A college-level video on "halo orbits" around Lagrange points in the three-body problem. In this video, we’ll explore why these fascinating orbits exist, how they are used in real-world missions, and the key steps to derive and numerically compute them.
We first discuss some history of missions using CR3BP solutions, particularly the halo orbits. We then go into more detail about halo orbits, three-dimensional periodic orbits about the Lagrange points, and discuss how to compute them. The computation of halo orbits follows standard nonlinear trajectory computation algorithms based on differential correction. Due to the sensitivity of the problem, an accurate first guess is essential, since the halo orbit is actually an unstable orbit (albeit with a fairly long time constant in the Sun-Earth system, about 180 days). This first guess is provided by a high order analytic expansion using the Lindstedt-Poincaré method. We then use the Lindstedt-Poincaré method to find a third-order approximation of a halo orbit (Richardson's approximation), and use this approximation as a first guess in a differential correction scheme.
We discuss halo orbits in the Sun-Earth system, like the James Webb Space Telescope's orbit about L2, and halo orbits in the Earth-Moon system, like the CAPSTONE mission.
We end with a MATLAB demonstration to compute halo orbits that you can download below.
💻 *MATLAB Code* Live Code File Format (.mlx). At the following link,
https://tinyurl.com/cr3bpmatlab
⬇️ *Download* cr3bp_halo_manifolds.mlx
You can then execute the file in MATLAB
▶️ Previous: Trajectories with Prescribed Itineraries and MATLAB Tutorial, 3 Body Problem, Topic 15
https://youtu.be/-QCIFmGYhTQ
▶️ *Lindsted-Poincare method* (a.k.a., method multiple scales) for estimating periodic orbits in nonlinear vector fields
https://youtu.be/AwMZH_hShPg
▶️ *Three-Body Problem Introduction*
https://youtu.be/ZE299fDuPjc
▶️ Related: Applications to Dynamical Astronomy
https://youtu.be/fV0kUmtQWZU
► Reference: Chapter 6, "Halo Orbits and Their Computation"
of the PDF book:
*Dynamical Systems, the Three-Body Problem and Space Mission Design*
Koon, Lo, Marsden, Ross (2022)
*Download for free* at https://ross.aoe.vt.edu/books
► *PDF Lecture Notes (Lecture 12 for this video)*
https://is.gd/3BodyNotes
The circular restricted 3-body problem (CR3BP) describes the motion of a body moving in the gravitational field of two primaries that are orbiting in a circle about their common center of mass, with trajectories such as Lagrange points, halo orbits, Lyapunov planar orbits, quasi-periodic orbits, quasi-halos, low-energy trajectories, etc.
• The two primaries could be Earth & Moon, Sun & Earth, the Sun & Jupiter, etc.
• The mass parameter μ = m2/(m1 + m2) is the main factor determining the type of motion possible for the spacecraft. It is analogous to the Reynold's number Re in fluid mechanics, determining the onset of new types of behavior.
► Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has a Caltech PhD, worked at NASA/JPL and Boeing on interplanetary trajectories, and is a world renowned expert in the 3-body problem. He has written a book on the subject (link above).
►Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Ross Dynamics Lab: http://rossdynamicslab.com
► *Follow me* https://x.com/RossDynamicsLab
► *Other Courses & Series Playlists by Dr. Ross*
📚3-Body Problem Orbital Dynamics
https://is.gd/3BodyProblem
📚Space Manifolds
https://is.gd/SpaceManifolds
📚Space Vehicle Dynamics
https://is.gd/SpaceVehicleDynamics
📚Lagrangian & 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics
📚Nonlinear Dynamics & Chaos
https://is.gd/NonlinearDynamics
📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics
📚Center Manifolds, Normal Forms, & Bifurcations
https://is.gd/CenterManifolds
► *CHAPTERS*
0:00 Introduction to halo orbit space missions
6:53 Mathematical theory
44:43 MATLAB demo
differential correction single and multiple shooting collocation state transition matrix variational equations
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