Analytical analysis on the orbits of Taiji spacecrafts

The unperturbed Keplerian orbits of Taiji spacecrafts are expanded to e³ order in the heliocentric coordinate system, where e is their orbital eccentricity. The three arm-lengths of the Taiji triangle and their rates of change are also expanded to e³ order, while the three vertex angles are expanded to e² order. These kinematic indicators of the Taiji triangle are further minimized, respectively, by adjusting the tilt angle of the Taiji plane relative to the ecliptic plane around ±π /3 , and thus, their corresponding optimized expressions are presented. Then, when the nominal trailing angle of the Taiji constellation following the Earth is χ (≈±π /9 ) from the viewpoint of the Sun, the influence of the Earth's perturbation on three spacecrafts is calculated according to the equations of motion in the problem of three bodies, and the perturbative solutions of the leading order and the next leading order are derived. With the perturbative solutions, the leading-order corrections to the above kinematic indicators of the Taiji triangle and the expression of the above trailing angle to the order of e³ are provided.