Sensitivity Analysis of Separation Time Along Weak Stability Boundary Transfers

Abstract

This study analyzes the sensitivity of the dynamics around Weak Stability Boundary Transfers (WSBT) in the elliptical restricted three-body problem. With WSBTs increasing popularity for cislunar transfers, understanding its inherently chaotic dynamics becomes pivotal for guiding and navigating cooperative spacecrafts as well as detecting non-cooperative objects. We introduce the notion of separation time to gauge the deviation of a point near a nominal WSBT from the trajectory's vicinity. Employing the Cauchy-Green tensor to identify stretching directions in position and velocity, the separation time, along with the Finite-Time Lyapunov Exponent are studied within a ball of state uncertainty scaled to typical orbit determination performances.

Figures (13)

• Figure 1: Illustration of separation time. The blue nominal trajectory originates from some point (usually the center) of the grid; the red neighbor trajectory stems from one of the gridded state, and its evolution in time is tracked. When the Euclidean distance of the position or velocity vector exceeds a threshold, the time stamp is recorded as the separation time for this neighbor trajectory.
• Figure 2: WSBT (a) shown in Earth-Moon rotatin-pulsating frame, with integer labels corresponding to time indices corresponding to locations where FTLE grid is constructed.
• Figure 3: WSBT (b) shown in Earth-Moon rotating-pulsating frame, with integer labels corresponding to time indices corresponding to locations where FTLE grid is constructed.
• Figure 4: WSBT (a): Separation Time vs. Time Index (unstable grid, OD-ball=(10 km, 10 cm/s))
• Figure 5: Contour FTLE and Separation Time Plots for WSBT (a) with unstable grid