Optimal low-thrust out-of-ecliptic trajectories

Low-thrust trajectories may offer a better means of entering out-of-ecliptic trajectories than ballistic maneuvers. Previous efforts examined families of trajectories resulting from the optimization of ballistic maneuvers with the expressed intent of reaching a given height above the ecliptic plane. This displacement serves to reduce the noise due to dust in the local zodiacal cloud. Low-thrust trajectories offer a more efficient means of reaching required out-of ecliptic orbits and also offer the possibility of increased observation time. The low-thrust trajectories developed in this paper were found by using an iterative genetic algorithm approach followed by a local optimizer. The paper examines the feasibility of using different thrust systems based on current and near term capabilities. INTRODUCTION In an ongoing effort to observe bodies outside the solar system, NASA has proposed a variety of spacebased observatories such as Next Generation Space Telescope (NGST), and Terrestrial Planet Finder (TPF). Scientists are searching for extra-solar planets in the habitable zone with the intent of locating liquid water. This search also targets candidates whose masses, densities, and temperatures do not prohibit life. Both the Earth’s local space environment and current launch vehicle capabilities limit the practical and accessible locations for space based observatories. The ecliptic plane, defined by the Earth’s orbit about the sun, is surrounded by a large amount of interplanetary dust (IPD), called the zodiacal cloud. This dust is the result of planetary construction and collisions between bodies in the solar system. Because the debris in the zodiacal cloud obscures the visible and infrared wavelengths, observatories close to Earth and in the ecliptic plane are required to deal with large amounts of noise in their measurement spectra. To avoid this noise, reduce mirror sizes, and reduce payload mass, the authors of Reference 2 proposed placing observatories in orbits reaching significant distances normal to the ecliptic plane. The orbits in Reference 2 consist of optimal ballistic maneuvers that maximize the distance from the ecliptic plane for a given energy. Another means of reaching high inclinations relative to the ecliptic plane is low thrust propulsion. Instead of a singular impulsive velocity change, low thrust propulsion systems remain active for a finite duration. Low-thrust propulsion’s efficiency addresses the large energy requirements of high normal displacements. Previous low-thrust propulsion studies examined optimization of interplanetary missions and interdiction of Earthcrossing asteroids. These works cite low thrust propulsion’s ability to decrease flight times, transfer energy and fuel/payload ratios. The optimal trajectories described below offer a more cost effective means of reaching target normal displacements. Low-thrust propulsion offers the added ability to “hover” by using control forces to maintain a steady-state position and velocity relative to Earth. This approach offers longer observation times than ballistic trajectories do. Because of the observation benefits, a low-thrust trajectory for a spacecraft with an integral propulsion system capable of station-keeping offers increased scientific payoff for a given duration mission. The purpose of this work is to optimize low thrust trajectories providing minimal time/energy solutions for a given mission duration and to identify promising propulsion systems. To ensure a global search space, a niching genetic algorithm is used to optimize control. The first version of the problem minimizes transfer time to a target normal displacement with limited thrust. Using these results, a minimum energy transfer to the same normal displacement is investigated. Difficulties in ensuring convergence are overcome by successive application of genetic algorithms to smaller subspaces centered on local minima. A local optimizer is applied to the resulting control histories to ensure compliance with terminal constraints. An analysis of current and near term low-thrust propulsion systems indicates that solar sails are the most promising candidates for missions studied in this paper. Since they do not require propellant to generate thrust, solar sails can hover for indefinite periods. State of the art Hall thrusters can realize the optimal transfers, but require AIAA/AAS Astrodynamics Specialist Conference and Exhibit 5-8 August 2002, Monterey, California AIAA 2002-4896 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 American Institute of Aeronautics and Astronautics too much propellant to allow extended hovering. Magnetoplasmadynamic (MPD) thrusters require improved efficiencies to become more practical means of reaching target orbits, but could hover for short periods due to high exhaust velocities. PROBLEM FORMULATION Coordinate System The circular restricted three body problem is used to model the spacecraft motion. This formulation approximates the Earth’s orbit as a circle with fixed radius of 1 AU rotating about the sun. The coordinate system used in this work is rotating and centered at the small primary, Earth, with the negative x-axis in the direction of the large primary, the Sun. The z-axis is normal to the ecliptic and the y-axis completes a righthanded orthogonal set (Figure 1.) References 2 and 7 are examples of earlier works examining trajectories normal to the ecliptic using the same coordinate system. The authors of these two papers cite the intuitive nature of orbit visualization as an advantage for using this system. Distances are normalized by the mean SunEarth distance (1AU=1.496·10 km) and time is normalized by the Earth mean heliocentric angular velocity, i.e. S AU t t μ / / 3 * = where μs is the Sun’s gravitational constant and t* is the time in seconds.