Trans. Japan Soc. Aero. Space Sci. Vol. 57, No. 4, pp. 238–244, 2014 Escape Trajectories from the L2 Point of the Earth-Moon System à By Keita TANAKA1Þ and Jun’ichiro KAWAGUCHI2Þ 1ÞDepartment of Aeronautics and Astronautics, The University of Tokyo, Tokyo, Japan 2ÞThe Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Japan (Received May 14th, 2013) To use the L2 point of the Earth-Moon (EM) system as a space transportation hub, it is absolutely necessary to know the available orbital energy of a spacecraft departing from EM L2 and how to obtain required excess velocity to the target planet. This paper presents a successful acceleration strategy using the combination of an impulsive maneuver and the Sun gravitational perturbation. Applying this strategy enables the spacecraft to efficiently convert the energy of the maneuver into its orbital energy. Key Words: Escape Trajectories, L2 Point of the Earth-Moon System, Sun-Earth-Moon Four-Body Problem Nomenclature vation and communication. Among them, EM L2 has re- cently attracted attention again as a new base for long-dura- a: semi-major axis tion habitation. It is considered as a desirable candidate site L: Lagrangian for extended ISS operations, which are very important to G: gravitational constant understand human space capabilities. M: mass 1.2. Problem statement r, R: distance It is a major topic of astrodynamics how to gain the nec- U: potential essary energy to navigate a spacecraft in the interplanetary ÁV: maneuver space. Over 3.0 km/s of the relative velocity with respect x; y; z: Cartesian coordinate to the Earth is required in order to reach even the nearest : angle outer planet, Mars. In considering the use of EM L2 as a : mass parameter hub for interplanetary transfer, it is absolutely necessary to !: frequency know the available orbital energy of the spacecraft departing Abbreviations from this point and the acceleration strategies to obtain pre- SE: Sun-Earth scribed relative velocity. EM: Earth-Moon 1.3. Related researches BCR4BP: bicircular restricted four-body problem Gobetz and Doll3) surveyed various types of escape Subscripts maneuvers in the two-body problem. They summarized S: Sun the available escape speed of one-/two-/three-impulsive E: Earth transfer from circular orbit as a function of the magnitude M: Moon of the impulse. Their results formed the basis of this L: libration point study. As to the escape from L2 of the Sun-Earth system, 1. Introduction Matsumoto and Kawaguchi4) investigated one-impulse modes and derived effective acceleration patterns. They 1.1. Background conducted an exhaustive search on different magnitude- The locations where the gravitational and centrifugal ac- and different direction-impulsive trajectories and showed celeration balance each other in the restricted three-body desirable control strategy. They also showed trajectories system are known as the libration points (Euler1) and with multi-impulse or low continuous acceleration. Lagrange2)). There exist five equilibria in the system and Nakamiya and Yamakawa5) conducted optimization calcu- they are constant with regard to the rotating two bodies. lations on escape trajectories from SE L2 and calculated Three of them, L1,L2 and L3, are located on the line passing how the control should be to attain a certain energy at the through the two bodies and the other two, L4 and L5, com- boundary of the Earth gravitational sphere. plete equilateral triangles with the bodies. The L1 and L2 One common explicit solution from their studies was that points of the Sun-Earth (SE) and Earth-Moon (EM) systems the maneuver works most efficiently when it is applied at the attract lots of attention for various space uses, such as obser- position where the spacecraft has highest velocity. This property held true with regard to the escape trajectories from Ó 2014 The Japan Society for Aeronautical and Space Sciences EM L2. ÃResearch Fellow of the Japan Society for the Promotion of Science Jul. 2014 K. TANAKA and J. KAWAGUCHI: Escape Trajectories from the L2 Point of the Earth-Moon System 239 unit of time is selected such that the orbital period of the Sun and the Earth-Moon barycenter becomes 2 and the orbital mean motion is 1. In addition, the following mass parame- ters are defined. M þ M ¼ E M SE M þ M þ M S E M ð2Þ MM EM ¼ ME þ MM SE is a mass ratio of the Earth-Moon system with regard to the Sun-Earth-Moon system and EM is a mass ratio be- tween the Moon and the Earth-Moon system. In the normalized rotating coordinate where the origin is Fig. 1. Sun-EM rotating coordinate. The origin is located at the barycen- located at the Earth-Moon barycenter and the negative x axis ter of the Earth-Moon system. The x axis lies along the line connecting lies along the line toward the Sun as shown in Fig. 1, the po- the Sun with the EM barycenter. The Earth and Moon revolve counter- clockwise about the origin. sition vectors can be expressed as 0 1 0 1 0 1 À1 RE cos E RM cos M r ¼ @ 0 A; r ¼ @ R sin A; r ¼ @ R sin A 1.4. Contribution S E E E M M M 0 0 0 In this study, the escape trajectories from the L2 point of the Earth-Moon system to the interplanetary space are con- ð3Þ sidered. The paper begins with the direct escape schemes, with which are the easiest to achieve. The spacecraft leaves the RM ¼ð1 À EMÞaM libration point with an impulsive ÁV and directly arrives R ¼ a at the boundary of the Earth gravitational region. Next, E EM M ð4Þ the technique of the trajectory-correction using the Sun per- M ¼ !Mt þ M0 turbation is proposed. The spacecraft is guided to pass close E ¼ M þ to the Earth by the Sun gravity and experiences the Earth where aM denotes the distance between the Earth and Moon. gravitational assist with an impulsive acceleration, realizing !M is the mean motion of the Earth-Moon system in the the high escape energy. The ÁV of 400 m/s at the perigee rotating frame. Normalizing Eq. (1) and substituting can generate escape velocity of 3.0 km/s at the boundary Eq. (3) into Eq. (1) yield the simplified form of U as of the Earth gravity. 1 SEð1 À EMÞ SEEM U ¼ð1 À SEÞ þ x À À rS rE rM 2. Preliminary Development ð5Þ with 2.1. Equations of motion r2 ¼ðx þ 1Þ2 þ y2 þ z2 In the Sun-Earth-Moon bicircular restricted four-body S r2 ¼ðx À R cos Þ2 þðy À R sin Þ2 þ z2 ð6Þ problem (BCR4BP), three bodies are assumed to move in E E E E E 2 2 2 2 a circular motion in the same plane. The Earth and Moon re- rM ¼ðx À RM cos MÞ þðy À RM sin MÞ þ z volve around their barycenter and the Earth-Moon barycen- Then the Lagrangian L in the rotating frame is expressed as ter revolves around the barycenter of all the system. Here is 1 ÀÁ L ¼ ðx_ À yÞ2 þðy_ þ xÞ2 þ z_ À U ð7Þ a brief description of the development of the equations of 2 the motion of the BCR4BP. Finally, the Euler-Lagrange equation gives the equations of The potential of the Sun-Earth-Moon BCR4BP is ex- motion of the BCR4BP pressed in the form x€ À 2y_ À x ¼Ux 1 rS Á r GME GMM y€ þ 2x_ À y ¼Uy ð8Þ U ¼GMS À 3 À À ð1Þ jrS À rj jrSj jrE À rj jrM À rj z€ ¼Uz where G is the gravitational constant. r is the position vector 2.2. Libration points with respect to the Earth-Moon barycenter and Mi stands for The position of the collinear libration point is represented the mass of the body i. The subscripts, S, E and M, indicate as a relative position with regard to the primary bodies. The that the value relates to the Sun, Earth and Moon, respec- distance between the smaller mass and L2 is obtained by tively. Szebehely6) and is expressed in the form For simplicity, the system is normalized with the follow- 1 1 2 31 3 ing values. The unit of mass and length are selected as the L2 ¼ rh 1 þ rh À r À r þ ... ð9Þ 3 9 h 81 h sum of the masses of the Sun, Earth and Moon and the dis- tance between the Sun and the Earth-Moon barycenter. The with the Hill radius rh defined by 240 Trans. Japan Soc. Aero. Space Sci. Vol. 57, No. 4 1 of EM L2 can be represented by that of the Moon as an angle 3 r ¼ ð10Þ measured from the Sun-EM line M as shown in Fig. 3. In h 3ð1 À Þ the following discussion, the term ‘‘initial Moon age’’ shall Thus the position of EM L2 in the Earth-Moon rotating indicate the position of the Moon when the spacecraft frame is obtained as departs from EM L2. T xL2 ¼ðxM þ L2; 0; 0Þ ð11Þ 3. Escape from EM L2 where xM denotes the position of the Moon in the x direc- tion. 3.1. Natural escape 2.3. Earth gravitational boundary The trajectories from EM L1 require maneuvers to open On an ordinary planning of the interplanetary missions, up the energetic barrier around EM L2 and to reach the the method of the patched conics is employed.