Analytical analysis on the orbits of Taiji spacecrafts

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 ±π/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 χ(≈ ±π/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.

Analytical analysis on the orbits of Taiji spacecrafts to infinite order of the orbital eccentricity

“…with Ω as the average angular velocity of SC1, r + 1 manifestly represents the orbit of SC1 in the original configuration [16], and in this case, SC1 is at the aphelion when t = 0, namely, the point C+ presented in FIG. 1.…”

“…For the original configuration of Taiji, rotating the orbit of SC1 by 2π/3 and 4π/3 about the z axis, respectively, gives those of SC2 and SC3, where their phases need to be adjusted correspondingly [8,9]. The expressions of r + 2 and r + 3 , representing the orbits of SC2 and SC3, respectively, have been obtained in our previous paper [16]. As for the new configuration of Taiji, above method can also be applied to derive the orbits of SC2 and SC3, denoted by r − 2 and r − 3 , respectively.…”

“…1, φ ± are the tilt angles of Taiji planes relative to the ecliptic plane at t = 0 in both configurations, respectively. For the original configuration [16], φ + can take values of ±|φ + |, and thus, there are two choices for orbits of SCs, which are symmetry about the ecliptic plane. Further, seeing that Taiji can follow or precede the Earth by π/9 from the viewpoint of the Sun, the original configuration, in fact, provides four feasible orbit designs for Taiji.…”

“…The general idea is originated from the fact that the Kepler's Eq. ( 11) can be expanded to infinite order of e when e ≈ 5.789 × 10 −3 1 for Taiji [16] according to the method of Lagrange [21], and then, the combination of Eqs. ( 2), ( 4), (5), and ( 7)-( 10) can bring about the expansions of the unperturbed Keplerian orbits of SCκ (κ = 1, 2, 3),…”

“…Moreover, optimizing the orbits of SCs also helps to reduce the adverse effect brought about by the Doppler shift of the laser frequency. For the original configuration of Taiji [16], by adjusting the tilt angle φ + around π/3 at e 1 order, the orbits of SCs are optimized at the next leading orders of all the kinematic indicators. This result can be generalized, in this paper, by slightly modifying the above algorithm, i.e., by adjusting φ ± around π/3 to any order of e, the orbits of SCs in both configurations of Taiji can be optimized, respectively, and that is to say, Taiji triangles can become as stable as possible with the different specific problem involved.…”