- The three-body system
in the animation
is probably
**NOT**in a periodic solution (of the three-body problem), but the original caption of the animation made**NO**definite statement. The solution, in fact, is probably chaotic. The solution was probably calculated by an N-body simulation. - Image 2 Caption: An animation
of the
figure-8 orbit solution
for a
gravitationally bound
three-body system
over a single
orbital period
of T = 6.3259 in some time
unit
(see Wikipedia:
Three-body problem: Special-case solutions).
- The solution in this case is periodic.
Thus, it is an exact periodic solution of the
three-body system.
However, there is
**NO**closed-form formula (i.e.,**NO**formula one can just write down). The figure-8 orbit solution is (neutrally) stable to small astronomical perturbations, but its range of (neutral) stability is very small. So it probably occurs only rarely and fleetingly in nature and has**NEVER**been observed. - Note stability
for orbits
is usually neutral stability
in yours truly's understanding.
So astronomical perturbations
cause changes that are
**NOT**damped out as for stable equilibriums, but the changes do**NOT**grow without bound as for unstable equilibriums. The changes are in a vague sense stay roughly proportional to the astronomical perturbations. However, stable equilibriums and unstable equilibriums do occur for orbits. For example, consider the Lagrange point orbits. The L4 and L5 points are stable equilibriums and the L1, L2, and L3 points are unstable equilibriums. For more on the Lagrange points, see File lagrange_points.html. - See also the definition
of figure eight (AKA figure 8).
- Credit/Permission: ©
User:Dnttllthmmnm,
2017 /
CC BY-SA 4.0.

Image link: Wikimedia Commons: File:Three-body Problem Animation with COM.gif.

- Credit/Permission: ©
User:MaxwellMolecule,
2019 /
CC BY-SA 4.0.

Image link: Wikimedia Commons: File:Three body problem figure-8 orbit animation.gif.

Image 1 Caption: An animation of a gravitationally-bound three-body system showing the approximate orbital trajectories of three identical point masses initially at rest in the inertial frame defined by their mutual barycenter (i.e., center of mass). The point masses are, of course, located at the vertices a triangle which at almost all times is a scalene triangle. The barycenter stays at rest in obedience to the law of conservation of momentum.

Features:

File: Orbit file: three_body_system.html.