Analytical solutions for the relative motion of spacecraft subject to Lorentz-force perturbations

A spacecraft capable of producing higher-than-natural electrostatic charges may achieve propellantless orbital maneuvering via the Lorentz-force interaction with a planetary magnetic field. Development of maneuver strategies for these propellantless vehicles is complicated by the fact that the perturbative Lorentz force acts along only a single line of action at any instant. Relative-motion dynamical models are developed that lead to approximate analytical solutions for the motion of charged spacecraft subject to the Lorentz force. These solutions indicate that the principal effects of the Lorentz force on a spacecraft in a circular orbit are to change the intrack position and to change the orbit plane. A rendezvous example is presented in which a spacecraft with a specific charge of ?3.81 $\times$ 10^?4 C/kg reaches a target vehicle initially 10 km away (on the same equatorial low-Earth orbit) in 1 day. Fly-around maneuvers may be achieved in low-Earth orbit with specific charges on the order of 0.001 C/kg.     

State propagation in an uncertain asteroid gravity field

Various spacecraft have been and will be sent to asteroids to characterize them. Generally, an asteroid's gravity field is very irregular and not accurately known when compared to the gravity field of a major planet, Earth in particular. It has been well studied that the irregularity significantly affects the trajectory of an orbiting spacecraft, and causes it to impact or to escape from the asteroid. Complementary to that, this paper focuses on the influence of the limited knowledge of this gravity field on the evolution of the spacecraft's orbit. It develops a general method by which this influence can be quantified. This method comprises specific Monte Carlo simulations with a discrete set of low-altitude orbits, taking into account the uncertainties in the gravity-field parameters. For illustration purposes, it is applied to two different asteroids. Already after three revolutions, the gravity-field uncertainties propagate to significant position uncertainties; this specifically holds for prograde orbits, and around the smaller asteroid. Applying this robust and accurate method helps mission designers and planners to assess the risk posed by gravity uncertainties, and take appropriate measures such as choosing the most favorable orbital geometries and/or lowering the orbit more slowly.     

On the possibility of long-term maintenance of geostationary satellite attitude without using the external information sensors and inertial sensors

The problem of maintaining spacecraft attitude during the loss of information from attitude sensors and inertial sensors is considered. The problem is solved on the basis of the force gyro stabilization principle with producing an angular momentum in the plane of orbit. The redundant mode of attitude maintenance is developed for spacecraft of the Yamal series. The results of testing the mode during the in-flight tests of the Yamal-200 spacecraft are presented.     

A Concept of Guidance for an Air-Launch Vehicle with Compensation of Initial Downrange and Launch Time Errors at Direct Insertion into a Rendezvous Point in the Orbit

A flying launcher (airplane carrier) can generate initial errors in position and time of launch. In order to compensate for these errors, one should have two control parameters in addition to those that provide for a spacecraft's injection into a preset orbit. We suggest the concept of controlling the trajectory of injection by choosing thrust values (within allowable regions of control) of second-stage engines or/and of a space booster of the Polyot carrier launcher. As an example, a rendezvous of the spacecraft at the end of its boost phase with the International Space Station (ISS) is considered. The methodology of the suggested approach can be extended to other mobile systems of launch to rendezvous orbits.     

Design of optimal earth pole-sitter transfers using low-thrust propulsion

Recent studies have shown the feasibility of an Earth pole-sitter mission using low-thrust propulsion. This mission concept involves a spacecraft following the Earth's polar axis to have a continuous, hemispherical view of one of the Earth's poles. Such a view will enhance future Earth observation and telecommunications for high latitude and polar regions. To assess the accessibility of the pole-sitter orbit, this paper investigates optimum Earth pole-sitter transfers employing low-thrust propulsion. A launch from low Earth orbit (LEO) by a Soyuz Fregat upper stage is assumed after which solar electric propulsion is used to transfer the spacecraft to the pole-sitter orbit. The objective is to minimize the mass in LEO for a given spacecraft mass to be inserted into the pole-sitter orbit. The results are compared with a ballistic transfer that exploits manifold-like trajectories that wind onto the pole-sitter orbit. It is shown that, with respect to the ballistic case, low-thrust propulsion can achieve significant mass savings in excess of 200 kg for a pole-sitter spacecraft of 1000 kg upon insertion. To finally obtain a full low-thrust transfer from LEO up to the pole-sitter orbit, the Fregat launch is replaced by a low-thrust, minimum time spiral, which provides further mass savings, but at the cost of an increased time of flight.     

Orbit prediction and shuttle rendezvous strategy for eureca

The European Retrievable Carrier (EURECA) is a platform to be launched, deployed and retrieved in low Earth orbit by the Space Shuttle.A newly developed analytical orbit prediction method is described which meets the severe requirements for EURECA's orbit propagation. It is based on an averaging procedure including the Earth's zonal harmonics J₂, J₃ and J₄ and a refined treatment of the air drag perturbation where EURECA's large solar panels are taken into account. Some orbit prediction results are included.In order to offer more flexibility for the Shuttle retrieval of EURECA, it is proposed to execute a part of the rendezvous manoeuvres by EURECA. A corresponding strategy is described.     

Orbit design for future SpaceChip swarm missions in a planetary atmosphere

The effect of solar radiation pressure and atmospheric drag on the orbital dynamics of satellites-on-a-chip (SpaceChips) is exploited to design equatorial long-lived orbits about the oblate Earth. The orbit energy gain due to asymmetric solar radiation pressure, considering the Earth's shadow, is used to balance the energy loss due to atmospheric drag. Future missions for a swarm of SpaceChips are proposed, where a number of small devices are released from a conventional spacecraft to perform spatially distributed measurements of the conditions in the ionosphere and exosphere. It is shown that the orbit lifetime can be extended and indeed selected through solar radiation pressure and the end-of-life re-entry of the swarm can be ensured, by exploiting atmospheric drag.     

All-propulsion design of the drag-free and attitude control of the European satellite GOCE

This paper concerns the drag-free and attitude control (DFAC) of the European Gravity field and steady-state Ocean Circulation Explorer satellite (GOCE), during the science phase. GOCE aims to determine the Earth's gravity field with high accuracy and spatial resolution, through complementary space techniques such as gravity gradiometry and precise orbit determination. Both techniques rely on accurate attitude and drag-free control, especially in the gradiometer measurement bandwidth (5–100 mHz), where non-gravitational forces must be counteracted down to micronewton, and spacecraft attitude must track the local orbital reference frame with micro-radian accuracy. DFAC aims to enable the gravity gradiometer to operate so as to determine the Earth's gravity field especially in the so-called measurement bandwidth (5–100 mHz), making use of ion and micro-thruster actuators. The DFAC unit has been designed entirely on a simplified discrete-time model (Embedded Model) derived from the fine dynamics of the spacecraft and its environment; the relevant control algorithms are implemented and tuned around the Embedded Model, which is the core of the control unit. The DFAC has been tested against uncertainties in spacecraft and environment and its code has been the preliminary model for final code development. The DFAC assumes an all-propulsion command authority, partly abandoned by the actual GOCE control system because of electric micro-propulsion not being fully developed. Since all-propulsion authority is expected to be imperative for future scientific and observation missions, design and simulated results are believed to be of interest to the space community.     

Sсheme of rendezvous mission to lunar orbital station by spacecraft launched from Earth

In recent years, great experience has been accumulated in manned flight astronautics for rendezvous in near-Earth orbit. During flights of Apollo spacecraft with crews that landed on the surface of the Moon, the problem of docking a landing module launched from the Moon’s surface with the Apollo spacecraft’s command module in a circumlunar orbit was successfully solved. A return to the Moon declared by leading space agencies requires a scheme for rendezvous of a spacecraft launched from an earth-based cosmodromee with a lunar orbital station. This paper considers some ballistic schemes making it possible to solve this problem with minimum fuel expenditures.     

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.     

Pressing Issues of the Day in Studying Deep Space

Two problems in studying deep space are discussed that are, in the author's opinion, the most important. The first is soil sampling from the smaller bodies of the Solar System, such as the Martian satellite Phobos and asteroids of groups C and S of the Main Asteroid Belt. This soil (so-called primordial substance) can help to elucidate some problems of the Solar System's formation; in particular, to construct a reliable model of the internal structure of the Earth. The second problem is to reveal all sufficiently large asteroids penetrating inside the Earth's orbit and to catalog those asteroids that are hazardous from the viewpoint of collision with the Earth. To this end, it is suggested to launch five or six Earth-orbiting spacecraft with telescopes capable of recording objects down to a brightness of 22– 25 ^m . It is pointed out that both problems can be solved in the near future using comparatively cheap standardized space vehicles launched into near-Earth orbits by the Soyuz carrier rocket and boosted further by electro-jet engines of small thrust.     

Analysis and implementation of in-plane stationkeeping of continuously perturbed Walker constellations

The stationkeeping of symmetric Walker constellations is analyzed by considering the perturbations arising from a high order and degree Earth gravity field and the solar radiation pressure. These perturbations act differently on each group of spacecraft flying in a given orbital plane, causing a differential drift effect that would disrupt the initial symmetry of the constellation. The analysis is based on the consideration of a fictitious set of rotating reference frames that move with the spacecraft in the mean sense, but drift at a rate equal to the average drift rate experienced by all the vehicles over an extended period. The frames are also allowed to experience the J₂-precession such that each vehicle is allowed to drift in 3D relative to its frame. A two-impulse rendezvous maneuver is then constructed to bring each vehicle to the center of its frame as soon as a given tolerance deadband is about to be violated. This paper illustrates the computations associated with the stationkeeping of a generic Walker constellation by maneuvering each leading spacecraft within an orbit plane and calculating the associated velocity changes required for controlling the in-plane motions in an exacting sense, at least for the first series of maneuvers. The analysis can be easily extended to lower flying constellations, which experience additional perturbations due to drag.     

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.     

Trajectory correction of the Spektr-R spacecraft motion

The results of refining the parameters of the Spektr-R spacecraft (RadioAstron project) motion after it was launched into the orbit of the Earth’s artificial satellite in July 2011 showed that, at the beginning of 2013, the condition of staying in the Earth’s shadow was violated. The duration of shading of the spacecraft exceeds the acceptable value (about 2 h). At the end of 2013 to the beginning of 2014, the ballistic lifetime of the spacecraft completed. Therefore, the question arose of how to correct the trajectory of the motion of the Spektr-R satellite using its onboard propulsion system. In this paper, the ballistic parameters that define the operation of onboard propulsion system when implementing the correction, and the ballistic characteristics of the orbital spacecraft motion before and after correction are presented.     

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.     

Fine structure of interplanetary shock fronts from the data of the BMSW instrument of the PLASMA-F experiment

The paper is concerned with studying the thickness of fronts of 38 interplanetary shocks detected by the BMSW instrument, which is a part of the scientific payload of the SPEKTR-R spacecraft, which was launched into a highly elliptical orbit in 2011. The main parameters of the interplanetary shocks have been calculated as follows: the ratio of thermal pressure to magnetic pressure before the front β, the angle between the shock front normal and the undisturbed magnetic field θ_Bn, the ratio of the shock propagation velocity to the magnetosonic velocity in the undisturbed region M_ms, and the shock front velocity relative to the Earth. It has been shown that the front thickness determined from the plasma parameters approximately matches the front thickness obtained from the magnetic field measurements and lies between 0.5 and 5 proton inertial lengths. In some events, the oscillations have been observed (upstream and downstream of the shock) in plasma parameters and in the magnetic field data. The length has been found to be between 0.5 and 6 proton inertial lengths for the preceding oscillations and between 0.5 and 10 proton inertial lengths for the following oscillations. The average value of the proton inertial length is 62 km.     

MESSENGER at Mercury: Early orbital operations

The MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) spacecraft, launched in August 2004 under NASA's Discovery Program, was inserted into orbit about the planet Mercury in March 2011. MESSENGER's three flybys of Mercury in 2008–2009 marked the first spacecraft visits to the innermost planet since the Mariner 10 flybys in 1974–1975. The unprecedented orbital operations are yielding new insights into the nature and evolution of Mercury. The scientific questions that frame the MESSENGER mission led to the mission measurement objectives to be achieved by the seven payload instruments and the radio science experiment. Interweaving the full set of required orbital observations in a manner that maximizes the opportunity to satisfy all mission objectives and yet meet stringent spacecraft pointing and thermal constraints was a complex optimization problem that was solved with a software tool that simulates science observations and tracks progress toward meeting each objective. The final orbital observation plan, the outcome of that optimization process, meets all mission objectives. MESSENGER's Mercury Dual Imaging System is acquiring a global monochromatic image mosaic at better than 90% coverage and at least 250 m average resolution, a global color image mosaic at better than 90% coverage and at least 1 km average resolution, and global stereo imaging at better than 80% coverage and at least 250 m average resolution. Higher-resolution images are also being acquired of targeted areas. The elemental remote sensing instruments, including the Gamma-Ray and Neutron Spectrometer and the X-Ray Spectrometer, are being operated nearly continuously and will establish the average surface abundances of most major elements. The Visible and Infrared Spectrograph channel of MESSENGER's Mercury Atmospheric and Surface Composition Spectrometer is acquiring a global map of spectral reflectance from 300 to 1450 nm wavelength at a range of incidence and emission angles. Targeted areas have been selected for spectral coverage into the ultraviolet with the Ultraviolet and Visible Spectrometer (UVVS). MESSENGER's Mercury Laser Altimeter is acquiring topographic profiles when the slant range to Mercury's surface is less than 1800 km, encompassing latitudes from 20°S to the north pole. Topography over the remainder of the southern hemisphere will be derived from stereo imaging, radio occultations, and limb profiles. MESSENGER's radio science experiment is determining Mercury's gravity field from Doppler signals acquired during frequent downlinks. MESSENGER's Magnetometer is measuring the vector magnetic field both within Mercury's magnetosphere and in Mercury's solar wind environment at an instrument sampling rate of up to 20 samples/s. The UVVS is determining the three-dimensional, time-dependent distribution of Mercury's exospheric neutral and ionic species via their emission lines. During each spacecraft orbit, the Energetic Particle Spectrometer measures energetic electrons and ions, and the Fast Imaging Plasma Spectrometer measures the energies and mass per charge of thermal plasma components, both within Mercury's magnetosphere and in Mercury's solar-wind environment. The primary mission observation sequence will continue for one Earth year, until March 2012. An extended mission, currently under discussion with NASA, would add a second year of orbital observations targeting a set of focused follow-on questions that build on observations to date and take advantage of the more active Sun expected during 2012–2013. MESSENGER's total primary mission cost, projected at $446 M in real-year dollars, is comparable to that of Mariner 10 after adjustment for inflation.     

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.     

Broad-search algorithms for the spacecraft trajectory design of Callisto–Ganymede–Io triple flyby sequences from 2024 to 2040, Part II: Lambert pathfinding and trajectory solutions

Triple-satellite-aided capture employs gravity-assist flybys of three of the Galilean moons of Jupiter in order to decrease the amount of ΔV$\mathrm{\Delta }V$ required to capture a spacecraft into Jupiter orbit. Similarly, triple flybys can be used within a Jupiter satellite tour to rapidly modify the orbital parameters of a Jovicentric orbit, or to increase the number of science flybys. In order to provide a nearly comprehensive search of the solution space of Callisto–Ganymede–Io triple flybys from 2024 to 2040, a third-order, Chebyshev's method variant of the p-iteration solution to Lambert's problem is paired with a second-order, Newton–Raphson method, time of flight iteration solution to the _V∞$V_{\infty }$-matching problem. The iterative solutions of these problems provide the orbital parameters of the Callisto–Ganymede transfer, the Ganymede flyby, and the Ganymede–Io transfer, but the characteristics of the Callisto and Io flybys are unconstrained, so they are permitted to vary in order to produce an even larger number of trajectory solutions. The vast amount of solution data is searched to find the best triple-satellite-aided capture window between 2024 and 2040.