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:
- 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).
Images
- 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.
Local file: local link: three_body_system.html.
File: Orbit file:
three_body_system.html.