  In the preliminary design of space missions it can be useful to identify
regions of dynamics that drive the system's behaviour or separate qualitatively
different dynamics. The Lagrangian Coherent Structure (LCS) has been widely
used in the analysis of dynamical systems, and generalises the concept of the
stable and unstable manifolds to systems with arbitrary time-dependence.
However, the use of three-dimensional LCS in astrodynamics has thus far been
limited. This paper presents the application of a new numerical method
introduced by the authors, DA-LCS, to astrodynamics systems using the
Elliptic-Restricted Three-body Problem (ER3BP) as a test case. We are able to
construct the full, three-dimensional LCS associated with the Sun-Mars ER3BP
directly from the variational theory of LCS even for numerically challenging
initial conditions. The LCS is analysed in detail, showing how it in this case
separates regions of qualitatively different behaviour without any a priori
knowledge. The paper then studies the effect of integration time and the
parameterisation of the initial condition on the LCS found. We highlight how
round-off errors arise from limits of floating-point arithmetic in the most
challenging test cases and provide mitigating strategies for avoiding these
errors practically.
