Stable configuration design for libration point gravitational wave observatory

The Sun-Earth $L_2$ libration point configuration is one of the options for space-based gravitational wave detection. Long-term configuration stability is crucial for high-precision measurements, challenged by the strong nonlinear dynamics of the Sun-Earth three-body system. This paper proposes an efficient design method and determines the feasible parameter domain for the libration point gravitational wave observatory. First, the dynamic model for the libration point configuration is established, and the stability indexes are defined. The sensitive parameters that affect the relative geometric configuration are discussed and the phase angle is found to be the key factor. Then, an efficient design method is proposed, and the procedure is divided into two steps. The phase angle of the Earth phase offset orbit and the libration point configuration are optimized successively. Finally, the proposed method is applied to the LAGRANGE mission concept. The results show that the three stability indexes decrease by 59%, 42% and 23%, respectively. Moreover, a mapping between configuration parameters and stability indexes is established. The feasible parameter domain for the stable libration point configuration is discussed. The feasible amplitudes domain in the $x$ and $z$ directions of the libration point orbit should be less than 6200 km and 42000 km, respectively, to guarantee configuration stability. This research could provide a reference for the stable design and implementation of gravitational wave detection missions utilizing libration point configuration in the future.