SCIENCE CHINA Information Sciences . RESEARCH PAPER . November 2017, Vol. 60 112301:1–112301:10 doi: 10.1007/s11432-016-9069-9 Effect of orbital shadow at an Earth-Moon Lagrange point on relay communication mission Yuhua TANG1,2, Weiren WU2, Dong QIAO1,3* & Xiangyu LI1,3 1School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China; 2Center of Lunar Exploration and Aerospace Engineering, Beijing 100037, China; 3Key Laboratory of Autonomous Navigation and Control for Deep Space Exploration, Ministry of Industry and Information Technology, Beijing 100081, China Received November 16, 2016; accepted February 9, 2017; published online September 19, 2017 Abstract The shadow effect is an important constraint to be considered during the implementation of explo- ration missions. In this paper, for the Earth-Moon Lagrange point L2 relay communication mission, shadow effect issues on a periodic orbit about L2 are investigated. A systematic analysis based on the time domain and phase space is performed including the distribution, duration, and frequency of shadows. First, the Lindstedt- Poincare and second-order differential correction methods are used in conjunction with the DE421 planetary ephemeris to achieve a mission trajectory family in a high-precision ephemeris model. Next, on the basis of a conical shadow model, the influence of different orbital phases and amplitudes on the shadow is analyzed. The distribution of the shadow is investigated as well. Finally, the configuration of the shadow and its characteristics are studied. This study provides an important reference and basis for mission orbit design and shadow avoidance for relay satellites at an Earth-Moon Lagrange point. Keywords Earth-Moon system, equilibrium point, relay orbit, shadow effect, motion control Citation Tang Y H, Wu W R, Qiao D, et al. Effect of orbital shadow at an Earth-Moon Lagrange point on relay communication mission. Sci China Inf Sci, 2017, 60(11): 112301, doi: 10.1007/s11432-016-9069-9 1 Introduction Chang’E-4 will perform the first landing, roaming, and exploration tasks on the far side of the Moon. During the mission, the lander and the rover, which will land on the lunar far side, will not be able to communicate directly with Earth because of the synchronous rotation of the Moon. Therefore, such communications need to be transmitted by a relay satellite. A periodic orbit about the Earth-Moon Lagrange point L2, owing to its unique dynamic characteristics, provides advantages, such as a relatively constant geometric relationship between the satellite and the Moon, long duration in Earth’s field of view, and low-cost orbital maintenance. It is, therefore considered to be an ideal position for relay communication. The concept of deploying relay satellites at the Earth-Moon L2 point was proposed in the 1970s [1]. Since then many studies concerning Earth-Moon low-energy transfer [2–4] and station keeping have been conducted [5–9]. In contrast, little research on illumination and shadow effects on satellites operating in * Corresponding author (email: [email protected]) c Science China Press and Springer-Verlag Berlin Heidelberg 2017 info.scichina.com link.springer.com Tang Y H, et al. Sci China Inf Sci November 2017 Vol. 60 112301:2 a periodic orbit of the Earth-Moon L2 point has been reported. As the angle between the Moon’s orbital plane and the ecliptic plane varies over time, relay satellites in a periodic orbit about an equilibrium point will not be illuminated in certain periods owing to the shadow of the Earth or the Moon. The duration of these shadows plays an important role in the design of a relay satellite power supply system. Long-term shadows will result in insufficient power supply and consequently affect the success of relay tasks [10]. This paper is intended to address shadow effects on relay satellites in a halo orbit about the Earth- Moon L2 point. In particular, we focus on analyzing the duration and distribution of the shadow effect for different orbital phases and orbital amplitudes, on the basis of which suggestions for mission orbit design are proposed. Further, a systematic analysis of the configuration, frequency and duration of a single shadow is presented. 2 Relay satellite mission orbit in precise ephemeris models 2.1 Restricted three-body model, equilibrium point and periodic orbit The circular restricted three-body problem (CRTBP) can be employed to describe the movement of a relay satellite in the Earth-Moon system. The dimensionless dynamic equations in the centroid rotation system can be written as follows: (1 − µ)(x + µ) µ(x − 1+ µ) x¨ − 2y ˙ − x = − − , r3 r3 − 1 2 − −(1 µ)y − µy y¨ + 2x ˙ y = 3 3 , (1) r1 r2 − −(1 µ)z − µz z¨ = 3 3 , r1 r2 where µ is the system mass ratio. In the Earth-Moon system, µ = 0.01215. In addition, r1, r2 are 2 2 2 the distances from the satellite to the Earth and to the Moon respectively. r1 = (x + µ) + y + z 2 2 2 q and r2 = (x − 1+ µ) + y + z . A normalized length, mass, and time are applied here, which are defined asq the average Earth-Moon distance, system mass, and inverse of the Moon orbital angular speed respectively. Five dynamic equilibrium points, i.e., Lagrange points, exist for the CRTBP: three collinear equilibrium points (L1, L2, and L3) and two triangular equilibrium points (L4 and L5). The L2 point, which can be used for relay communication is located on the opposite side of the Moon from the Earth. According to the Lindstedt-Poincare method [11], the motion equation near the Earth-Moon L2 point can be written as ∂ ξ ξ¨− 2η ˙ − (1+2c )ξ = c (µ)ρnP , 2 ∂ξ n n ρ n> X3 ∂ ξ η¨ +2ξ˙ + (c − 1)η = c (µ)ρnP , 2 ∂η n n ρ (2) n>3 X ∂ n ξ ζ¨ + c ζ = cn(µ)ρ Pn , 2 ∂ζ ρ n>3 X 2 2 2 2 where ρ = x + y + z , and c2(µ),cn(µ) are functions of only the mass and can be expressed as 3 1 − γ c2 = 3 µ + (1 µ) 3 , γ " (1 + γ) # n+1 (3) 1 n n γ cn(µ)= (−1) µ + (−1) (1 − µ) , n > 3, γ3 1+ γ " # where γ is the distance between the L2 point and the Moon, and pn is the nth-order Legendre polynomial. Tang Y H, et al. Sci China Inf Sci November 2017 Vol. 60 112301:3 0.02 0 Moon -axis Z −0.02 −0.04 0.05 0 0 −0.05 Y-axis 1.0 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 X-axis 1.18 Figure 1 (Color online) Southern family of halo orbit in the Earth-Moon three-body system. The high-order approximate analytic solution of (2) can be expressed as ∞ i j ξ(t)= ξijkm cos(kθ1 + mθ2) α β , i,j |k|6i,|m|6j X X ∞ i j η(t)= ηijkm sin(kθ + mθ ) α β , (4) 1 2 i,j |k|6i,|m|6j X X ∞ i j ζ(t)= ζijkm cos(kθ + mθ ) α β , 1 2 i,j X |k|6Xi,|m|6j where α and β represent the in-plane and out-of-plane amplitudes, respectively. θ1 = ωt + φ1, and θ2 = vt + φ2. φ1, φ2 are initial phases. ω, v are power functions, that describe the orbital vibration amplitude: ∞ ∞ i j i j ω = ωp + ωij α β , v = ωv + vij α β . (5) i,j i,j X X The approximate analytic solution for a periodic orbit around the Earth-Moon L2 point can be obtained using (4). An accurate numerical solution can be obtained by the differential correction method. The southern family of halo orbits with different amplitudes around the L2 point in the Earth-Moon system obtained using the above process is shown in Figure 1. For convenience, the phase angle θ of the halo orbit is defined as the angle between the projection of any orbital point on the XY -plane and the X axis, and this angle has a positive value along the clockwise direction. The initial point of the phase angle θ is selected as the peak of the halo orbit on the negative Y axis (as indicated by the marks in Figure 1). 2.2 Periodic orbit in a high-fidelity ephemeris model To systematically analyze the shadow effect on relay satellites, the relative positions of the Sun, Earth, and Moon have to be considered. Therefore, a high-fidelity ephemeris model is employed in this study [12]. In this paper, we adopt the DE 421 planetary ephemeris. Assume that the position vector of a relay satellite in the selenocentric inertial coordinate system is [xi,yi,zi], the velocity vector is [x ˙ i, y˙i, z˙i], and the instantaneous angular velocity of the Earth-Moon rotation system relative to the inertial coordinate system is [0, 0,ω]. According to the ephemeris, the relative position and speed of the Moon in the geocentric frame are obtained as L = a(1 − e cos(E)). (6) If we take the average Earth-Moon distance D as the normalized unit length, the actual Earth-Moon distance can be expressed as R = L/D. Under the ephemeris model, the Earth’s position in the Earth- Moon rotation system is [R(1 − µ), 0, 0], and the instantaneous equilibrium point position is Rλ, where Tang Y H, et al. Sci China Inf Sci November 2017 Vol. 60 112301:4 Figure 2 Geometrical relations of the shadow effect. λ is the position of the equilibrium point in the CRTBP.