INFLUENCE OF THE MOON ON ORBIT MANEUVERINGS NEAR THE EARTH-MOON SYSTEM S. Jeong1, N. Baddour 2 1 2
Graduate Student, Department of Mechanical Eng., University of Ottawa, sjeon045@uottawa.ca Assistant Professor, Department of Mechanical Eng., University of Ottawa, nbaddour@uottawa.ca
Abstract Orbit maneuverings near the earth-moon system are mostly influenced by gravitational forces of the earth and moon. In this paper, the effect of gravitational force of the moon is calculated in terms of specific energy based on three bodies consisting of the earth, the moon, and a satellite maneuvering near the earth-moon system. To develop the three-body problem model, one assumption is added to the four restrictions of the twobody problem instead of using the restricted three-body problem. The specific numerical model is chosen to quantify the influence of gravitational force of the moon acting on orbit maneuverings. 1. Introduction Because of its great simplicity, a satellite orbiting around the earth is often considered as a two-body problem. In reality, analyses based on the three-body problem are more consistent and appropriate for modern space engineering. Increased rocket carrying capacity has triggered the development of bulky satellites which require handling orbit maneuvering near the earth-moon system as the three-body problem. This paper aims to shows the difference between the two-body and three-body based analyses and the twobody and three-body models are compared in terms of the specific energy to show the influence of the gravitational force of the third body. To derive the specific energy equation of the three-body problem, the need for the simplest model has arisen. Near the earth, the gravitational forces of the earth, sun and moon govern the orbit of a satellite. A simple calculation shows that that the gravitational force of the moon is stronger than that of the sun. Orbit maneuvering near the earth is therefore considered as a three-body problem consisting of the earth, moon and satellite. Based on these bodies, the equations of specific energy are developed. 2. Equations of Motion of the Bodies The motions of celestial or artificial bodies are one of long-standing problems of celestial mechanics as well as astrodynamics. Despite their long history, no analytic
solutions of the individual motions of the bodies under the influence of the others’ gravitational forces have been found [1]. To simplify its complexity, a two-body system is introduced which consists of the body with the most influence and the body affected only by the primary body. This model, however, is only applicable with restrictions: (a) The mass of the satellite is negligible compared to that of the attracting body; (b) the coordinate system chosen for a particular problem is inertial; (c) the bodies of the system are spherically symmetrical, with uniform density and (d) gravitational forces acting on the two bodies are the only forces that exist [2]. With these four restrictions, the basic two-body equation, called the relative form, is derived: G rG = − μ ⎛⎜ r ⎞⎟ es 2 r ⎝r⎠ The subscript, es, indicates the vector from the earth to satellite and μ is equal to G multiplied by the mass of the earth. In a similar manner, the equation of the threebody problem can be developed as: G G rG = − Gm e rG + Gm ⎛⎜ rsm − rem ⎞⎟ es es m⎜ 3 3 ⎟ res3 ⎝ rsm rem ⎠ where the subscripts ‘sm’ and ‘em’ represent directions from the satellite to the moon and from the earth to the moon, respectively. In the both equations, the mass of the satellite is neglected. 3. Specific Energy of the Three-Body Problem In the two-body problem, the angular momentum of the system remains perpendicular to the plane of the orbit. The directions of angular momenta of the three-body system, however, vary according to the positions of each body. To obtain the simplest specific energy equation of the three-body problem and to avoid applying the restricted three-body problem, orbits of the satellite and moon are restricted such that both bodies have a common plane. By adding this assumption to the previous four restrictions on the two-body problem, the specific energy of the three-body problem can be derived in an analogue manner to the two-body problem as ⎡m v2 m m ⎤ ξ = s − G⎢ e − m + m ⎥ 2 r r rem ⎦ sm ⎣ es
where the subscript ‘es’ refers to the direction from the satellite to earth and vs is the velocity of the satellite. The first term of the right-hand side of the derived equation expresses the kinetic energy of the system and the first term in the brackets is the potential energy. These two terms are correspond exactly to those of the two-body system but the last two terms are the effects of the third body as perturbations. The specific energy fields of the three-body system can be drawn over a range of velocities and altitudes. Circular orbits near the earth-moon system having radii of 7000 to 57000 km and velocities from 3.00 to 8.00 km/s are selected to for plotting. The level difference of two points of the field indicates energy required for certain orbit maneuvering.
the influence of the moon should be taken into account near the earth orbit. The three-body system consisting of the earth, moon, and satellite is proposed for more precise calculations. The advantageous aspect of the model developed in this paper is that the specific energy of the three-body system can be simply derived from that of the two-body system by adding an assumption on the restrictions of the classical two-body problem. The specific energy equations derived based on the assumptions shows that the effect of the third body can be expressed as two perturbation forms. The numerical results verify the influence of the gravitational force of the moon. An arbitrary orbit was chosen as an example; its specific energy difference between the two-body and three-body system is 9 J/kg. Based on this calculation and given the size of a modern satellite, the effect of the moon should not be underestimated. Since it is given in terms of the specific energy, the difference will become larger as the mass of the satellite increases; with heavier satellites leading to larger energy differences. In the earth-moon system, therefore, the influence of the moon on orbit maneuverings should not be neglected for precise analyses. References
Figure 1: The Specific Energy Field To quantify the influence of the third body, a specific orbit maneuver is chosen; the initial altitude of the orbit is 191 km and the final altitude is 36781 km. Since the specific energy equations are derived based on certain restrictions, the selection of the orbit should be done carefully. The chosen orbit is assumed to be in the same plane as the orbit of the moon and the satellite only performs co-planar Hohmann transfer between the chosen orbits in order to keep the direction of the angular momentum. Finally, the orbits of the moon and satellite are assumed to be circular, since this allows for simpler modeling. The specific energy difference between the chosen twobody and three-body cases is 9 J/kg which is 4 percent of the specific energy required for the chosen maneuver. This value is much smaller than expected, but it is evident that the moon obviously influences orbit maneuvers performed in the earth-moon system. It also must be noticed that the difference is given in terms of specific energy. The specific energy difference, therefore, depends on the mass of the satellite performing orbit maneuvers. 4. Conclusion The analysis of orbit maneuvers based on the two-body equation is not adequate for orbit maneuvers of modern satellites since the increased size of satellites influences the specific energy of the system. To reflect this feature,
[1] Franz T. Gelyling, H. Robert Westerman, Introduction to orbital mechanics, Addison-Wesley, 1971. [2] David A. Vallado, Fundamentals of astrodynamics and applications 2nd ed., Microcosm Press, 2001.