end-of-life strategy is evaluated in terms of time of flight and propellant budget.     

Properties of transit trajectory in the restricted three and four-body problem

In the restricted three-body problem if the Jacobi constant is just below the value corresponding to Lagrangian point only a little neck exists around the equilibrium point and capture trajectories are indicated as low-energy. Capture properties depend on the dynamics around these critical points and qualitative results can be obtained using linearized systems. In this paper, to study transit trajectory properties in the restricted three and four-body problem, the Earth–Moon–Sun–Satellite system is considered as example and studied using different models. In the circular restricted three-body problem (Earth–Moon–Satellite), transit, non transit and asymptotic trajectories, are easily identified by using the principal reference frame. Dynamics around Lagrangian point are then studied introducing the Moon eccentricity into the elliptical restricted three-body model. A preferential region for transit orbit is individuated and studied as a function of eigenvalue properties. To introduce the Sun effect, the bi-circular four-body model is considered and dynamics around Lagrangian points studied as a function of angular distance between Earth–Sun and Earth–Moon line. Finally, results obtained in the elliptical three-body model and bi-circular four-body model, are compared with numerical simulations using real Sun–Moon–Earth ephemeris.     

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.     

Energy spectrum of proton flux near geomagnetic equator at low altitudes

The results of proton energy (tens keV – several MeV) spectrum measurements near geomagnetic equator (L < 1.15) at low altitudes (<1000 km) are presented. We used data of experiments onboard ACTIVE, SAMPEX, NOAA TIROS-N satellites and SPRUT-VI (MIR station) and cover a time range of about 30 years (including previous measurements). It was found that the kappa-distribution function fits the experimental spectrum with the best correlation coefficient. A comparison of energy spectra of near-equatorial protons and ring-current protons was made. Using the estimation of the life time of near-equatorial protons we explain the difference in spectral indices of radiation belt and near-equatorial proton formation. We conclude that the ring current is the main source of the near-equatorial protons.     

Fast low-energy halo-to-halo transfers between Sun–planet systems

In this paper, the problem of fast low-energy halo-to-halo transfers between Sun–planet systems is discussed under ephemeris constraints. According to the structure of an invariant manifold, employing an invariant manifold and planetary gravity assist to save fuel consumption is analyzed from the view of orbital energy. Then, a pseudo-manifold is introduced to replace the invariant manifold in such a way that more transfer opportunities are allowed. Fast escape and capture can be achieved along the pseudo-manifold. Furthermore, a global searching method that is based on patched-models is proposed to find an appropriate transfer trajectory. In this searching method, the trajectory is divided into several segments that can be designed under simple dynamical models, and an analytical algorithm is developed for connecting the segments. Earth–Mars and Earth–Venus halo-to-halo transfers are designed to demonstrate the proposed approach. Numerical results show that the transfers that combine the pseudo-manifolds and planetary gravity assist can offer significant fuel consumption and flight time savings over traditional transfer schemes.     

雷诺数对跨声速压气机转子内部流动失稳触发机理的影响

利用带有先进转捩模型的数值模拟方法,对高低两种雷诺数下的跨声速压气机转子NASA Rotor67的内部流动进行了数值模拟.对比了不同雷诺数下叶片内部复杂三维流动,剖析了雷诺数影响风扇转子流动失稳的机制.研究发现:雷诺数降低使得叶片表面低能流体增多,径向迁移加剧,造成叶片顶部吸力面分离加剧;且雷诺数降低使得叶顶间隙泄漏流强度减弱,间隙泄漏流和主流相互作用造成的叶片顶部流场堵塞减弱.雷诺数通过上述两种作用影响压气机转子的失稳机制.