# Analytical analysis on the orbits of Taiji spacecrafts

## TL;DR

This paper provides a detailed analytical expansion of Taiji spacecraft orbits and triangle geometry up to third order in eccentricity, optimizing configuration parameters and analyzing Earth's perturbative effects.

## Contribution

It introduces third-order analytical expansions of Taiji orbits and triangle parameters, including optimization of the spacecraft formation and perturbation analysis.

## Key findings

- Analytical expressions for orbit and triangle parameters up to e^3 order.
- Optimized tilt angle for minimal arm-length variations.
- Perturbative solutions for Earth's influence on spacecraft orbits.

## Abstract

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.