Minimum Fuel Three-Dimensional Trajectories for an Asteroid Explorer

Paper number

IAC-05-C1.5.09

Author

Dr. Seiya Ueno, Yokohama National University, Japan

Coauthor

Mr. Hironori Asaoka, Yokohama National University, Japan

Year

2005

Abstract

This paper discusses on minimum fuel trajectories for an asteroid explorer accelerated by solar electric propulsion. Exploration of asteroids in the solar system is a very interesting project in astronomy, because it is expected that asteroids have different features from large planets or satellites. Thousands of asteroids have been found from the ground observation and new asteroids are found almost everyday. Thus it is very difficult to choose one asteroid as a target for a new project. The purpose of this paper is to provide effective information for the selection of a target asteroid using their orbital elements. The authors have shown the existence of ‘critical eccentricity’1), which divides orbits with same semi-major axis into two groups. Fuel consumption almost dose not depend on the eccentricity when the eccentricity of target orbit is less than the critical eccentricity. This conclusion, however, is adopted for two-dimensional problem, in which the target orbit is in the same plane of the earth’s orbit. This paper discusses on the three-dimensional problem. The effect of the inclination of target orbit is also discussed.

The problem in this paper is to minimize the fuel consumption of an explorer from the earth in a circle orbit to an asteroid in an elliptic orbit. Each orbits is in different plane. To simplify the problem, the following assumptions are used in this paper. The explorer exists in a central force field in which the sun locates at the center of earth’s orbit. The maximum thrust of explorer is in inverse-square proportion to the distance between the explorer and the sun. The specific impulse of solar electric propulsion is constant. The initial conditions are given as the same motion of the earth and the terminal conditions are of the target asteroid. The arrival time and position are not specified. The problem is to find the optimal amplitude and direction of thrust to minimize the fuel consumption. This nonlinear optimal control problem is solved numerically. According to optimal control theory, the amplitude of thrust is bang-bang control, in which maximum or minimum value is optimal. To obtain highly accurate solution, a number of burn period is assumed as two. Thus the explorer leaves from the earth with acceleration and coasting period follows. After that the explorer accelerates for arrival at the target asteroid.

The optimal trajectories are calculated for different eccentricity, semi-major axis, inclination of target orbit and different departure time. The following relations between the fuel consumption and the eccentricity of target orbit are shown from the numerical results. In the case of low inclination, the fuel consumption is minimum when the eccentricity of target orbit is the critical eccentricity. In the case of high inclination, the fuel consumption is monotonously decreasing as the eccentricity is increasing. This is a main conclusion of this paper.

The conclusion is explained from two reasons. One is orbital energy depending on ‘critical eccentricity’ shown in two-dimensional problem and the other one is torque to change angular momentum. Critical eccentricity in a two-dimensional plane is defined as an eccentricity when two orbits are tangent at a point. When two orbits are not crossing, the semi-major axis of transfer orbit is larger than the inner orbit and smaller than the outer orbit. The orbital energy is only a function of semi-major axis, thus the energy of explorer increases monotonously from the initial orbit to the target orbit. However, when the eccentricity of target orbit is larger than the critical eccentricity, two orbits are crossing. In this case, the semi-major axis of transfer orbit must be smaller than the smaller orbit or larger than the larger orbit. Thus the explorer decreases the energy lower than the lower orbital energy or increases the energy higher than the higher orbital energy. More fuel consumption is necessary for higher eccentricity of target orbit. The second reason is fuel consumption to change the orbital plane. The explorer is orbiting the sun, so it has angular momentum. To change the orbital plane is to change the angular momentum vector by torque, which is a product of force and distance. The distance is from the sun and the explorer. In the case of high eccentricity of target orbit, transfer orbit also has high eccentricity and the explorer can reach far from the sun in the transfer orbit. Thus larger distance is available in this case. Less fuel consumption is necessary for higher eccentricity of target orbit. Fuel consumption to change the orbital plane depends on the inclination of target orbit. Thus the first reason is dominant in the case of low inclination and the second reason is dominant in the case of high inclination.

1) S.Ueno and Y.Aoki, "Minimum Fuel Trajectories to Elliptic Orbits Using Solar Electric Propulsion", Advances in the Astronautical Sciences Vol.117, American Astronautical Society, pp.389-399, 2004.

Abstract document

IAC-05-C1.5.09.pdf

Manuscript document

IAC-05-C1.5.09.pdf (🔒 authorized access only).

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