Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

Solar Sail Halo Orbits at the Sun–Earth Artificial L 1 Point

Halo orbits for solar sails at artificial Sun–Earth L₁ points are investigated by a third order approximate solution. Two families of halo orbits are explored as defined by the sail attitude. Case I: the sail normal is directed along the Sun-sail line. Case II: the sail normal is directed along the Sun–Earth line. In both cases the minimum amplitude of a halo orbit increases as the lightness number of the solar sail increases. The effect of the z-direction amplitude on x- or y-direction amplitude is also investigated and the results show that the effect is relatively small. In case I, the orbit period increases as the sail lightness number increases, while in case II, as the lightness number increases, the orbit period increases first and then decreases after the lightness number exceeds ~0.01.     

Controlled Solar Sail Transfers into Near-Sun Regions Combined with Planetary Gravity-Assist Flybys

Results of numerical simulations of 'local-optimal' (or 'instantaneously optimal') trajectories of a space probe with a flat solar sail which moves from the circular Earth orbit to near-Sun regions are presented. We examine planar (ecliptic) solar sail transfer with gravity-assist flybys of Earth, Venus and Mercury. Several complex control modes of the sail tilt orientation angle for near-Sun orbits and for some 'falling onto the Sun' trajectories are investigated. The numerical simulations are used to examine the flight duration of some sail missions and to investigate the evolution of osculating elliptical orbits.     

Non-Keplerian orbits for electric sails

An electric sail is capable of guaranteeing the fulfilment of a class of trajectories that would be otherwise unfeasible through conventional propulsion systems. In particular, the aim of this paper is to analyze the electric sail capabilities of generating a class of displaced non-Keplerian orbits, useful for the observation of the Sun’s polar regions. These orbits are characterized through their physical parameters (orbital period and solar distance) and the spacecraft propulsion capabilities. A comparison with a solar sail is made to highlight which of the two systems is more convenient for a given mission scenario. The optimal (minimum time) transfer trajectories towards the displaced orbits are found with an indirect approach.     

Periodic and quasi-periodic motions of a solar sail close to SL 1 in the Earth–Sun system

Solar sails are a proposed form of spacecraft propulsion using large membrane mirrors to propel a satellite taking advantage of the solar radiation pressure. To model the dynamics of a solar sail we have considered the Earth–Sun Restricted Three Body Problem including the Solar radiation pressure (RTBPS). This model has a 2D surface of equilibrium points parametrised by the two angles that define the sail orientation. In this paper we study the non-linear dynamics close to an equilibrium point, with special interest in the bounded motion. We focus on the region of equilibria close to SL ₁, a collinear equilibrium point that lies between the Earth and the Sun when the sail is perpendicular to the Sun–sail direction. For different fixed sail orientations we find families of planar, vertical and Halo-type orbits. We have also computed the centre manifold around different equilibria and used it to describe the quasi-periodic motion around them. We also show how the geometry of the phase space varies with the sail orientation. These kind of studies can be very useful for future mission applications.     

The Sun–Earth saddle point: characterization and opportunities to test general relativity

The saddle points are locations where the net gravitational accelerations balance. These regions are gathering more attention within the astrophysics community. Regions about the saddle points present clean, close-to-zero background acceleration environments where possible deviations from General Relativity can be tested and quantified. Their location suggests that flying through a saddle point can be accomplished by leveraging highly nonlinear orbits. In this paper, the geometrical and dynamical properties of the Sun–Earth saddle point are characterized. A systematic approach is devised to find ballistic orbits that experience one or multiple passages through this point. A parametric analysis is performed to consider spacecraft initially on \(L_{1,2}\) Lagrange point orbits. Sun–Earth saddle point ballistic fly-through trajectories are evaluated and classified for potential use. Results indicate an abundance of short-duration, regular solutions with a variety of characteristics.     

New families of Sun-centered non-Keplerian orbits over cylinders and spheres

This paper introduces new families of Sun-centered non-Keplerian orbits (NKOs) that are constrained to a three-dimensional, cylindrical or spherical surface. As such, they are an extension to the well-known families of displaced NKOs that are confined to a two-dimensional plane. The cylindrical and spherical orbits are found by investigating the geometrically constrained spacecraft dynamics. By imposing further constraints on the orbit’s angular velocity and propulsive acceleration, the set of feasible orbits is defined. Additionally, the phase spaces of the orbits are explored and a numerical analysis is developed to find periodic orbits. The richness of the problem is further enhanced by considering both an inverse square acceleration law (mimicking solar electric propulsion) and a solar sail acceleration law to maintain the spacecraft on the three-dimensional surface. The wealth of orbits that these new families of NKOs generate allows for a range of novel space applications.     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².     

A Hopf variables view on the libration points dynamics

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.     

Three-dimensional time optimal double angular momentum reversal trajectory using solar sails

A new concept of three dimensional non-Keplerian trajectories with double angular momentum reversal is investigated with high performance solar sails. The main discussion of this paper is about such 3D solar inverse orbits with inner constraints. The problem is addressed in a time optimal control framework solved by an indirect method. Two typical solar inverse orbits have been achieved and presented in a 3D non-dimensional dynamic model in the Heliocentric Inertial Frame. Starting from the Earth orbit ecliptic plane, a sailcraft in the inverse orbit exhibits a butterfly shape trajectory. As such, the new orbits are symmetrical with respect to a plane which contains the Sun-perihelion line. The relation of the sail attitude angles between the two symmetrical parts of the orbits are used to reduce the simulation effort. The quasi-heliostationary property at its aphelia is demonstrated with variation of the orbital radius. Evolutions of the orbital velocity and optimal sail orientations are also outlined and discussed to benefit future design work. As is suited for space observation guaranteed by its butterfly shape, the inverse orbits are thoroughly studied in terms of the concerned parameters. The discussion of the parametric influence is ranked in order as perihelion distance r _E , required maximum position z _max, perihelion position z _f and the sail lightness number β. Suitable ranges of each parameter are adopted to illustrate the orbital variation trend. Through numerical simulations the features of such inverse orbits are further emphasized to provide an initial reference for future researchers.     

Utilization of an H-reversal trajectory of a solar sail for asteroid deflection

Near Earth Asteroids have a possibility of impacting the Earth and always represent a threat. This paper proposes a way of changing the orbit of the asteroid to avoid an impact. A solar sail evolving in an H-reversal trajectory is utilized for asteroid deflection. Firstly, the dynamics of the solar sail and the characteristics of the H-reversal trajectory are analyzed. Then, the attitude of the solar sail is optimized to guide the sail to impact the target asteroid along an H-reversal trajectory. The impact...     

Delta-v requirements for earth co-orbital rendezvous missions

Earth co-orbital asteroids present advantages as potential targets for future asteroid rendezvous missions. Their prolonged proximity to Earth facilitates communication, while their Earth-like orbits mean a steady flux of solar power and no significant periodic heating and cooling of the spacecraft throughout the course of the mission. Theoretical studies show that low-inclination co-orbital orbits are more stable than high-inclination orbits. As inclination is the most significant indicator of low delta-v rendezvous orbits, there is the potential for a large population of easily accessible asteroids, with favorable engineering requirements. This study first looks at phase-independent rendezvous orbits to a large number of objects, then looks in more detail at the phase-dependent orbits to the most favorable objects. While rendezvous orbits to co-orbital objects do not have a low delta-v necessarily, some objects present energy requirements significantly less than previous rendezvous missions. Currently we find no ideal co-orbital asteroids for rendezvous missions, although theoretical Earth Trojans present very low-energy requirements for rendezvous.     

Evolution of the \mathcal {L}_1 halo family in the radial solar sail circular restricted three-body problem

We present a detailed investigation of the dramatic changes that occur in the \(\mathcal {L}_1\) halo family when radiation pressure is introduced into the Sun–Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. in Int J Bifurc Chaos 1:493–520, 1991) the families can be fully mapped out as the parameter \(\beta \) is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at \(\beta \approx 0.0387\) acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at \(\beta \approx 0.289\) , resulting in a third new family. The linear stability of the families changes rapidly at low values of \(\beta \) , with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter \(\beta \) and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of \(\beta \) there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.     

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.     

Symmetries in the Optimal Control of Solar Sail Spacecraft

The theory of optimal control is applied to obtain minimum-time trajectories for solar sail spacecraft for interplanetary missions. We consider the gravitational and solar radiation forces due to the Sun. The spacecraft is modelled as a flat sail of mass m and surface area A and is treated dynamically as a point mass. Coplanar circular orbits are assumed for the planets. We obtain optimal trajectories for several interrelated problem families and develop symmetry properties that can be used to simplify the solution-finding process. For the minimum-time planet rendezvous problem we identify different solution branches resulting in multiple solutions to the associated boundary value problem. We solve the optimal control problem via an indirect method using an efficient cascaded computational scheme. The global optimizer uses a technique called Adaptive Simulated Annealing. Newton and Quasi-Newton Methods perform the terminal fine tuning of the optimization parameters.     

Fast Earth–Moon transfers with ballistic capture

This contribution deals with fast Earth–Moon transfers with ballistic capture in the patched three-body model. We compute ensembles of preliminary solutions using a model that takes into account the relative inclination of the orbital planes of the primaries. The ballistic capture orbits around the Moon are obtained relying on the hyperbolic invariant structures associated to the collinear Lagrangian points of the Earth–Moon system, and the Sun–Earth system portion of the transfers are quasi-periodic orbits obtained by a genetic algorithm. The trajectories are designed to be good initial guesses to search optimal cost-efficient short-time Earth–Moon transfers with ballistic capture in more realistic models.     

Astronomical Engineering Revisited: Planetary Orbit Modification Using Solar Radiation Pressure

As the Sun evolves along the main sequence its luminosity will grow, leading to a steadily increasing solar flux at the Earth with corresponding catastrophic consequences for the biosphere. A novel means of avoiding this terminal route to human evolution has recently been proposed by Korycansky et al. which utilises a series of grazing fly-pasts of the Earth with a small solar system body to increase the orbit radius of the Earth over a timescale of order 10⁹ years. This short paper will propose an alternative strategy which utilises a large reflective sail to generate a propulsive thrust due to solar radiation pressure. It will be shown that if the sail is configured to be in static equilibrium relative to the Earth, the centre-of-mass of the Earth-sail system slowly accelerates. This scheme offers some advantages in that the mass of the sail is four orders of magnitude less than the mass to be processed in the scheme of Korycanskyet al. for trajectory correction manoeuvres alone. In addition, the severe hazard posed by multiple grazing fly-pasts of the Earth by a small solar system body is avoided. Although offering significant advantages, any thoughts of engineering on an astronomical scale clearly requires a leap of the imagination and a ready use of liberal assumptions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constructing ballistic capture orbits in the real Solar System model

A method to design ballistic capture orbits in the real Solar System model is presented, so extending previous works using the planar restricted three-body problem. In this generalization a number of issues arise, which are treated in the present work. These involve reformulating the notion of stability in three-dimensions, managing a multi-dimensional space of initial conditions, and implementing a restricted \(n\) -body model with accurate planetary ephemerides. Initial conditions are categorized into four subsets according to the orbits they generate in forward and backward time. These are labelled weakly stable, unstable, crash, and acrobatic, and their manipulation allows us to derive orbits with prescribed behavior. A post-capture stability index is formulated to extract the ideal orbits, which are those of practical interest. Study cases analyze ballistic capture about Mercury, Europa, and the Earth. These simulations show the effectiveness of the developed method in finding solutions matching mission requirements.     

New Challenges to Trajectory Design by the Use of Electric Propulsion and Other New Means of Wandering in the Solar System

The design of spacecraft trajectories is a crucial part of a space mission design. Often the mission goal is tightly related to the spacecraft trajectory. A geostationary orbit is indeed mandatory for a stationary equatorial position. Visiting a solar system planet implies that a proper trajectory is used to bring the spacecraft from Earth to the vicinity of the planet. The first planetary missions were based on conventional trajectories obtained with chemical engine rockets. The manoeuvres could be considered 'impulsive' and clear limitations to the possible missions were set by the energy required to reach certain orbits. The gravity-assist trajectories opened a new way of wandering through the solar system, by exploiting the gravitational field of some planets. The advent of other propulsion techniques, as electric or ion propulsion and solar sail, opened a new dimension to the planetary trajectory, while at the same time posing new challenges. These 'low thrust' propulsion techniques cannot be considered 'impulsive' anymore and require for their study mathematical techniques which are substantially different from before. The optimisation of such trajectories is also a new field of flight dynamics, which involves complex treatments especially in multi-revolution cases as in a lunar transfer trajectory. One advantage of these trajectories is that they allow to explore regions of space where different bodies gravitationally compete with each other. We can exploit therefore these gravitational perturbations to save fuel or reduce time of flight. The SMART-1 spacecraft, first European mission to the Moon, will test for the first time all these techniques. The paper is a summary report on various activities conducted by the project team in these areas.