On the Interplanetary Flight from the Deep Space Port Michihiro Matsumoto Department of Aeronautics and Astronautics, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JAPAN. E-mail:[email protected] Abstract Recently, various kinds of planetary explorations have become more feasible, taking the advantage of low thrust propulsion means such as ion engines that have come into practical use. The field of space activity has now been expanded even to the rim of the outer solar system. In this context, the Japan Aerospace Exploration Agency (JAXA) has started investigating a Deep Space Port built at the L2 Lagrange point in the Sun-Earth system. For the purpose of making the deep space port practically useful, there is a need to establish a method to making spaceship depart and return from/to the port. This paper first discusses the escape maneuvers originating from the L2 point under the restricted three-body problem. Impulsive maneuvers from the L2 point are extensively studied here, and using the results, optimal low-thrust escape strategies are synthesized. Furthermore, this paper proposes the optimal escape and acceleration maneuvers schemes using Electric Delta-V Earth Gravity Assist (EDVEGA) technique. 太陽-地球系 L2 点に深宇宙港を設置することを想定し，その脱出方法について検討を行った．前年の発表で， L2 点から惑星間へは容易に脱出させることができるが，ポテンシャルの観点から脱出方向は，太陽から地球 に向かう線上，すなわち半径方向におおよそ限定されることを述べた．これは，逆の言い方をすれば，半径 方向に脱出させることが，加える増速量と獲得できる脱出速度の比の効率の観点で有利であることを示して いる．この結果をもとに，複数インパルスでの脱出軌道について考察し，また、低推力推進を使用した脱出 軌道を求め評価する． space port practically useful, there is a need to establish a 1. Introduction method to making spaceship depart and return from/to the port. Since interplanetary voyage generally needs delta-V of over 10 km/s, interplanetary spacecraft uses high Isp 2. Dynamics propulsion, such as electric propulsion. On the contrary, to escape Earth gravity field, the spacecraft needs the In this study, we consider the restricted three body propulsion of high thrust level, such as chemical system (see Fig. 1). The Sun and the Earth revolve propulsion. This means that deep space exploration needs around their center of mass in circular orbits under the a couple of propulsions which differ in kind. And so influence of their mutual gravitational attraction and a construction of bases of interplanetary flights and change spacecraft moves in the plane defined by them. The of the spacecrafts at this port are effective. rt. equation of motion of the spacecraft is written as follows, The L2(L1) point is considered as the leading &x& − 2y& = Ω x candidate site for deep space port, for this point has some (1) &y& + 2x& = Ω y advantages that are applicable for the deep space port. L2(L1) point is located at the potential hill in the where boundary of the gravity influence. And although L2 point x2 + y 2 1− µ µ µ(1− µ) is unstable point, this point is stable enough with keeping Ω = Ω(x, y) = + + + (2) 2 r r 2 the position with respect to the Earth and the Sun with 1 2 the small delta-V. In this context, the Japan Aerospace and the subscripts of Ω in Eqa.1, denote the partial Exploration Agency (JAXA) has started investigating a derivatives with respect to the coordinates of the Deep Space Port built at the L2 Lagrange point in the spacecraft (x, y). Also, Sun-Earth system. For the purpose of making the deep - 1 - m µ = E mS + mE (3) 2 2 2 2 r1 = (x + y) + y , r2 = (x −1+ µ) + y ⊿V (Departure delta-V) mE mS where is the mass of the Earth and is the Sun Earth mass of the Sun. Eqa.1 are written in a rotating reference L2 α frame with the following conventions, (Departure angle) ・The sum of the masses of the Sun and the Earth is m + m =1 s e . 3 million km η Spacecraft r2 y Fig. 2. Setting of the departure velocity and angle x r1 Earth This problem is calculated with the following assumptions, r a ・The Earth’s sphere of influence is assumed 3 million km. (In this study, we define this distance as “infinity”, and nt = θ this boundary sphere is used evaluation point of potential ξ Sun C energy.) ・The maximum flight time is one year. Fig. 1. The restricted three body system 3.2 Results “Fig. 3,” shows inertial velocity at infinity and “Fig. 4,” shows the direction of inertial velocity at infinity. The ・ The distance between the Sun and the Earth is region A of “Fig. 3,” is only one color. This means that, normalized to 1. in the region of low departure delta-V, the escape ・The angular velocity of the Earth around the origin is direction is limited to one direction. The V-infinity of normalized to 1. 0.6km/s can be obtained from departure delta-V of a few The system has a first integral of motion, called Jacobi m/s. In other words, if the escape direction is limited to integral, which is given by one direction, the spacecraft can escape from L2 point by 1 2 2 2 infinitely small delta-V. And this direction is +x axis. On (x& + y& + z& ) − Ω(x, y, z) = −C = const (4) 2 the contrary, the escape trajectory in the tangential direction of Earth’s orbit needs high departure delta-v. Therefore, these trajectories are inefficient. 3. Escape trajectory using impulsive transfer First of all, impulsive escape paths from the equilibrium point where the quay anchored is are examined. A single impulse trajectory is not necessarily an optimal in terms of delta-V budget. This examination is aimed at examining the features of those trajectories. 3.1 Single impulse trajectory I II III Single impulse trajectories from the L2 point are examined. The departure velocity and angle are taken for A the parameters (see Fig. 2). Fig. 3. Single impulse trajectory : Inertial velocity at infinity (Vinf) [km/s] - 2 - at the infinity. 1.2 1 0.8 0.6 0.4 0.2 1 impulse < 200m/s Relative velocityrespect with [km/s] to Earth 0 0 1 2 3 4 5 6 Fig. 4. Single impulse trajectory : Direction of inertial Escape velocity angle [deg] velocity at infinity [deg] Fig.6. Result of total delta-V of 200 m/s 3.3 Double impulse trajectory 4. Escape trajectory using low thrust Transfer A single impulse trajectory is not necessarily an optimal in terms of delta-V budget. And so, here is In this section, based on the result of single impulse considered the double impulsive maneuvers. The trajectory, escape trajectory using low-thrust transfer is spacecraft is given the first delta-V (departure delta-V) optimized. In this study, the steering law is optimized by at the L2 point and the spacecraft is given the second using DCNLP (Direct Collocation with Nonlinear delta-V at the perigee where the spacecraft can obtain Programming) method. the energy efficiently. The departure velocity and angle are taken for the parameters (see Fig. 5). 4.1 Escape trajectory optimization 1 First, escape trajectory optimization from Sun-Earth L2 point to the infinity is considered. ⊿V1 The assumptions for this problem ⊿V2 α Sun – Initial velocity ： 0km/s (The relative velocity of the spacecraft with respect to L2 point is 0 km/s.) L2 Earth – Terminal position ： infinity (Radius of 3 million km from the center of the Earth) – Flight time ： Free – Terminal velocity at infinity ： given Formulation Fig. 5. Setting of the parameters • Equation of motion The equation of motion of the spacecraft is given by Eqn. (1). This problem is calculated with the following assumptions, • Criterion function ・The Earth’s sphere of influence is assumed 3 million km. The minimum fuel consumption problem is considered, (In this study, we define this distance as “infinity”, and so the criterion is written as follow, this boundary sphere is used evaluation point of potential tf 2 2 energy.) J = (ax + a y )dt → min (5) ∫t0 ・The maximum flight time is one year. a , a where x y are the control input for x and y direction ・Total⊿V(=⊿V1+⊿V2)=200m/s respectively. • Thrust constraint 3.2 Results Suppose the spacecraft is equipped with electric The conclusion is relatively simple and double propulsion, so that the thrust is very small. This thrust is impulse trajectories expand the direction of velocity at assumed by the infinity with less fuel. Fig.6 shows the relation a = ( a 2 + a 2 ) =1.5 ×10−4 [m / s] (6) between the escape velocity and escape velocity angle max x y max - 3 - • Boundary conditions According to the assumptions for this problem, the Escape L2 Escape spacecraft remain stationary with respect to L2 point at initial time. And terminal position of the spacecraft is L2 Earth L2 infinity. Therefore the boundary conditions can be written as follows, Type A-1 TypeA-2 TypeA-3 ⎛ x0 ⎞ ⎛1− µ + γ ⎞ ⎛ x f − (1− µ)⎞ ⎛ xe ⎞ Fig. 7. Initial estimate for this problem ⎜ ⎟ ⎜ L2 ⎟ ⎜ ⎟ ⎜ ⎟ -4 x 10 ⎜ y0 ⎟ ⎜ 0 ⎟ ⎜ y f ⎟ ⎜ ye ⎟ 0.02 1.6 = , = (7) 6 ⎜ ⎟ ⎜ ⎟ ⎜ x ⎟ ⎜V ⎟ 0.015 1.4 x&0 0 & f xe Earth′s sphere of influence 5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0.01 1.2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ y&0 ⎠ ⎝ 0 ⎠ y& f Vye 0.005 1 4 ⎝ ⎠ ⎝ ⎠ Earth L2 0 0.8 3 γ L2 where is the distance between the Sun and the -0.005 Thrust [m/s] 0.6 Escape trajectory 2 y(rotating frame) [AU] y(rotating Earth. The subscripts “0, f” mean initial and final -0.01 0.4 Steering Angle [rad] 1 -0.015 0.2 condition respectively. And the subscript “ε” means -0.02 0 0 0.98 0.99 1 1.01 1.02 0 20 40 60 80 100 condition at infinity.