Orbital manoeuvres of free flying platforms for retrieval by space shuttle

In a typical future mission a free flying platform will be released to space by Space Shuttle. After performing its active mission, it will have to wait for a suitable later Shuttle flight for retrieval at its original orbital altitude. To allow for the orbital descent during the total mission time of typically several months, one or several orbit raise manoeuvres have to be performed with the platform's own propulsion system. In the paper, the velocity-requirements Δv for these orbital transfers, depending on Sun activity, rendezvous-altitude, ballistic coefficient and longest expected mission time are treated.The simplest manoeuvre, consisting of one initial ascent transfer and one descent transfer at the actual retrieval date, is shown to be not optimal. Up to 25% of Δv can be saved, if several orbit raising transfers in a certain sequence are applied. A straightforward analytical treatment is presented for the optimization, while a computer program with the CIRA-atmosphere model is used for actual mission planning.     

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.     

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities e_S and e_J, respectively; the mean motion of Jupiter n_J is also small. We obtain the averaged Hamiltonian function for a small parameter of ε = n_J and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I₂; (β) the occurrence of new libration zones of oscillations I₂ near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I₂ at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).     

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.     

Results of the early orbit manoeuvres of the ampte-uks

The Active Magnetospheric Particle Tracer Explorers (AMPTE) program consists of three satellites which were launched on 16th August 1984. The scientific aim of the mission is to inject lithium and barium tracer ions inside and outside the Earth's magnetosphere and to detect and monitor these ions as they diffuse through the inner magnetosphere. The first of these three satellites, the U.S. Charge Composition Explorer (CCE) was launched into an elliptical orbit of apogee 8 R_e. The other two satellites are the West German Ion Release Module (IRM) and the U.K. Subsatellite (UKS), both of which were launched on the same vehicle into a highly elliptical orbit of apogee 18 R_e. At discreet intervals during the mission the IRM will release ions into the solar wind, and the movement of these ions will be monitored by the UKS. Depending on the particular scientific requirement, the UKS has to be positioned accurately at a given distance behind the IRM. Initially the UKS has to be located 100 km behind the IRM, and held there for ～9 months. It will then be moved a distance of ～1 R_e behind the IRM. In order to manoeuvre the UKS around its orbit, a cold gas jet system is incorporated on the satellite, allowing impulses to be applied both along and perpendicular to the orbit velocity vector. The orbit control system also has to cater for relative orbit changes due to air drag at perigee, as the IRM and the UKS have different $\text{area}\text{mass}$ ratios. This paper presents an account of the orbit control system implemented on the UKS, together with the mathematical approach adopted, and results from manoeuvres made in the first weeks of the mission.     

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.     

Prediction of turbulent diffusion flames with a four-equation turbulence model

For turbulent flame prediction a four-equation turbulence model consisting of the turbulent kinetic energy, the dissipation rate and the density-velocity correlations using unweighted statistics is suggested. The fluctuating concentrations are taken into account by using Spalding's (1971) equation for the variance of the mixture fraction f and by defining the probability density function of f as beta-function. The thermodynamic model for the combustion of H₂ with air assumes infinitely fast reaction in a global single step. Preliminary results are shown for the calculation of Kent and Bilger (1972) turbulent H₂-air diffusion flame. For this flame with fluctuating density the inclusion of the velocity-density correlations appears to be essential, because they have a strong effect especially on the development of the turbulent viscosity and the mean velocity and mass fraction profiles. Consequently the mean concentrations are in good agreement with the experimental results.     

Survey of mission evolution and flexibility in the Space Shuttle program

Given the diversity of missions it has accomplished and the myriad of adaptations it has undergone, the US Space Shuttle is widely regarded as a highly flexible space vehicle. With the Shuttle’s upcoming 2011 retirement, it is instructive to survey the history of this vehicle’s flexibility for the insights it can provide to the design and characterization of flexibility in future space systems. Data are presented on the evolution of mission requirements over time for 120 missions performed by the Space Shuttle over a period of some 27 years. Distinct trends in the time domain – as well as their causes – are identified and discussed, and early manifest plans from 1982 serve as a confirmation that these trends were not originally anticipated. Eight examples are then presented of engineering modifications that allowed the Shuttle to adapt and accommodate these requirement changes. Several additional instances of Shuttle flexibility are explored, such as post-Columbia disaster modification, upgrade programs and derived vehicles, and one case in which flexibility was inhibited by an early design decision.     

The shuttle development and its growth potential

During the next two decades, we will establish the foundation for the 21st century's accomplishments in space. The Space Shuttle vehicle will become the cornerstone for that foundation by providing economical opportunities for space exploration and utilization.Reusability of the Shuttle vehicle is the key to its economy. The major developmental challenges encountered in the Shuttle program are typified by the complexities involved in designing the reusable propulsion and thermal protection subsystems. We successfully met such challenges and are nearing the launch of the first Shuttle orbital flight.Our immediate goal is to enter the Space Shuttle operational phase because only then will we fully understand the unique capabilities of the Shuttle. Concurrent with our effort to begin Shuttle operations are our initial efforts to expand Shuttle capabilities, extending them significantly beyond those of the current baseline system.Shuttle payload capacity and mission-duration capabilities are to increase considerably during the next decade. Just as present Shuttle performance specifications and development timetables were guided by the space program plans and forecasts of the 1960s, so will the development of long-range space programs be determined by our near-future achievements. We anticipate that the Space Shuttle will play a critical role in those achievements.     

Microdisturbances on the International Space Station during dynamic operations

The results of analysis of microdisturbances on the International Space Station (ISS) at performing various dynamic operations are presented. Docking of transfer manned and cargo vehicles Progress and Soyuz to various docking modules of the ISS, docking of the Space Shuttle Discovery, the ISS orbit correction and, also, disturbances at “EVA” (Extra Vehicular Activity) operations during astronauts working on the external ISS surface are considered. The results of measuring microaccelerations by sensors of both Russian and American segments are analyzed.     

Halo orbits in the vicinity of the L 2 libration point in the Sun-Earth system

Methods are proposed for constructing the orbits of spacecraft remaining for long periods of time in the vicinity of the L ₂ libration point in the Sun-Earth system (so-called halo orbits), and the trajectories of uncontrolled flights from low near-Earth orbits to halo orbits. Halo orbits and flight trajectories are constructed in two stages: A suitable solution to a circular restricted three-body problem is first constructed and then transformed into the solution for a restricted four-body problem in view of the real motions of the Sun, Earth, and Moon. For a halo orbit, its prototype in the first stage is a combination of a periodic Lyapunov solution in the vicinity of the L ₂ point and lying in the plane of large-body motion, with the solution for the linear second-order system describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is found as the solution to the three-body problem best approximating the prototype in the mean square. The constructed orbit serves as a similar prototype in the second stage. In both stages, the approximating solution is constructed by continuation along a parameter that is the length of the approximation interval. Flight trajectories are constructed in a similar manner. The prototype orbit in the first stage is a combination of a solution lying in the plane of large-body motion and a solution for a linear second-order system describing small deviations of the spacecraft from this plane. The planar solution begins near the Earth and over time tends toward the Lyapunov solution existing in the vicinity of the L ₂ point. The initial conditions of both prototypes and the approximating solutions correspond to the spacecraft’s departure from a low near-Earth orbit at a given distance, perigee, and inclination.     

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.     

Potential cost reductions for three STS/Spacelab mission modes

Based on the results of studies carried out by ESA several possibilities are discussed to achieve mission cost reductions for large Spacelab instrument facilities as compared to their flight on several 7-day duration Spacelab missions. As an example three scientific telescope facilities are selected (LIRTS, EXSPOS, GRIST) which are defined to a Phase A level.Three new mission modes are considered:

• —Shuttle attached Spacelab mission mode with extended flight duration (up to 30 days) for which the application of planned capability extensions and new elements of the STS/Spacelab (e.g. Short Spacelab Pallets, Power Extension Package) are investigated.

• —Shuttle deployed mission mode, for which the telescope, accommodated on a Spacelab pallet, is docked to the Power Module, a new element of the Space Transportation System under study by NASA.

• —Free-flying mission mode, for which Shuttle launched dedicated missions of the facilities are considered, assuming varying degrees of autonomy with respect to supporting services of the Shuttle.

Reduction of costs have been considered on the levels of single mission cost and total programme cost. Fundamentally the charges for the instrument can be reduced by constraining the mass/volume factors with respect to the Shuttle capability. However, the instrument as part of a payload is only viable if an acceptable resource sharing including observation time can be achieved. Any single instrument will require several mission opportunities or one mission which achieves a similar or longer total observation programme.Based on an identification of instrument modifications of the Phase A baseline designs to favour cost reductions and on a derivation of technical requirements, constraints and finally budgetary cost comparisons an attempt is made to assess the advantages and disadvantages of the different mission modes.The favoured option for GRIST is a 2–3 weeks sortie mission followed after refurbishment by a longer Power Module docked mission. For LIRTS and EXSPOS the free-flying pallet modes are very attractive in terms of the longer durations achieved and in terms of cost per unit operating time.     

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.     

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.     

Forecasting the velocity of quasi-stationary solar wind and the intensity of geomagnetic disturbances produced by it

A brief review is given of contemporary approaches to solving the problem of medium-term forecast of the velocity of quasi-stationary solar wind (SW) and of the intensity of geomagnetic disturbances caused by it. At the present time, two promising models of calculating the velocity of quasi-stationary SW at the Earth’s orbit are realized. One model is the semi-empirical model of Wang-Sheeley-Arge (WSA) which allows one to calculate the dependence V(t) of SW velocity at the Earth’s orbit using measured values of the photospheric magnetic field. This model is based on calculation of the local divergence f _S of magnetic field lines. The second model is semi-empirical model by Eselevich-Fainshtein-Rudenko (EFR). It is based on calculation in a potential approximation of the area of foot points on the solar surface of open magnetic tubes (sources of fast quasistationary SW). The new Bd-technology is used in these calculations, allowing one to calculate instantaneous distributions of the magnetic field above the entire visible surface of the Sun. Using predicted V(t) profiles, one can in EFR model calculate also the intensity of geomagnetic disturbances caused by quasi-stationary SW. This intensity is expressed through the K _p index. In this paper the EFR model is discussed in detail. Some examples of epignosis and real forecast of V(t) and K _p (t) are discussed. A comparison of the results of applying these two models for the SW velocity forecasting is presented.     

Single-impulse transfers from orbits of artificial satellites to orbits around <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> or <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> libration points

In the context of the restricted circular three-body problem a method for constructing families of periodic orbits is described. Each orbit contains a segment of transfer from artificial satellite orbit of a smaller body to an orbit around L ₁ or L ₂ points of the Sun-Earth and Earth-Moon systems, a segment of multiple flyby of this libration point, and a segment of return to the artificial satellite orbit. Dependences of velocities at the pericenter on the pericenter radius are given.     

Equilibrium attitude and stability of a spacecraft on a stationary orbit around an asteroid

The stationary orbits around an asteroid, if exist, can be used for communication and navigation purposes just as around the Earth. The equilibrium attitude and stability of a rigid spacecraft on a stationary orbit around a uniformly-rotating asteroid are studied. The linearized equations of attitude motion are obtained under the small motion assumption. Then, the equilibrium attitude is determined in both cases of a general and a symmetrical spacecraft. Due to the higher-order inertia integrals of the spacecraft, the equilibrium attitude is slightly away from zero Euler angles. Then necessary conditions of stability of this conservative system are analyzed based on the linearized equations of motion. The effects of different parameters, including the harmonic coefficients C₂₀ and C₂₂ of the asteroid and higher-order inertia integrals of the spacecraft, on the stability are assessed and compared. Due to the significantly non-spherical shape and rapid rotation of the asteroid, the effects of the harmonic coefficients C₂₀ and C₂₂ are very significant, while effects of the third- and fourth-order inertia integrals of the spacecraft can be neglected. Considering a spacecraft on a stationary orbit around an example asteroid, we show that the classical stability domain predicted by the Beletskii–DeBra–Delp method on a circular orbit in a central gravity field is modified due to the non-spherical mass distribution of the asteroid. Our results are confirmed by a numerical simulation.