Overview of orbital mechanics for space-based gravitational wave observatories

Gravitational waves (GWs) are “ripples in space-time”. The detection of gravitational waves (GWs) is critical to the understanding of the origin and evolution of stars, galaxies, and the Universe. At present, the laser interferometry is the most commonly use technology to detect GWs by measuring the phase change between two beams of coherent light. Due to the limitations of arm length, the ground-based GWs measurement is hard to detect the low-frequency GWs. While the space-based GWs observation is capable of longer arm length of the interferometer, the detection of GWs in space is expected to cover a greater number and variety of GWs sources. Configuration design and stability control for space-based GWs observatories are one of the key factors to realizing the detection of GWs in space. In a research paper recently published in Space: Science & Technology, Dong Qiao etc. from School of Aerospace Engineering, Beijing Institute of Technology, summarize and analyze the research progress in the orbital mechanics of the space-based GWs observatory.

First, authors introduce the principle and the proposed GWs observatories of space-based GWs detection. 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. Generally, the geocentric configuration has the smallest configuration size among the three types. The probes are also easy to deploy as they are close to the Earth. However, the space environments in the vicinity of Earth is complicated. The heliocentric configuration has the medium size. The transfer duration and fuel consumption is relatively high as the probes are far from Earth. The libration point configuration can cover the widest frequency band theoretically. But the probe on the libration point is hard to control. Therefore, most of libration configuration missions are still in concept phase.

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 , the arm length variation rate L’, and the breathing angle variation  (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. Scholars have carried out research on the analysis of factors affecting configuration stability, numerical configuration optimization based on intelligent algorithms, and semi-analytical optimization based on average orbital elements. Through model simplification and layer-by-layer iteration, the dimension of optimization variables is reduced, and the optimization efficiency and convergence are improved. The newly proposed semi-analytical double-layer iterative optimization algorithm can greatly improve the optimization efficiency of geocentric configuration and realize the configuration optimization in the whole space. Future study will be focusing on the evolution of configuration error and discussing the stable domain of the geocentric gravitational wave detection configuration.

Although the arm length of the heliocentric configuration is large, it is still small value compared to the heliocentric distance is still small. Therefore, the design of the heliocentric configuration can be regarded as the formation configuration design. But compared with the traditional formation, the dimension of configuration design parameters is high and there are many optimization objectives. The requirements for long-period configuration stability is also extremely high. Aiming to the above difficulties, scholars have established a relative motion dynamics model considering the regenerative force to determine the initial value of heliocentric configuration. In addition, the research on gravitational wave configuration optimization parameter selection, parameter domain segmentation search, and multi-objective optimization is carried out. It is found that the relative distance or relative phase angle between the formation center and Earth is the key factor affecting the stability of heliocentric configuration. At present, heliocentric configuration optimization relies on high-precision numerical methods, which have the problems of low optimization efficiency and weak robustness. The relative motion of multiple probes configuration cannot be revealed.  The analytical heliocentric configuration optimization method needs to be further explored.

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.

Reference

Authors: Dong Qiao, Feida Jia, Xiangyu Li, and Xingyu Zhou

Title of Original Article: A Review of Orbital Mechanics for Space-Based Gravitational Wave Observatories

Link of Original Article: https://spj.science.org/doi/10.34133/space.0015

Journal: Space: Science & Technology

Affiliations:
School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China

Key Laboratory of Autonomous Navigation and Control for Deep Space Exploration, Beijing Institute of Technology, Beijing, China.