Preliminary designs of low-thrust transfer trajectories are developed in the Earth-moon three-body problem with variable specific impulse engines and fixed engine power. The solution for a complete time history of the thrust magnitude and direction is initially approached as a calculus of variations problem to locally maximize the final spacecraft mass. The problem is then solved directly by sequential quadratic programming, using either single or multiple shooting. The coasting phase along the transfer exploits invariant manifolds and, when possible, considers locations along the entire manifold surface for insertion. Such an approach allows for a nearly propellant-free final coasting phase along an arc selected from a family of known trajectories that contract to the periodic libration point orbit. This investigation includes transfer trajectories from an Earth parking orbit to some sample libration point trajectories, including L 1 halo orbits, L 1 and L 2 vertical orbits, and L 2 butterfly orbits. Given the availability of variable specific impulse engines in the future, this study indicates that fuel-efficient transfer trajectories could be used in future lunar missions, such as south pole communications satellite architectures. Nomenclature c = constraint vector G = endpoint function H = Hamiltonian I sp = engine specific impulse i = instantaneous inclination angle m = total spacecraft mass P = engine power magnitude R = rotation matrix r, v = position and velocity vectors, Earth-moon rotating frame S = design variable vector T = engine thrust magnitude t = time U = pseudopotential function u c = control vector u T = thrust direction unit vector X = uncontrolled state vector, Earth-moon rotating framê = eigenvector at a fixed point on a periodic orbit = control constraint Lagrange multiplier vector = instantaneous argument of latitude M = anglelike manifold parameter = costate vector M = timelike manifold parameter = state transition matrix = kinematic boundary condition vector = instantaneous right ascension angle $, = kinematic boundary condition Lagrange multiplier vectors Subscripts c = control variable f = final condition FP = fixed point condition MS = multiple shooting phase p = powered phase PO = parking orbit condition s = stable manifold SS = single shooting phase u = unstable manifold 0 = initial condition
A potential ground station at the lunar south pole has prompted studies of orbit architectures that ensure adequate coverage. Constant communications can be achieved with two spacecraft in different combinations of Earth-Moon libration point orbits. Halo and vertical families, as well as other orbits near L1 and L2 are considered. The investigation includes detailed results using nine different orbits with periods ranging from 7 to 16 days. Natural solutions are generated in a full ephemeris model, including solar perturbations. A preliminary station-keeping analysis is also completed.
Potential orbits for continuous surveillance of the lunar south pole with just one spacecraft and a solar sail are investigated. Periodic orbits are first computed in the Earth-Moon Restricted Three-Body Problem, using Hermite-Simpson and seventh-degree Gauss-Lobatto collocation schemes. The schemes are easily adapted to include path constraints favorable for lunar south pole coverage. The methods are robust, generating a control history and a nearby solution with little information required for an initial guess. Five solutions of interest are identified and, using collocation, transitioned to the full ephemeris model (including the actual Sun-to-spacecraft line and lunar librations). Of the options investigated, orbits near the Earth-Moon L2 point yield the best coverage results. Propellant-free transfers from a geosynchronous transfer orbit to the coverage orbits are also computed. A steering law is discussed and refined by the collocation methods. The study indicates that solar sails remain a feasible option for constant lunar south pole coverage.
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