Transfers between libration-point orbits in the elliptic restricted problem

A strategy is formulated to design optimal time-fixed impulsive transfers between three-dimensional libration-point orbits in the vicinity of the interiorL ₁ libration point of the Sun-Earth/Moon barycenter system. The adjoint equation in terms of rotating coordinates in the elliptic restricted three-body problem is shown to be of a distinctly different form from that obtained in the analysis of trajectories in the two-body problem. Also, the necessary conditions for a time-fixed two-impulse transfer to be optimal are stated in terms of the primer vector. Primer vector theory is then extended to non-optimal impulsive trajectories in order to establish a criterion whereby the addition of an interior impulse reduces total fuel expenditure. The necessary conditions for the local optimality of a transfer containing additional impulses are satisfied by requiring continuity of the Hamiltonian and the derivative of the primer vector at all interior impulses. Determination of the location, orientation, and magnitude of each additional impulse is accomplished by the unconstrained minimization of the cost function using a multivariable search method. Results indicate that substantial savings in fuel can be achieved by the addition of interior impulsive maneuvers on transfers between libration-point orbits.An earlier version was presented as Paper AAS 92–126 at the AAS/AIAA Spaceflight Mechanics Meeting, Colorado Springs, Colorado, February 24–26, 1992.     

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.     

On the Dynamics of Weak Stability Boundary Lunar Transfers

Recent studies demonstrate that lunar and solar gravitational assists can offer a good reduction of total variation of velocity Vneeded in lunar transfer trajectories. In particular the spacecraft, crossing regions of unstable equilibrium in the Earth—Moon—Sun system, can be guided by the Sun towards the lunar orbit with the energy needed to be captured ballistically by the Moon. The dynamics of these transfers, called weak stability boundary (WSB) transfers, will be studied here in some detail. The crucial Earth—Moon—Sun configurations allowing such transfers will be defined. The Sun's gravitational effect and lunar gravitational capture will be analyzed in terms of variations of the Jacobi constants in the Earth—Sun and Earth—Moon systems. Many examples will be presented, supporting the understanding of the dynamical mechanism of WSB transfers and analytical formulas will be obtained in the case of quasi ballistic captures.This revised version was published online in October 2005 with corrections to the Cover Date.     

Impulsive time-free transfers between halo orbits

A methodology is developed to design optimal time-free impulsive transfers between three-dimensional halo orbits in the vicinity of the interior L ₁ libration point of the Sun-Earth/Moon barycenter system. The transfer trajectories are optimal in the sense that the total characteristic velocity required to implement the transfer exhibits a local minimum. Criteria are established whereby the implementation of a coast in the initial orbit, a coast in the final orbit, or dual coasts accomplishes a reduction in fuel expenditure. The optimality of a reference two-impulse transfer can be determined by examining the slope at the endpoints of a plot of the magnitude of the primer vector on the reference trajectory. If the initial and final slopes of the primer magnitude are zero, the transfer trajectory is optimal; otherwise, the execution of coasts is warranted. The optimal time of flight on the time-free transfer, and consequently, the departure and arrival locations on the halo orbits are determined by the unconstrained minimization of a function of two variables using a multivariable search technique. Results indicate that the cost can be substantially diminished by the allowance for coasts in the initial and final libration-point orbits.An earlier version was presented as Paper AIAA 92-4580 at the AIAA/AAS Astrodynamics Conference, Hilton Head Island, SC, U.S.A., August 10–12, 1992.     

Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

Optimal transfers between unstable periodic orbits using invariant manifolds

This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.     

Optimal Mars transfers for small payload transportation

Interplanetary transfers represent one of the most interesting themes of astrodynamics, because of its complexity and outcomes for human exploration of the Solar System. A wide number of works concerning different aspects of the interplanetary mission have been developed. The examination of these works leads to the conclusion that, by far, there is not a preferential propulsion system or an optimal trajectory to perform an interplanetary mission, but a precise kind of transfer according to a given mission profile. Here, minimum time trajectories to Mars for small payload transportation with different electric propulsion systems have been analyzed; results have been obtained considering the initial impulse given by the Ariane 5 upper stage. Additionally, an adaptative, multiple shooting optimization algorithm is proposed to solve the problem of optimality in interplanetary transfers with a low continuous thrust. The algorithm searches for the optimal set of initial Lagrange multipliers solving the two point problem by adapting the search intervals according to the unsmooth shape of the augmented cost function.     

Tracking a Ulysses High-latitude ICME Event Back to Its Solar Origins

High-latitude interplanetary mass ejections (ICMEs) observed beyond 1 AU are not studied very often. They are useful for improving our understanding of the 3D heliosphere. As there are only few such events registered by the Ulysses spacecraft, the task of detecting their solar counterparts is a challenge, especially during high solar activity periods, because there are dozens coronal mass ejections (CMEs) registered by SOHO that might be chosen as candidates. We analyzed a high-latitude ICME registered by the Ulysses spacecraft on 18 January 2002. Our investigation focused on the correlation between various plasma parameters that allow the identification to be made of the ICME and its components such as the forward shock, the magnetic cloud and the reverse shock.     

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.     

The gravitation theory in bimetric formulation

The gravitation theory is considered in bimetric formulation with a Lagrangian of the most common type and quadratic with respect to the derivativesg _ik/l of the metric tensor. Restrictions on the dimensionless parameters of the theory are found, basing on the comparison with the Newtonian theory and the analysis of corresponding observational data in the vicinity of the solar system. Propagation of the weak gravitation waves is investigated in the outside gravitational field.     

Minimum fuel control of the planar circular restricted three-body problem

The circular restricted three-body problem is considered to model the dynamics of an artificial body submitted to the attraction of two planets. Minimization of the fuel consumption of the spacecraft during the transfer, e.g. from the Earth to the Moon, is considered. In the light of the controllability results of Caillau and Daoud (SIAM J Control Optim, 2012), existence for this optimal control problem is discussed under simplifying assumptions. Thanks to Pontryagin maximum principle, the properties of fuel minimizing controls is detailed, revealing a bang-bang structure which is typical of L¹-minimization problems. Because of the resulting non-smoothness of the Hamiltonian two-point boundary value problem, it is difficult to use shooting methods to compute numerical solutions (even with multiple shooting, as many switchings on the control occur when low thrusts are considered). To overcome these difficulties, two homotopies are introduced: One connects the investigated problem to the minimization of the L²-norm of the control, while the other introduces an interior penalization in the form of a logarithmic barrier. The combination of shooting with these continuation procedures allows to compute fuel optimal transfers for medium or low thrusts in the Earth–Moon system from a geostationary orbit, either towards the L ₁ Lagrange point or towards a circular orbit around the Moon. To ensure local optimality of the computed trajectories, second order conditions are evaluated using conjugate point tests.     

Aero-gravity assist maneuvers: controlled dynamics modeling and optimization

The aero-gravity assist maneuver is here proposed as a tool to improve the efficiency of the gravity assist as, thanks to the interaction with the planetary atmospheres, the angular deviation of the velocity vector can be definitely increased. Even though the drag reduces the spacecraft velocity, the overall Δυ gain could be remarkable whenever a high lift-to-drag vehicle is supposed to fly. Earlier studies offer simplified approaches according to both the dynamics modeling and the atmospheric trajectory constraints. In this paper a 3D dynamical model is adopted and a more realistic L/D performance for the hypersonic vehicle is assumed. Some relevant aspects related to the multidisciplinary design have been considered such as heating rates and structural loads bounding. Comparisons between in and out of plane maneuvering have been performed by assuming, as control variables, either the angle of attack or the bank angle, respectively. The optimal control problem has been solved by selecting a direct method approach. The dynamics has been transcribed into a set of non-linear constraints and the arising non-linear programming problem has been solved with a sequential quadratic programming solver. To gain the global optimum convergence the initial guess has been supplied by solving the same problem by a direct shooting technique and a genetic optimizer.     

Halo orbit transfer trajectory design using invariant manifold in the Sun-Earth system accounting radiation pressure and oblateness

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.     

Spacecraft orbit analysis

The true orbit of a spacecraft differs from its reference orbit by small deviations due to erros in planetary and initial data, radiation and impact accelerations, and correction terms to Newton's theory of gravitation. These distortions are usually small enough to haveadditive effects on range, at the 10 m accuracy level, throughout the mission. Closed form expressions are derived (in cylindrical coordinates) for thedeviation inrange due to any given perturbing acceleration.     

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.     

Solar Wind Effects Associated with a Recurrent,High-Latitude Coronal Brightness Enhancement

During the descent of Ulysses following the 2001 solar north pole passage, the SOHO LASCO C2 telescope recorded a particularly strong sequence of recurrent polarization brightness (pB) features at latitudes of around 55°. As Ulysses passed overhead, solar rotation swept the interplanetary extensions of these persistent coronal structures over the spacecraft. Comparison of solar remote sensing and Ulysses in situ observations through 2002 reveals the solar wind effects of very bright and recurrent K-coronal structures at high solar latitudes and of a steeply inclined heliospheric neutral sheet (HNS). Despite the high level of solar activity, the HNS at high latitude still organizes solar wind stream structure much as it did near the previous solar minimum. The recurrent coronal streamers originate slow solar wind and mark the northern extremity of a very tilted HNS whose passage at Ulysses is accompanied by slow, dense solar wind, enhanced temperature, depressed α abundance, enhanced magnetic fields, and magnetic field directional changes that evolve with spacecraft latitude.     

Planar heliocentric roto-translatory motion of a spacecraft with a solar sail of complex shape

A complete treatment of the general motion of rotation and translation of a solar-sail spacecraft is proposed for the non-flat sail of complex shape. The planar heliocentric roto-translatory motion is considered, orbit-rotational coupling in the problem of altitude and orbital sail motion is investigated for the two-folding sail formed by two unequal reflective rectangular plates oriented at a right angle. The problem of orbit-rotational coupling is essentially a planar one: both sail plates are orthogonal to the orbital plane. The possibility of the non-controlled interplanetary transfer with such two-folding sail at its passive radiational orientation is established analytically from point of view of orbit-rotational coupling. Optimal geometric proportions of this sail are found at minimum-time interplanetary transfers.     

The use of invariant manifolds for transfers between unstable periodic orbits of different energies

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.     

A Study of Correlation between Solar Wind Data Observed at Two Points in Interplanetary Space During the Recent Solar Maximum

Identifying co-rotating structures in solar wind enables us to predict solar wind variation at the Earth and, hence, geomagnetic disturbances. However, co-rotating structures during solar maximum are sometimes difficult to see. We correlated solar wind data obtained by two spacecraft, Nozomi heading towards Mars and ACE at the L1 point, from late 1999 through early 2002. There were intervals when the solar wind showed specific co-rotating structures even in the midst of the solar maximum, whereas no correlation was found during the other intervals. The coefficient was generally higher between Nozomi and ACE than for the 27-day recurrence at ACE, while there was some correlation, especially when the difference in longitude between the two spacecraft was less than 120°. Although frequency of occurrence of CMEs is partly responsible for the correlation, the results can be interpreted in terms of rapid changes in co-rotating high-speed streams from near-equatorial coronal holes at the solar maximum.     

On the overtaking interaction of typical shock waves in the solar wind flow

The interaction of traveling fast solar shock waves with other fast shock waves generated previously is considered in terms of magnetohydrodynamics for various solar wind parameters. The shocks are not piston ones and move freely in the flow. The magnetic structure in the interplanetary magnetic field emerging after the shock interaction is shown to correspond to the well-known magnetic configuration commonly observed on spacecraft or the classical Hundhausen R model. A head-on collision of solar shock waves with the boundary of a magnetic cloud is considered. It is pointed out that a slow shockwave refracted into the magnetic cloud can appear at an oblique collision of the shock with the cloud boundary. The results clarify our understanding of the available spacecraft data.