MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

Applications 83
Application 7 (Lagrange’s Equilateral Triangle Solution (Three-Body
Problem)). The three-body problem requires the solution of the equations of
motion of three mutually attracting masses confined to a plane. One of the few
known analytical solutions is Lagrange’s Equilateral Triangle Solution [171].
As illustrated in Figure 3.7 (m1 : m2 : m3 = 1 : 2 : 3), the particles
sit at the vertices of an equilateral triangle as this triangle changes size and
rotates. Each particle follows an elliptical path of the same eccentricity but
oriented at different angles with their common center of mass located at a focal
point of all three orbits. The motion is periodic with the same period for all
three particles. This solution is stable if and only if one of the three masses is
much greater than the other two. However, very special initial conditions are
required for such a configuration.
Figure 3.8: Lagrangian Points [1]
Application 8 (Lagrangian Points (Restricted Three-Body Problem)).
In the circular restricted three-body problem, one of the three masses is taken
to be negligible while the other two masses assume circular orbits about their
center of mass.
There are five points (Lagrangian points, L-points, libration points) where
the gravitational forces of the two large bodies exactly balance the centrifugal
force felt by the small body [1]. An object placed at one of these points
would remain in the same position relative to the other two. Points L4 and
L5 are located at the vertices of equilateral triangles with base connecting
the two large masses; L4 lies 60 ◦ ahead and L5 lies 60 ◦ behind as illustrated
in Figure 3.8. These two Lagrangian Points are (conditionally) stable under
small perturbations so that objects tend to accumulate in the vicinity of these
points. The so-called Trojan asteroids are located at the L4 and L5 points
of the Sun-Jupiter system. Furthermore, the Saturnian moon Tethys has two
smaller moons, Telesto and Calypso, at its L4 and L5 points while the Saturn-
Dione L4 and L5 points hold the small moons Helene and Polydeuces [296, p.
222].