On optimal LISA orbit design

TL;DR

The paper treats LISA orbit design as a nonlinear optimization problem using exact Kepler orbital dynamics to minimize inter-spacecraft distance deviations from a designed separation. It introduces an arc-search infeasible interior-point method to solve the problem and compares DNKV, NKDV, and an optimal solution where the eccentricity $e$ and inclination $i$ are optimized (either identically across spacecraft or independently). The identical-$e,i$ optimum yields $e^* \approx 0.004824$ and $i^* \approx 0.008356$, with peak distance deviation around $1.2\times 10^4$ km and a mean radius centered near the designed 2.5 million km; allowing independent $e_k,i_k$ converges to essentially the same solution, indicating a likely global optimum. The method generalizes to $n$ spacecraft by rotating the reference orbit, offering a scalable framework for designing LISA-like formation flies with reduced Doppler and breathing-angle effects, and demonstrating improvements over prior DNKV/NKDV approximations.

Abstract

The ESA/NASA joint LISA (laser interferometer space antenna) mission is designed to detect gravitational waves, which relies crucially on maintaining three-spacecraft constellation as close to an equilateral triangle with a designed distance as possible. Efforts have been made to achieve this goal by using various simplified models to make it easy to approximately solve the complex problem. In this paper, the problem is formulated as a nonlinear optimization problem using exact nonlinear Kepler's orbit equations. It is shown that the optimal solution based on the exact nonlinear Kepler's orbit equations gives a better solution than the previously obtained ones.

Figures (7)

• Figure 1: The geometry of the LISA spacecraft orbit with respect to the ecliptic plane.
• Figure 2: Spacecraft distances in one period of DNKV orbit. The horizontal line is the desired distance between any spacecraft pair. The solid line is the distance variation between spacecraft 1 & 2. The dashed line is the distance variation between spacecraft 1 & 3. The dashed-dot line is the distance variation between spacecraft 2 & 3.
• Figure 3: Spacecraft distances in one period of NKDV orbit. The horizontal line is the desired distance between any spacecraft pair. The solid lines are the distance variation between spacecraft 1 & 2. The dashed lines are the distance variation between spacecraft 1 & 3. The dashed-dot lines are the distance variation between spacecraft 2 & 3.
• Figure 4: The constraint of HS-19.
• Figure 5: The arc used for searching optimizer at current iterate of Problem (\ref{['simpleEx']}).