Implementation of a semianalytic satellite theory with recovery of short period terms

First order averaging is applied to the artificial satellite problem to obtain the averaged orbit which includes the secular, long and medium period effects of the oblateness of the Earth and the third body perturbations of the moon and sun. Perturbation theory is then used to recover the short period effects due to J₂, the moon, and sun. The perturbation analysis is carried out by means of Lie series and is developed through the first order. Optimization of the resulting short period series was then accomplished in several steps: first all separate algebraic coefficients were precalculated and stored; then all redundant SIN/COS calls were eliminated; next all repetition of numeric and algebraic coefficients were precalculated in pairs; application of the distributive principle allowed a significant reduction in additions and multiplications; finally trigonometric identities were used to further reduce the SIN/COS computations. The result of this optimization along with an interpolator for the averaged equations of motion results in a computer program which requires only $\text{1}\text{6}$ the CPU time (with no loss in accuracy) of the original non-optimized test program.     

Collisions of stars by oscillating orbits of the second kind

Ordinary estimations of the number of star collisions in our galaxy—by simple kinematic considerations—lead to a very small number of such collisions: about one or even less every millions of years. However star collisions can occur through the following indirect way which has a much higher probability. (a) Binary stars are very common in our galaxy, about 30–50% of the stars. (b) If two binary stars meet a triple system can be formed by an ordinary exchange type motion. (c) A triple system is generally decomposed into the “inner orbit” (i.e. the relative orbit of the two nearest stars) and the “outer orbit” (i.e. the relative orbit of the third star with respect to the center of mass of the two nearest stars). The major axes of these two orbits have generally small perturbations and it is the same for the eccentricity of the outer orbit. On the contrary, if the relative inclination of the two orbits is large, the perturbations of the eccentricity of the inner orbit are important and can even in some cases lead to an eccentricity equal to one, that is to a collision of the two stars of the inner orbit.Such orbits can be called “oscillating orbits of the second kind”, indeed the first oscillating orbits—conceived by Khilmi and described for the first time in an example by Sitnikov—have unbounded mutual distances r_ij, but the system always come back to small sizes, it has an infinite number of very large expansions followed by strong contractions and, in the three-body case, an upper bound of lim inf (r_1.2 + r_1.3 + r_2.3) can be given in terms of the three masses and the integrals of motion. For the oscillating orbits of the second kind the mutual distances r_ij are bounded, but the velocities are unbounded (i.e. lim inf r_ij = 0 for at least one r_ij) and the system goes to a collision if the bodies have non-zero radius even small. The analytical study of the oscillating orbits of the second kind is a part of the general analytical study of the three-body problem, a part which must be valid for large eccentricities and large inclinations. The use of Delaunay's variables and of a Von Zeipel transformation lead to a first order integrable approximation, valid for any eccentricities and any inclinations, and giving the following results: (a) The oscillating orbits of the second kind occur when the angular momentum of the outer orbit has a modulus sufficiently close to the modulus of the total angular momentum of the three-body system. Hence these orbits occur for inclinations in the vicinity of 90°. (b) The oscillating orbits represent a set of positive measure of phase space and the first order study allows to give a rough estimation of the probability of collisions—even for stars of infinitely small radius. This probability, for given initial major axes and eccentricities and for isotropic arbitrary initial orientations, is generally of the order of m₃$\text{R}\text{M}$ (m₃ being the mass of the outer star, M the total mass and R the ratio of the period of the inner orbit to that of outer orbit).One question remains to be solved: how many collisions of stars are due to that phenomenon? That question is difficult because the probability of formation of a triple system by a random meeting of two binaries is very uneasy to estimate. However it seems that, compared to the usual evaluations based on pure kinematic considerations without gravitational effects, the number of collisions must be multiplied by a factor between one thousand and one million.     

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.     

Decay of high eccentricity satellite orbits with air drag using uniformly regular KS canonical elements

A non-singular analytical theory for the motion of high eccentricity satellite orbits under the influence of air drag is developed in terms of the Uniformly Regular Kustaanheimo and Stiefel (URKS) canonical elements, by assuming the atmosphere to be oblate with variation of density scale height with altitude. The series expansions include up to fourth-order terms of an independent variable Δ=λ² (function of eccentric anomaly) and c (a small parameter dependent on the flattening of the atmosphere). Only two of the nine equations are solved analytically due to symmetry in the equations of motion. Comparison of the important orbital parameters semi-major axis and eccentricity up to 1000 revolutions, obtained with the present analytical solution and the KS theory, shows the superiority of the present solution over the KS elements analytical solution. The theory can be used effectively for the orbital decay of aero-assisted orbital transfer orbits during mission planning.     

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.     

Lorentz-force-perturbed orbits with application to J2-invariant formation

The Lorentz force acting on an electrostatically charged spacecraft in the Earth's magnetic field provides a new propellantless means for controlling a spacecraft's orbit. Assuming that the Lorentz force is much smaller than the gravitational force, the perturbation of a charged spacecraft's orbit by the Lorentz force in the Earth's magnetic field, which is simplified as a titled rotating dipole, is studied in this article. Our research starts with the derivation of the equations of motion in geocentric equatorial inertial Cartesian coordinates using Lagrange mechanics, and then derives the Gauss variational equations involving Lorentz-force perturbation using a set of nodal inertial coordinates as an intermediate step. Subsequently, the approximate averaged changes in classical orbital elements, including single-orbit-averaged and one-day-averaged changes, are obtained by employing orbital averaging. We have found that the approximate analytic one-day-averaged changes in semi-major axis, eccentricity, and inclination are nearly zero, and those in the other three angular orbital elements are affected by J₂ and Lorentz-force perturbations. This characteristic is applied to model bounded relative orbital motion in the presence of the Lorentz force, which is termed Lorentz-augmented J₂-invariant formation. The necessary condition for J₂-invariant formation is derived when the chief spacecraft's reference orbit is either circular or elliptical. It is shown that J₂-invariant formation is easier to implement if the deputy spacecraft is capable of establishing electric charge. All conclusions drawn from the approximate analytic solutions are verified by numerical simulation.     

卫星轨道摄动频谱分析

本文基于Ｌｉｅ级数的方法对卫星轨道摄动进行了频谱分析。这种摄动是由地球引力场的所有带谱调和项和田谐调和项引起的。为了适用小偏心率轨道，采用了Ｈｉｌｌ变量来描述卫星轨道，并在所获得的谱分析式中保留了所有与偏心率ｅ无关的项。在这个基础上又经过简单的变换给出了卫星位置摄动三分量的谱分析式。     

<Emphasis Type="Italic">Luna-5</Emphasis> (1965): Some Results of a Failed Mission to the Moon

Luna-5 was the second Soviet spacecraft to reach the Moon. During the first decade of space exploration of the Moon, the Luna probe series was the main part of the Soviet scientific program. The tasks of the Luna-5 probe launched to the Moon in May 1965 were to land softly on the lunar surface, take photos, and study the surface. Before the Luna-5 landing, the prospective coordinates of the landing site were telegraphed to observatories so that they would observe the event. However, during its descent, the braking engine failed and the probe crash landed at 22 h 13 min on May 12, 1965. Later, new supposed coordinates of the impact were reported. All the experiments were undoubtedly lost; nevertheless, successive television images of the failed landing made at the Abastumani Astrophysical Observatory (AbAO) of the Georgian Academy of Sciences can be considered a specific scientific result of the mission. In the images, a changeable object was detected near the large Lansberg crater; for obscure secrecy reasons, almost nothing was reported to specialists about this object. It has been identified as a small, gradually spreading impact cloud. An analysis of the reprocessed images taken at the AbAO has revealed the exact coordinates of the Luna-5 impact for the first time to be 1.35° S, 25.48° W, which differ substantially the calculation data published earlier. Some properties of the regolith at the Luna-5 impact site are compared to the results of the Lunar Crater Observation and Sensing Satellite (LCROSS) related to the region near the south pole of the Moon and reported in 2010.     

Probabilistic assessment of radiation risk for astronauts in space missions

Accurate estimations of the health risks to astronauts due to space radiation exposure are necessary for future lunar and Mars missions. Space radiation consists of solar particle events (SPEs), comprised largely of medium energy protons (less than several hundred MeV); and galactic cosmic rays (GCR), which include high-energy protons and heavy ions. While the frequency distribution of SPEs depends strongly upon the phase within the solar activity cycle, the individual SPE occurrences themselves are random in nature. A solar modulation model has been developed for the temporal characterization of the GCR environment, which is represented by the deceleration potential, ?. The risk of radiation exposure to astronauts as well as to hardware from SPEs during extra-vehicular activities (EVAs) or in lightly shielded vehicles is a major concern for radiation protection. To support the probabilistic risk assessment for EVAs, which could be up to 15% of crew time² on lunar missions, we estimated the probability of SPE occurrence as a function of solar cycle phase using a non-homogeneous Poisson model [1] to fit the historical database of measurements of protons with energy>30 MeV, Φ₃₀. The resultant organ doses and dose equivalents, as well as effective whole body doses, for acute and cancer risk estimations are analyzed for a conceptual habitat module and for a lunar rover during space missions of defined durations. This probabilistic approach to radiation risk assessment from SPE and GCR is in support of mission design and operational planning for future manned space exploration missions.     

Perturbation effect of the Coulomb drag on the orbital elements of the earth satellite moving in the ionosphere

The perturbation effects of the Coulomb drag on the orbital elements of the earth satellite moving in the ionosphere are studied. The theoretical results show that the Coulomb drag results in both the secular and periodic variation in the semi-major axis and eccentricity. However, the argument of the perigee exhibits no secular variation, but only periodic variation. The inclination and the ascending node remain no variation. As an example, the secular effects of the Coulomb drag on the semi-major axis and the eccentricity of an ionosphere satellite Alouette (S-27) are calculated in the ionosphere with the mean height 1000 km. It can be shown that the semi-major axis contracts and the eccentricity decreases for the case of the Coulomb drag under the interaction of the ions with the electric field of an earth satellite.     

Artificial satellites orbits in 2:1 resonance: GPS constellation

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J₂₀ and J₄₀ and the tesseral harmonics J₂₂ and J₄₂. Through an averaging procedure and successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, three critical angles are studied. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. Through the resonance overlap criterion the possible regular and irregular motions are investigated by the time behavior of the semi-major axis, argument of perigee and eccentricity. The largest Lyapunov exponent is used as tool to verify the chaotic motion.     

Artificial satellites orbits in 2:1 resonance: GPS constellation

In this work, the resonance problem in the artificial satellites motion is studied. The development of the geopotential includes the zonal harmonics J₂₀ and J₄₀ and the tesseral harmonics J₂₂ and J₄₂. Through an averaging procedure and successive Mathieu transformations, the order of dynamical system is reduced and the final system is solved by numerical integration. In the simplified dynamical model, three critical angles are studied. The half-width of the separatrix is calculated through a linearized model which describes the behavior of the dynamical system in a neighborhood of each critical angle. Through the resonance overlap criterion the possible regular and irregular motions are investigated by the time behavior of the semi-major axis, argument of perigee and eccentricity. The largest Lyapunov exponent is used as tool to verify the chaotic motion.     

Stability of resonance rotation of a satellite with respect to its center of mass in the orbit plane

The stability of resonance oscillations and rotations of a satellite in the plane of its orbit in the case when the difference of the moments of inertia with respect to the principal axes lying in the orbit plane is small is determined at a given rotation number m by the sign of function Φ_m(e), introduced by F.L. Chernous’ko in 1963. In this paper, convenient analytical representations of functions Φ_m(e) are described in the form of integrals and series of Bessel functions regular at e → 1^?. Values of Φ_m(1) are calculated in explicit form. A theorem about the double asymptotic form of functions Φ_m(e) at m → ∞ and e → 1^? is proved by the saddlepoint method.     

Fuel-optimal eccentricities for precise formation keeping of spacecraft on Keplerian orbits

In this paper, the problem of precise relative position keeping associated with the formation flying of two spacecraft is discussed. Taking into account of the applications to astronomic observation, the relative position of spacecraft is controlled to be precisely retained on a part of an orbit rather than on the entire orbit; moreover, the relative position is retained in the inertial coordinates rather than in the local vertical local horizontal coordinates. The fuel-optimal eccentricities are computed for each length of control time. Then, it is shown that the fuel-optimal eccentricity is close to 1 when the control time is shorter than approximately 90% of the orbit period; on the contrary, the fuel-optimal eccentricity is in the neighborhood of 0 when the control time is close to the orbit period. Moreover, several analytic expressions including the propellant consumptions at the eccentricities of 0 and 1 are obtained.     

Physicochemical model of the auroral ionosphere

A physicochemical model of excited polar ionosphere has been presented. The model makes it possible to calculate vertical profiles of concentrations of the following excited and ionized constituents: O₂ ⁺, N₂ ⁺, O⁺(⁴S), O⁺(²D), O⁺(²P), O(¹D), O(¹S), N(⁴S), N(²D), N(²P), NO, NO⁺, N⁺, N₂(A³Σ_u ⁺), N₂(B³П_g), N₂(W³Δ_u), and N₂(B′³Σ_u ^-) and the electron concentration during electron precipitations. The energy spectrum of the electrons at the upper boundary of the ionosphere and concentrations of neutral constituents are the input parameters of the model. A model has been compiled based on available publications and includes 56 physicochemical reactions that influence concentrations of the aforementioned constituents in the polar ionosphere. The method of calculating vertical profiles of the excitation rates of atmospheric gases and proper allowance for the electron-vibrational kinetics in the processes of exciting the triplet states of N₂ are specific features of the presented model. The ionospheric model has been approbated using the results of the coordinated rocket–satellite experiment. The agreement between the modeling results and experimental data best for the time being is achieved.     

SETI terminology: Do we interpret SETI terms correctly?

Many important SETI terms are either ambiguously defined or interpreted by different experts differently. Based on the author's experience with astronautical terminology (IAA multilingual space dictionary) a summary of the usual problems connected with an uniform definition of fundamental terms is attempted. In the second part several examples are quoted from the SETI literature—including the terms ETI, SETI and METI themselves, the definition of a habitable zone, of alien life, of an extraterrestrial artifact, of the Drake equation, of the Fermi-paradox, etc. In the third part of the paper a new task for the SETI social sciences community is raised, namely to collect “Lasting Universal Terms”; i.e. terms the meaning of which did not change since millennia, are independent on geographic position and also on the terrestrial environment and biology. Such terms might be preferably used in interstellar communication. All these questions are related to the manner how we might think about ETI and SETI in new ways. The paper tries to summarize the problems connected with exact SETI terminology and its potential implications for the future.     

基于角锥体原理的月面巡视探测器定位方法

为了获取月面巡视探测器在月球表面的相对位置信息,提出了一种基于角锥体原理的相对定位方法。此方法无需初始值,仅利用巡视探测器获取的包含着陆器目标的影像,就可以实现巡视探测器高精度定位。通过嫦娥二期工程试验,验证了该算法的有效性,可为月面巡视探测器开展科学考察任务所应用。     

A methodology for system-of-systems design in support of the engineering team

Space missions have experienced a trend of increasing complexity in the last decades, resulting in the design of very complex systems formed by many elements and sub-elements working together to meet the requirements. In a classical approach, especially in a company environment, the two steps of design-space exploration and optimization are usually performed by experts inferring on major phenomena, making assumptions and doing some trial-and-error runs on the available mathematical models. This is done especially in the very early design phases where most of the costs are locked-in.With the objective of supporting the engineering team and the decision-makers during the design of complex systems, the authors developed a modelling framework for a particular category of complex, coupled space systems called System-of-Systems. Once modelled, the System-of-Systems is solved using a computationally cheap parametric methodology, named the mixed-hypercube approach, based on the utilization of a particular type of fractional factorial design-of-experiments, and analysis of the results via global sensitivity analysis and response surfaces.As an applicative example, a system-of-systems of a hypothetical human space exploration scenario for the support of a manned lunar base is presented. The results demonstrate that using the mixed-hypercube to sample the design space, an optimal solution is reached with a limited computational effort, providing support to the engineering team and decision makers thanks to sensitivity and robustness information. The analysis of the system-of-systems model that was implemented shows that the logistic support of a human outpost on the Moon for 15 years is still feasible with currently available launcher classes. The results presented in this paper have been obtained in cooperation with Thales Alenia Space—Italy, in the framework of a regional programme called STEPS.¹     

An inversion approach for lunar regolith layer thickness using optical albedo data and microwave emission simulation

An approach to inversion of the lunar regolith layer thickness by using multi-channel brightness temperature observation in passive microwave remote sensing is developed. To first make simulation of brightness temperature from the lunar layered media, the lunar regolith layer thickness (d) is proposed being constructed by available lunar DEM (digital elevation mapping) and on site measurements. The physical temperature distribution (T) over the lunar surface is also empirically assumed as a monotonic function of the latitude. Optical albedo of the lunar nearside from the telescopic observation is employed to construct the spatial distribution of the FeO+TiO₂ content (S) in the lunar regolith layer. A statistic relationship between the DEM and S of the lunar nearside is further extended to construction of S of the lunar farside. Thus, the dielectric permittivity (ε) of global lunar regolith layer can then be determined. Based on all these conditions (d,T,ε), brightness temperature of the lunar regolith layer in passive microwave remote sensing, which is planned for China's Chang-E lunar project, is numerically simulated by a parallel layering model using the strong fluctuation theory of random media.Then, taking these simulations with random noise as observations, an inversion method of the lunar regolith layer thickness is developed by using three- or two-channels brightness temperatures. When the S is low, and the four channels brightness temperatures in China's Chang-E project are well distinguishable, the regolith layer thickness and physical temperature of the underlying lunar rock media can be inverted by the three-channels approach. When the S becomes high that the brightness temperature at high frequency channels such as 19.35, 37 GHz are saturated, the regolith layer thickness is alternatively inverted only by the two-channels approach.Numerical simulation and inversion approach in this paper make an evaluation of the performance for lunar passive microwave remote sensing, and for future data calibration and validation.