# Trajectory Design

## Circular Restricted Three-Body Problem (CR3BP)

Many problems in orbital mechanics can be solved using the two-body approximation, which yields analytic solutions in the form of conic sections. However, when a spacecraft moves close enough to a third body for its gravity to become significant, the two-body approximation breaks down. While we lose the ability to analytically solve for our spacecraft's position and velocity, we gain flexibility when we can take advantage of the dynamics of the three-body problem.

## Rotating Reference Frame

Normally when we think of orbits, we think of circles, ellipses, and hyperbolas. Nice conic sections that are centered on the Earth or the Sun. However, when working in the CR3BP, there are two central bodies, so it is often more convenient to plot things in the rotating frame. We define the x-axis as the line between the two large bodies and the origin as their barycenter. The two bodies are stationary in our plots and orbits can now follow all sorts of strange paths.

In the rotating frame, there are five equilibrium points, referred to as Lagrange or Libration points. These are the points where the gravity from both large bodies cancels out the centrifugal force. Rather than orbiting one of the large bodies, spacecraft can orbit the Lagrange points. These periodic orbits may exist purely in the plane of motion of the large bodies or may exhibit out-of-plane motion. The figure below shows a portion of the L2 halo orbit family around Enceladus. These orbits reach low periapsis over the south pole of Enceladus, making them attractive candidates for missions to observe the plumes there.

## Invariant Funnels

## Quasi-Periodic Orbits (QPOs)

Just like periodic orbits can be considered oscillations around the Lagrange points, quasi-periodic orbits (QPOs) are oscillations around periodic orbits. For most periodic orbits, there exist regions around them where a spacecraft will oscillate rather than escaping from the region entirely. This gives the spacecraft another degree of freedom, moving on the surface of a torus rather than on a closed curve. QPOs can give engineers more options to work with when designing science orbits as they spread out over a wider region of space.

## ThreeBodyProblem.jl : Open-source Julia Package

We've developed an open-source Julia Package for working in the Three-Body Problem. It contains equations of motions for several useful astrodynamics models as well as useful functions for analysis. Additionally, we've put together a set of examples to get new users started with the package, even those who are unfamiliar with the Julia language.

## People

- Aerospace Corporation (PhD 2023)

## Publications

- Blanchard, J., Lo, M., Anderson, B., Landau, D., Restrepo, R., & Elschot, S. (2023).
*Using NRHO invariant funnels to target Enceladus south pole*. AAS/AIAA Spaceflight Mechanics Meeting. - Blanchard, J., & Elschot, S. (2022). Model Predictive Control in the Three-Body Problems Using Invariant Funnels as Terminal Sets.
*AAS/AIAA Space Flight Mechanics Meeting*. - Blanchard, J., Lo, M., Landau, D., Anderson, B., & Close, S. (2021). Invariant funnels for resonant landing orbits.
*AAS/AIAA Space Flight Mechanics Meeting*. - Blanchard, J., Anderson, B., Lo, M., & Close, S. (2020). Low energy capture into high inclination orbits for ocean worlds missions.
*AAS/AIAA Astrodynamics Specialist Conference*. - Swenson, T. (2018).
*The topology of transit orbits in multibody systems with applications to dynamical astronomy and mission design*[Doctoral Dissertation, Stanford University]. https://purl.stanford.edu/jm914rm5569 - Li, A., & Close, S. (2013). Orbital debris parameter estimation from vertical pointing radar.
*International Astronautical Congress*. http://iafastro.directory/iac/paper/id/18546/summary/ - Li, A., Close, S., & Markannen, J. (2012). EISCAT space debris after the International Polar Year (IPY).
*International Astronautical Congress*. http://commercialspace.pbworks.com/w/file/fetch/87182083/Li,%20et%20al%202012.pdf