Design of transfer trajectories between resonant orbits in the Earth–Moon restricted problem

The application of dynamical systems techniques to mission design has demonstrated that employing invariant manifolds and resonant flybys enables previously unknown trajectory options and potentially reduces the ΔV$\mathrm{\Delta }V$ requirements. In this investigation, planar and three-dimensional resonant orbits are analyzed and cataloged in the Earth–Moon system and the associated invariant manifold structures are computed and visualized with the aid of higher-dimensional Poincaré maps. The relationship between the manifold trajectories associated with multiple resonant orbits is explored through the maps with the objective of constructing resonant transfer arcs. As a result, planar and three-dimensional homoclinic- and heteroclinic-type trajectories between unstable periodic resonant orbits are identified in the Earth–Moon system. To further illustrate the applicability of 2D and 3D resonant orbits in preliminary trajectory design, planar transfers to the vicinity of L₅ and an out-of-plane transfer to a 3D periodic orbit, one that tours the entire Earth–Moon system, are constructed. The design process exploits the invariant manifolds associated with orbits in resonance with the Moon as transfer mechanisms.     

Low-energy Earth–Moon transfers involving manifolds through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for decades. Exterior and interior transfers, based on the transit through the regions where the collinear libration points 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 concerned with a geometrical approach for low-energy Earth-to-Moon mission analysis, based on isomorphic mapping. The isomorphic mapping of trajectories allows a visual, 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. Two types of Earth-to-Moon missions are considered. The first mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a periodic orbit around the Moon. The second mission is composed of the following arcs: (i) transfer trajectory from a circular low Earth orbit to the stable invariant manifold associated with the Lyapunov orbit at L₁ (corresponding to a specified energy level) and (ii) transfer trajectory along the unstable manifold associated with the Lyapunov orbit at L₁, with final injection in a capture (non-periodic) orbit around the Moon. In both cases three velocity impulses are needed to perform the transfer: the first at an unknown initial point along the low Earth orbit, the second at injection on the stable manifold, the third at injection in the final (periodic or capture) orbit. The final goal is in finding the optimization parameters, which are represented by the locations, directions, and magnitudes of the velocity impulses such that the overall delta-v of the transfer is minimized. This work proves how isomorphic mapping (in two distinct forms) can be profitably employed to optimize such transfers, by determining in a geometrical fashion the desired optimization parameters that minimize the delta-v budget required to perform the transfer.     

Horseshoe orbits in the Earth–Moon system

Horseshoe orbits in the restricted three-body problem have been mostly considered in the Sun–Jupiter system and, in recent years, in the Sun–Earth system. Here, these orbits have been used to find asteroids that have orbits of this kind. We have built a planar family of horseshoe orbits in the Earth–Moon system and determined the points of planar and 1/1 vertical resonances on this family. We have presented examples of orbits generated by these spatial families.     

Evolution of the out-of-plane amplitude for quasi-periodic trajectories in the Earth–Moon system

The out-of-plane amplitude along quasi-periodic trajectories in the Earth–Moon system is highly sensitive to perturbations in position and/or velocity as underscored recently by the ARTEMIS spacecraft. Controlling the evolution of the out-of-plane amplitude is non-trivial, but can be critical to satisfying mission requirements. The sensitivity of the out-of-plane amplitude evolution to perturbations due to lunar eccentricity, solar gravity, and solar radiation pressure is explored and a strategy for designing low-cost deterministic maneuvers to control the amplitude history is also examined. The method is sufficiently general and is applied to the L₁ quasi-periodic orbit that serves as a baseline for the ARTEMIS P2 trajectory.     

Earth–Moon libration point orbit stationkeeping: Theory,modeling, and operations

Collinear Earth–Moon libration points have emerged as locations with immediate applications. These libration point orbits are inherently unstable and must be maintained regularly which constrains operations and maneuver locations. Stationkeeping is challenging due to relatively short time scales for divergence, effects of large orbital eccentricity of the secondary body, and third-body perturbations. Using the Acceleration Reconnection and Turbulence and Electrodynamics of the Moon's Interaction with the Sun (ARTEMIS) mission orbit as a platform, the fundamental behavior of the trajectories is explored using Poincaré maps in the circular restricted three-body problem. Operational stationkeeping results obtained using the Optimal Continuation Strategy are presented and compared to orbit stability information generated from mode analysis based in dynamical systems theory.     

A Study of the Luminescence of the Lyman–Birge–Hopfield Bands in the Atmospheres of the Earth and Titan

Cosmic Research - Calculations were made of the volume and integral emission intensities of the Lyman–Birge–Hopfield (LBH) bands of molecular nitrogen at 146.4, 138.4, 135.4, and 132.5...     

CHANG'E-2 lunar escape maneuvers to the Sun–Earth L2 libration point mission

This paper addresses lunar escape maneuvers of the first Chinese Sun–Earth L2 libration point mission by the CHANG'E-2 satellite, which is also the world's first satellite to reach the L2 point from a lunar orbit. The lunar escape maneuvers are heavily constrained by the remaining propellant and the condition of telemetry, track and command, among others. First, these constraints are analyzed and summarized to design a target L2 Lissajous orbit and an initial transfer trajectory. Second, the maneuver mathematical models are studied. The multilevel maneuver schemes which consist of phasing maneuvers and a final lunar escape maneuver are designed for actual operations. Based on the scheme analysis and comparison, the 2-maneuver scheme with a 5.3-h-period phasing orbit is ultimately selected. Finally, the mission status based on the scheme is presented and the control operation results are discussed in detail. The methodology in this paper is especially beneficial and applicable to a future multi-mission instance in the deep space exploration.     

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.     

Spectral and spatial dependency of aerosol extinction in the upper troposphere based on the moon surface photometry in the Earth’s umbra

The work is based on photometry of the Moon’s surface during the total lunar eclipse of June 15, 2011. Photometric data are used to retrieve the value of aerosol extinction coefficient in the troposphere above the Earth’s limb in various spectral bands in the optical and near-IR ranges. The results are analyzed in comparison with the data of previous eclipses in 2004–2008. The main result consists in an increased aerosol concentration in the South-East Asia, which is maximum among all eclipses observed in recent years. At the same time, no relationship between eclipse brightness and solar activity is observed, as well as unified latitude dependency of umbra brightness noticed earlier.     

Relationship between the fluxes of relativistic electrons at geosynchronous orbit and the level of ULF activity on the Earth’s surface and in the solar wind during the 23rd solar activity cycle

We present the results of a cross-correlation analysis made on the basis of Spearman’s rank correlation method. The quantities to correlate are daily values of the fluence of energetic electrons at a geosynchronous orbit, intensities of ground and interplanetary ultra-low-frequency (ULF) oscillations in the Pc5 range, and parameters of the solar wind. The period under analysis is the 23rd cycle of solar activity, 1996–2006. Daily (from 6 h to 18 h of LT) magnetic data at two diametrically opposite observatories of the Intermagnet network are taken as ground-based measurements. The fluxes of electrons with energies higher than 2 MeV were measured by the geosynchronous GOES satellites. The data of magnetometers and plasma instruments installed on ACE and WIND spacecraft were used for analysis of the solar wind parameters and of the oscillations of the interplanetary magnetic field (IMF). Some results elucidating the role played by interplanetary ULF waves in the processes of generation of magneospheric oscillations and acceleration of energetic electrons are obtained. Among them are (i) high and stable correlation of ground ULF oscillations with waves in the solar wind; (ii) closer link of mean daily amplitudes of both interplanetary and ground oscillations with ‘tomorrow’ values of the solar wind velocity than with current values; and (iii) correlation of the intensity of ULF waves in the solar wind, normalized to the IMF magnitude, with fluxes of relativistic electrons in the magnetosphere.     

Effect of parametric and nonlinear perturbations on statistical dynamics of the Earth’s pole

We study the influence of additive and parametric slowly varying harmonic (at the Chandler frequency and doubled frequency) and stochastic Gaussian broadband perturbations on mathematical expectations, variances, and covariations of oscillations of the Earth’s pole. The influence of perturbations on both regular and irregular stochastic oscillations is considered in detail. Results of numerical experiments are presented. The developed models and software are included into information resources on the fundamental problem “Statistical dynamics of the Earth’s rotation” of Russian Academy of Sciences.     

Impact of helio-and geophysical disturbances on thermobaric and climatic characteristics of the Earth’s troposphere

We present the mechanism and the concept of a model of the solar activity impact on thermobaric and climatic characteristics of the troposphere. Both are based on the idea of parametric action. The results of analysis are presented concerning specific features and regularities of changes in temperature regime of the troposphere in the period of variable helio-and geophysical activity, as well as long-term variations of temperature and heat content of the troposphere. The influence of changes in circulation in the atmosphere and ocean on processes in the system atmosphere-ocean-cryosphere is considered: thermohaline circulation of the oceans and energy exchange between the atmosphere and ocean. The revealed regularities find their complete explanation within the context of a model and mechanism of solar activity impact on climatic characteristics of the troposphere that were suggested previously by the authors.     

A comparison between the 19th century early proposals and the 20th–21st centuries realized projects intended to contact other planets

Methods dealing with how to contact other planets that are supposed to be inhabited by “intelligent” civilizations have begun more than one century and a half ago. The historical question has been already treated in several studies and the aim of this paper is not to provide details on that aspect. On the other hand, it could be interesting to make a comparison between the different approaches to contact planets, formulated at different epochs (even if obviously techniques were not in the same state of advancement). The most important characteristics of the earliest messages, remained only on a theoretical form, will be presented. The main features of modern messages, which have been concretely realized, will also be emphasized. Drawing a parallel between these two series of projects could demonstrate what has been considered as unavoidable by both pioneer and modern messages creators, while it has not been proved that the first ones have had any influence on the second ones. The common points emerging from this comparison could then (perhaps) help to select adequate models for an intelligible message intended to ETs, particularly concerning the language forms. Besides this, the differences could illustrate the human cultural advances in the field of METI and underline the tendencies that have been chosen in that field since the last decades.     

Variations in the Parameters of Radio Waves in the Earth’s High-Latitude Ionosphere on the Satellite–Satellite Paths during the Geomagnetic Storm of June 22–23, 2015

Cosmic Research - This paper analyzes the results of about 100 radio occultation sessions of sounding of the high-latitude (>65 °N) lower ionosphere of the Earth’s Northern...     

A Study of the Reflection of Radio Waves from the Plasma Region on the Track behind an Axisymmetric Body Moving in the Earth’s Atmosphere

The results of numerical calculation of the dependences of the electron density, the eigenfrequency and the dielectric plasma permeability on the geometric parameters and the altitude of body motion in the near and far wake behind a thin conical body with a spherical nose blunting have been presented. The electron density maximum has been shown to be located in the region of the neck of the near wake behind the body, which determines the type of this region (supercritical or subcritical). This in turn affects the propagation of radio waves through this plasma region. A comparative analysis was performed for two different bodies with the same ballistic coefficient values. No characteristic distinctions were revealed in the values of electron density or the plasma eigenfrequency in the near and far wake behind these bodies. However, it has been shown that there are differences in the values of the distance from the bottom cross section to the neck of the near wake behind these bodies.     

Separation of the Galactic Cosmic Rays and Inner Earth Radiation Belt Contributions to the Daily Dose Onboard the International Space Station in 2005–2011

The DB-8 detectors of the ISS radiation monitoring system (RMS) have operated almost continuously onboard the ISS service module since August 2001 till December 2014. The RMS data obtained were used for the daily monitoring of the radiation environment aboard the station. This paper considers the technique of RMS data analysis that allows one to distinguish the contributions of galactic cosmic rays and the Earth’s inner radiation belt to the daily dose based on the dosimetry data obtained as a result of the station’s passage in areas of the highest geomagnetic latitudes. The paper presents the results of an analysis of the dosimetry data based on this technique for 2005–2011, as well as a comparison with similar results the authors obtained previously using the technique based on an analysis of the dosimetry data obtained during station passages in the area of the South Atlantic Anomaly.     

On increasing the spectral efficiency and transmissivity in the data transmission channel on the spacecraft–ground tracking station line

The algorithms for achieving a practical increase in the rate of data transmission on the space-craft–ground tracking station line has been considered. This increase is achieved by applying spectral-effective modulation techniques, the technology of orthogonal frequency compression of signals using millimeterrange radio waves. The advantages and disadvantages of each of three algorithms have been revealed. A significant advantage of data transmission in the millimeter range has been indicated.     

Erratum to: The Influence of the Earth’s Oblateness on the Energy Integral and Some Characteristics of a Spacecraft’s Orbit

Cosmic Research - An Erratum to this paper has been published: https://doi.org/10.1134/S0010952521120029     

Calculation and Study of Limited Orbits around the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> Libration Point of the Sun–Earth System

A procedure has been proposed for calculating limited orbits around the L₂ libration points of the Sun–Earth system. The motion of a spacecraft in the vicinity of the libration point has been considered a superposition of three components, i.e., decreasing (stable), increasing (unstable), and limited. The proposed procedure makes it possible to correct the state vector of the spacecraft so as to neutralize the unstable component of the motion. Using this procedure, the calculation of orbits around various types of libration points has been carried out and the dependence on the orbit type on the initial conditions has been studied.