Orbital blocking game near Earth–Moon L1 libration point

First, authors present the dynamical models and pursuit-evasion orbits used in this paper In this paper, the Earth-Moon rotating frame (EMR) and the circular restricted three body problem (CR3BP) is adopted as the reference frame and the dynamics, respectively. The characteristic energy (C₃) is used to describe the energy of the evader which remains constant under two-body dynamics and its change in value can be used to measure the effectiveness of LGA. The effectiveness of these LGAs is measured in terms of equivalent ΔV. The evader is supposed to maximize its orbit energy through an LGA. The relationship among C₃, ΔV, and the time the spacecraft arrives at the Moon periapsis is shown in Fig. 3. If the pursuer forces the evader to overcome the blockade at a higher cost than the number in Fig. 3, the evader may choose to completely abandon breaking/bypassing the blockade and forfeit the dynamic advantage of the Moon, opting to reach the required energy directly in low Earth orbit. The pursuer is assumed be able to only make a single impulsive maneuver. The minimum capability required for the pursuer to intercept the evader can be calculated by the algorithm depicted in Fig. 4, which brings the pursuer as close as possible to the evader under a certain ΔV and the given ΔV increases if the most intimate approach distance does not meet a given criterion (less than 1 km). Moreover, the evader does not consider the presence of the pursuer and continues using the most energy-efficient LGAs.

Fig. 3. Effect and arrival time of best LGA orbits.

Fig. 4. Algorithm of calculating the ΔV required by the pursuer.

Then, authors present results of blocking at the L₁ libration point. The pursuer is initially located at the L₁ libration point and the ΔV needed for the pursuer to intercept “dumb” evaders with different initial energies is shown in Fig. 5. According to Fig. 5 by considering the location of the interception point and the vital dynamical properties implicated, interceptions can be categorized into 4 energy-dependent regions: the distance and phase-dominated region (C_3,0 < −2.34), the CR3BP-dominated region (−2.34 < C_3,0 < −2.07), the time-dominated part (−2.07 < C_3,0 < − 0.10), and the speed-dominated region (−0.10 < C_3,0). The distance and phase-dominated region exhibits notable characteristics relevant to 2-body dynamics, in which the evader’s orbit is relatively low and experiences minimal influence from lunar gravity. In CR3BP-dominated region, the ΔV needed for intercepting evaders is much smaller under CR3BP than two-body dynamics. In the time-dominated region, the only opportunity for the pursuer to intercept the evader occurs near the Moon, and the primary factor influencing the pursuer’s ΔV is the time the evader reaches the Moon. In the speed-dominated region, the optimal interception point shifts near the L₁ libration point with an optimal ΔV and the main factor for ΔV is no longer the time of the evader to reach the lunar orbit.

Fig. 5. ΔV required by the L₁ pursuer to intercept evader with different energies.

Authors also present results of blocking at Lyapunov orbits. When the pursuer is initially orbiting in a planer Lyapunov orbit which is not stationary, the pursuer can have some initial velocity, reducing the cost of their movement. Moreover, Lyapunov orbits may directly intersect the evader’s transfer orbit, enabling the pursuer to achieve interception at a lower price. Three interception scenarios of a pursuer with a ΔV of 50 m/s deployed on a planar Lyapunov orbit with L_Lya = 10000 km are shown in Fig. 10. It can be observed that under the same circumstances, the pursuer deployed on the Lyapunov orbit can perform interceptions that the L₁ pursuer cannot. The figure also includes the mission capable rates (MCRs), which represent the percentage of time available to execute the mission within each orbital period. It can be found that the cost of expanding the mission capability is a decrease in the MCR. For different Lyapunov orbits, each Lyapunov orbit has its own “optimal intercept C_3,0”, and geometrically the optimal MCR corresponds to the top of the Lyapunov orbit. If the top of the Lyapunov orbit is too high, the MCR decreases because orbit is too large. Meanwhile, if the top of the Lyapunov orbit is too low, the pursuer needs to expend a large amount of ΔV to raise its orbit to intercept the evader, also leading to a decrease in the MCR. Therefore, when choosing the size of the Lyapunov orbit, it is necessary to consider the evaders’ energy and MCR simultaneously.

Fig. 10. Typical interception from Lyapunov orbit and their RDs.

Last but not least, authors introduce a method for the evader to bypass the pursuer’s blockade and continue reaching the Moon. As for a pursuer deployed on a Lyapunov orbit, if the evader’s orbit is “higher” than the Lyapunov orbit (in the EMR frame), the pursuer’s MCR will decrease. The evader may deliberately choose a more extensive, “incorrect” launch angle to direct its orbits away from the Lyapunov orbit. Subsequently, the evader applies a pulse to correct its orbit at a particular moment. It continues utilizing LGA and creating a more “arched” trajectory that avoids intersecting with the pursuer’s RD even at higher C_3,0, as illustrated in Fig. 16 (not to scale). According to this strategy, Fig. 20 (to scale) illustrates the optimal solution for a pursuer on a Lyapunov orbit with L_Lya = 3000 km. The evader acquires an equivalent energy of 140.9 m/s from LGA and utilizes 36.2 m/s for the correction impulse. Despite expending some energy to bypass the blockade, the overall trajectory remains profitable.

Fig. 20. Evader’s trajectory of bypassing the blockade.