Analytical analysis on the orbits of Taiji spacecrafts

The unperturbed Keplerian orbits of Taiji spacecrafts are expanded to $e^3$ order in the heliocentric coordinate system, where $e$ is their orbital eccentricity. The three arm-lengths of Taiji triangle and their rates of change are also expanded to $e^3$ order, while the three vertex angles are expanded to $e^2$ order. These kinematic indicators of Taiji triangle are, further, minimized, respectively, by adjusting the tilt angle of Taiji plane relative to the ecliptic plane around $\pm\pi/3$, and thus, their corresponding optimized expressions are presented. Then, under the case that the nominal trailing angle of Taiji constellation following the Earth is set to be $\chi(\approx\pm\pi/9)$ from the viewpoint of the Sun, the influence of the Earth 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 Taiji triangle and the expression of the above trailing angle to the order of $e^3$ are provided.