A review of orbital mechanics for space-based gravitational wave observatories

First, authors summarize the principle and the proposed GWs observatories of space-based GWs detection. In the space-based GWs observatory, the basic structure is the two-source (in a triangular configuration of spacecraft) laser interferometer, including telescope, laser, test mass, phasemeter, etc. The phase change of the interference signal is measured by the phasemeter, so that the distance change between the test masses in the primary-secondary spacecraft caused by GWs can be inversely obtained. At present, a few mission concepts have been put forward for the space-based laser interference GWs observatories and their configurations can be divided into three categories. (1) The geocentric configuration of space-based GWs observatories, which form a large constellation on the high Earth orbit, with the plane normal of the constellation configuration pointing to the direction of the target GWs source. A typical example is the TianQin mission. (2) The heliocentric configuration of space-based GWs observatories, in which the spacecraft are deployed on different heliocentric orbits with a semi-major axis of 1 AU to form a large-scale formation, making the Sun their center. Typical heliocentric space-based GWs observatories include LISA, the Taiji plan and DECIGO/BBO. (3) The libration point configuration of space-based GWs observatory, in which the spacecraft are deployed in the vicinity of libration points in the Three-Body system. Typical libration point space-based GWs observatories include the ASTROD mission, the full libration points LAGRANGE mission and the single libration point LAGRANGE mission. The typical space-based GWs observatories are listed in Table 1.

Then, the authors summarize the status of the existing constellation and formation design methods for the space-based GWs observatory. The configuration parameters considered mainly include the arm length variation or variation proportion ∆L, the arm length variation rate L’, and the breathing angle variation ∆θ (see Fig. 2). For the design of geocentric configuration GWs observatory, different from the traditional constellation design, which usually focuses on the performance of the target coverage, it mainly pays attention to the geometric stability of the configuration. Wan et al. and Ye et al. designed the highly stable mission orbit for the TianQin plan based on the particle swarm optimization and the combinatorial optimization algrithms to meet the requirements of the arm length variation, the arm length variation rate and the breathing angle variation. Tan et al. studied the influence of orbit direction and radius on the stability of inter-spacecraft motion. Zhou et al. investigated the stability of a geocentric GWs observatory from the view of the configuration uncertainty propagation. Jia et al. proposed a semi-analytical double-layer iterative optimization algorithm to solve the geocentric configuration optimization problem efficiently. The configuration design of the heliocentric space-based GWs observatories belongs to formation design. The key question is to overcome the perturbation difference in space environment caused by the large scale of the formation. Take LISA as an example, Marchi et al. obtained the initial conditions for a stable circular formation by removing the secular term and adding constraints in the C-W equations without considering perturbation. Nayak et al. adopted an improved form of C-W equations with the second-order correction to further improve the accuracy of the relative motion model. Li et al. derived the expression of the trailing angle between the formation center and the Earth based on the semi-analytical form. However, the existing constellation and formation design methods for the space-based GWs observatory are mainly based on numerical computation, bringing up a heavy computational burden. Thus, the efficient analytical optimization methods remain to be investigated.

At last, authors made prospects of future study on determination of configuration parameter design space in complex environment considering multiple perturbation effects, highly efficient optimization method for initial configuration of space-based GWs observatory, and error propagation and stability region evaluation of configuration.