Maximizing the value of an Earth observation satellite orbit

Earth observation satellites are platforms equipped with optical instruments that orbit the planet. During the course of an orbit, they take photographs of some regions of the Earth at the request of customers. Each photograph generates a profit but, due to the presence of several constraints, not all requests can be satisfied. The problem is to select a subset of requests of maximal profit for a given orbit. The problem is solved by means of a tabu search heuristic and computational results are reported. This work was initiated as part of a challenge organized by the French Operational Research Society. The algorithm won the second prize in the final round of the competition.     

Distance Preserving Recombination Operator for Earth Observation Satellites Operations Scheduling

The paper describes an adaptation of a memetic algorithm to the problem of scheduling operations of Earth observation satellites. The adaptation uses a systematic approach to the design of the recombination operator preserving important features common to both parents. The important features are identified experimentally on the basis of correlations between the value of the objective function and the similarity of good solutions. Our results indicate that this systematic approach reduces the effort needed to design a high quality recombination operator by avoiding not promising development directions.      

Satellites in the gravitational field of the moon

Expressions are derived correct to the first order in the ellipticities of the Moon for the radial shift and the rotation of the unperturbed orbit of a close satellite in the plane of its instantaneous motion. Criteria for the rotation to be a libration lead to the existence of a critical range of values of orbital inclination as against a single value cos^?1 (·2)^1/2=63·4° quoted for the Earth satellites. It appears that mean radial shift and the perturbation in the period of the satellite, also possess similar ranges. The perturbation in the velocity vector is then analysed and it is shown that the angle between the unperturbed and real trajectories vanishes only at points where the radial distance of the satellite from the selenocentre attains extreme values. The interesting case of nearly circular trajectories is considered next in some detail. It is deduced that the critical inclination for the orbital libration is given byα=60° for an Earth satellite in this case. The paper concludes with numerical results based on Jeffreys’ estimates.     

High-accuracy attitude control of satellites in elliptic orbits

A class of feedback attitude-control systems which efficiently result in high-accuracy (10^–4 radians), Earth-pointing motions of satellites of different configurations in elliptic orbits is developed. Gas jets provide the control torque. The state of the attitude motion is assumed to be completely known. Linear differential equations, which have time-varying coefficients and include terms for the gravity torque due to an oblate Earth and terms for the aerodynamic torque, are used to describe the attitude motion when the satellite is practically Earth-pointing. Pontryagin's maximum principle, the necessary conditions for exact solutions of optimal bounded-phase-coordinate problems, and guidelines obtained from the minimum-fuel attitude controls devised for single-axis systems are used in the development. The motions of somestable and someunstable satellites with the developed attitude-control systems are simulated on an analog computer. The overall control system, which is reliable as well as efficient, is simple to realize.This research was sponsored by the National Aeronautics and Space Administration under Grant No. G-133-61.     

Scheduling for single agile satellite,redundant targets problem using complex networks theory

Scheduling for the Earth observation satellites (EOSs) imaging mission is a complicated combinatorial optimization problem, especially for the agile EOSs (AEOSs). The increasing observation requirements and orbiting satellites have exacerbated the scheduling complexity in recent years. In this paper, the single agile satellite, redundant observation targets scheduling problem is studied. We introduce the theory of complex networks and find similarities between AEOS redundant targets scheduling problem and the node centrality ranking problem. Then we model this problem as a complex network, regarding each node as a possible observation opportunity, and define two factors, node importance factor and target importance factor, to describe the node/target importance. Based on the two factors, we propose a fast approximate scheduling algorithm (FASA) to obtain the effective scheduling results. Simulation results indicate the FASA is quite efficient and with broad suitability. Our work is helpful in the EOSs and AEOSs scheduling problems by using complex network knowledge.     

Well-Posedness of a Problem of Propagation of Nonlinear Acoustic-Gravity Waves in the Atmosphere Generated by Surface Pressure Variations

Currently there are many international microbarograph networks for high-resolution recording of wave pressure variations on the Earth’s surface. This arouses interest in wave propagation in the atmosphere generated by atmospheric pressure variations. A full system of nonlinear hydrodynamic equations for atmospheric gases with lower boundary conditions in the form of wavelike pressure variations on the Earth’s surface is considered. Since the wave amplitudes near the Earth’s surface are small, linearized equations are used in the analysis of well-posedness of the problem. With the help of a wave energy functional method, it is shown that in the non-dissipative case the solution to the boundary value problem is uniquely determined by the variable pressure field on the Earth’s surface. The corresponding dissipative problem is well-posed if, in addition to the pressure field, appropriate conditions on the velocity and temperature on the Earth’s surface are given. In the case of an isothermal atmosphere, the problem admits analytical solutions that are harmonic in the variables x and t. A good agreement between the numerical and analytical solutions is obtained. The study shows that the temperature and density can rapidly vary at the lower boundary of the boundary value problem. An example of solving the three-dimensional problem with variable pressure on the Earth’s surface taken from experimental observations is given. The developed algorithms and computer programs can be used to simulate atmospheric waves generated by pressure variations on the Earth’s surface.     

Fractal and multifractal analysis of the rise of oxygen in Earth’s early atmosphere

The rise of oxygen in Earth’s atmosphere that occurred 2.4–2.2 billion years ago is known as the Earth’s Great Oxidation, and its impact on the development of life on Earth has been profound. Thereafter, the increase in Earth’s oxygen level persisted, though at a more gradual pace. The proposed underlying mathematical models for these processes are based on physical parameters whose values are currently not well-established owing to uncertainties in geological and biological data. In this paper, a previously developed model of Earth’s atmosphere is modified by adding different strengths of noise to account for the parameters’ uncertainties. The effects of the noise on the time variations of oxygen, carbon and methane for the early Earth are investigated by using fractal and multifractal analysis. We show that the time variations following the Great Oxidation cannot properly be described by a single fractal dimension because they exhibit multifractal characteristics. The obtained results demonstrate that the time series as obtained exhibit multifractality caused by long-range time correlations.     

Two-echelon vehicle routing problem with satellite bi-synchronization

In considering route optimization at a series of express stages from pickup to delivery via the intercity linehaul, we introduce the two-echelon vehicle routing problem with satellite bi-synchronization (2E-VRP-SBS) from the perspective of modeling the routing problems of two-echelon networks. The 2E-VRP-SBS involves the inter-satellite linehaul on the first echelon, and the pickups from senders to origin satellites (i.e., satellites for cargo collection) and deliveries from destination satellites (i.e., satellites for cargo deliveries) to receivers on the second echelon. The 2E-VRP-SBS integrates satellite bi-synchronization constraints, multiple vehicles, and time window constraints on the two-echelon network and aims to find cost-minimizing routes for various types of trucks. Satellite bi-synchronization constraints, which synchronously guarantee the synchronization at origin satellites and the synchronization at destination satellites, provide an innovative method to formulate the two-echelon routing problem. In this study, we develop a mixed-integer programming model for the 2E-VRP-SBS. An exact method using CPLEX solver is presented and a modified adaptive large neighborhood search is conducted. Furthermore, the effectiveness of the 2E-VRP-SBS formulation and the applicability of the heuristic for various instances are experimentally evaluated.     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body. Original Russian Text ? V.I. Prokhorenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 156–173.     

Identification and orbit determination of a tethered satellite system

Orbital motion of a tethered satellite system, composed of two satellites and an inextensible tether, is considered by using a perturbed two-body model. This approach is adopted so that the determination of the orbit of one of the satellites can be attempted without using observations of the motion of the other satellite in the system. The identification of the tethered condition of the system using observations of only one of the satellites in the tethered satellite system is considered. The characteristics of the `tether perturbed' motion of the observed satellite are investigated. Estimation of the state of the system using near perfect data is also illustrated. Observations of one satellite provide the entire state of the system and a parameter involving the ratio of the masses of satellites and the tether length.     

On the Dynamics of Space Debris: 1:1 and 2:1 Resonances

We study the dynamics of the space debris in the 1:1 and 2:1 resonances, where geosynchronous and GPS satellites are located. By using Hamiltonian formalism, we consider a model including the geopotential contribution for which we compute the secular and resonant expansions of the Hamiltonian. Within such model we are able to detect the equilibria and to study the main features of the resonances in a very effective way. In particular, we analyze the regular and chaotic behavior of the 1:1 and 2:1 resonant regions by analytical methods and by computing the Fast Lyapunov Indicators, which provide a cartography of the resonances. This approach allows us to detect easily the location of the equilibria, the amplitudes of the libration islands and the main dynamical stability features of the resonances, thus providing an overview of the 1:1 and 2:1 resonant domains under the effect of Earth’s oblateness. The results are validated by a comparison with a model developed in Cartesian coordinates, including the geopotential, the gravitational attraction of Sun and Moon and the solar radiation pressure.     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body.     

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.     

Analysis of deformations of the Earth's crust by laser interferometers

Results of measurement of phugoid oscillations of the Earth by a laser interferometric complex, which contains two interferometers of Michelson type, are given. Using the results of 900 h of processing, the spectrum of oscillations of the Earth in the range of periods from 20 to 310 min is obtained as well as the spectrum of free oscillations of the Earth in the seismic quiet time. The multiplet spectrum of oscillations of the Earth in the interval of 160 min is isolated.Translated from Dinamicheskie Sistemy, No. 9, pp. 60–65, 1990.     

Optimization of expedition to Phobos using the impulse control and solution to Lambert problems taking into account attraction of the earth and mars

The trajectory of expedition to Mars and its satellite Phobos with return to the Earth is optimized. The attraction of the Sun, or the Earth, or Mars is considered at each part of the trajectory. The Earth and Mars positions correspond to ephemerides DE424, and the position of Phobos corresponds to ephemerides MAR097. Not more than 6 impulses are assumed on the trajectory. The spacecraft must start from the Earth in the period of 2020–2030 and stay at Phobos at least for 30 days. The total time of the expedition is limited to 1500 days. The characteristic velocity is minimized.     

Critical solution for a Hill's type problem

We study the problem of two satellites attracted by a center of force. Assuming that the motion of the center of mass of the two satellites is a Keplerian circular orbit around the center of force, we regularize the collision between them using the Levi-Civita procedure. The existence of a constant of motion in the extended phase space allows us to study the stability of the solution, where the two satellites are tied together in their circular motion around the center of force. We call this solution the critical solution. A theorem of M. Kummer is applied to prove, in specific conditions, the existence of two one-parametric families of almost periodic orbits for the satellites motion that bifurcates from the critical solution.     

Optimal Trajectories for Earth-to-Mars Flight

This paper deals with the optimal transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The transfer problem is formulated via a restricted four-body model in that the spacecraft is considered subject to the gravitational fields of Earth, Mars, and Sun along the entire trajectory. This is done to achieve increased accuracy with respect to the method of patched conics.The optimal transfer problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome numerical difficulties due to large changes in the gravitational field near Earth and near Mars. The optimization criterion is the total characteristic velocity, namely, the sum of the velocity impulses at LEO and LMO. The major parameters are four: velocity impulse at launch, spacecraft vs. Earth phase angle at launch, planetary Mars/Earth phase angle difference at launch, and transfer time. These parameters must be determined so that V is minimized subject to tangential departure from circular velocity at LEO and tangential arrival to circular velocity at LMO.For given LEO and LMO radii, a departure window can be generated by changing the planetary Mars/Earth phase angle difference at launch, hence changing the departure date, and then reoptimizing the transfer. This results in a one-parameter family of suboptimal transfers, characterized by large variations of the spacecraft vs. Earth phase angle at launch, but relatively small variations in transfer time and total characteristic velocity.For given LEO radius, an arrival window can be generated by changing the LMO radius and then recomputing the optimal transfer. This leads to a one-parameter family of optimal transfers, characterized by small variations of launch conditions, transfer time, and total characteristic velocity, a result which has important guidance implications. Among the members of the above one-parameter family, there is an optimum–optimorum trajectory with the smallest characteristic velocity. This occurs when the radius of the Mars orbit is such that the associated period is slightly less than one-half Mars day.     

Algorithm of orbit determination using one or two GPS satellites

The problem of orbit determination using one or two GPS satellites is discussed. Methods of getting initial values based on linear translation is presented; the Secant method and the descend Newton iterative procedure and the continuation algorithm are used synthetically to solve the nonlinear equations. Computer simulation shows that this algorithm can give preliminary state of satellite orbit with a certain precision in short time.     

Solution of the inverse magnetotelluric sounding problem by means of a method using the synthesized magnetic field

A method of solution of the inverse magnetotelluric sounding problem is considered. The method uses the known frequency-dependent magnetic field on the Earth’s surface. The magnetic field on the Earth’s surface is found with the help of a special technique from the known impedance on the Earth’s surface. The method is applied to solve typical inverse problems and provides for efficient determination of final solutions.     

Interannual variations in the length of day and ENSO events in 1982–1983 and 1997–1998

The length of day series during the period of 1962.0–2000.0, the atmospheric angular momentum and the Southern Oscillation Index are adopted to analyze the relationships among the ENSO events that have occurred since 1960, the changes in the length of day and the atmospheric angular momentum. Attention is particularly given to the different effects of the 1982–1983 and 1997–1998 ENSO events on the variations of Earth rotation. The synthetic excitation effects of multi-scale atmospheric oscillations on the anomalous variations of the interannual rates of Earth rotation are revealed by means of the time-frequency spectrum of the wavelet transform.