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A circular, restricted, three-body problem gives solutions
for the motion of a third body whose mass does not affect
the motion of the system due to the gravitational forces from
two other massive bodies rotating around their center of mass
under their mutual gravitational attraction. The problem takes
the orbits of the two massive bodies as coplanar and circular.
Even if the two orbits are not exactly coplanar and circular,
approximating their orbits in this manner and assuming no
influence to the third object's motion because of its mass
yields analytical solutions for qualitative analysis for the
motion. Forces from outside the system are neglected, assuming
the gravitational forces of the two massive objects determine
the behavior of third object's motion.
At five different locations in the plane of rotation the gravitational
forces the object feels is equal to the centripetal force
needed to rotate with the other two bodies. An object at one
of these points will not move in the plane of rotation. The
object will share the same orbital period as the other two
bodies do around the system's center of mass. The points of
equilibrium are called the Lagrange points.
L1, L2, L3, L4, and L5 are the labels for the individual
Lagrange points. L1, L2, and L3 are collinear with the axis
connecting the two massive bodies, with one between them and
the other two on the outside. In the Sun-Earth system the
L1 point is between the two massive bodies. L2 is past the
Earth, and past the Sun is L3. L4 and L5 are at the apex of
equilateral triangles with the massive bodies at the vertices
(Figure 1.) L4 usually is usually associated with the leading
triangle, L5 the trailing.
Outside forces ignored by the circular, restricted three-body
system become important to the motion of an object at a Lagrange
point. The forces due to gravity of the two massive bodies
in the system dwarf outside perturbations, but objects at
Lagrange points are in a delicate balance between those forces.
Despite their small magnitude, outside forces will disrupt
that balance, not allowing an object to stay at those points
for any length of time. Some small outside forces can be radiation
pressure or forces of gravity from an outlying massive body.
Objects can settle in an orbit around a Lagrange point. Orbits
around the three collinear points, L1, L2, and L3, are unstable.
They last but days before the object will break away. L1 and
L2 last about 23 days. Objects orbiting around L4 and L5 are
stable because the Coriolis force keeps them spinning around
the Lagrange point. |
