What is a chaotic orbit?

When many objects interfere (as in a n-body system) it is very difficult to
solve the problem. But if the determination of the orbits is
hard, it doesn't mean that it is impossible: with computers that
are powerful enough, a very complicated n-body system such
as the solar system is
completely solved, and the orbits of planets are today perfectly
known. In other words, the planets orbits are* stable*:
independently on how hard the calculus is, it is possible to
predict where the planets will exactly be in the future. In fact,
for these objects, the final state of the object (the position
and the speed of the planet at a precise moment in the future)
depends totally on the initial conditions and on the forces that
act on it.

When talking of the orbits of asteroids and
comets, an additional concept must be introduced: the concept of *chaotic
orbits.*

Stable and unstable positions

It is much easier to understand the nature
of chaotic orbits making a parallelism with stable
and unstable positions in the motion of a
pendulum. For a pendulum, this concept is very intuitive:
a position of stable equilibrium corresponds to a
position the body will come back to, when slightly
perturbed. For unstable equilibrium, a little deviation
from this initial position is sufficient to make the body
leave its initial condition. |

Forcing this parallelism we can say that
*stable* orbits will be highly predictable orbits (just
like the position of stable equilibrium is for the pendulum)
while* unstable *orbits will depend abruptly from the
initial conditions (as for the pendulum, starting from a position
of unstable equilibrium, the final position depends strongly on
the perturbation).

Chaotic orbits and the Lyapounov time

It is important to
understand that a chaotic orbit is unpredictable for its
nature and therefore, the position of the object can't be
perfectly determined in the future. In other words, a
chaotic orbit is so sensible to very small changes in the
initial conditions (or in the forces that act upon the
body) that the prevision of the trajectory over long
periods of time is impossible. This because different orbits are divergent:
a very little difference in the initial conditions leads
to totally different orbits in the future. Even if this
explanation is very intuitive, it is important to
understand that chaos is a mathematical concept, that can
be formalised and measured by a parameter called the Lyapounov time. |

NEOs' chaotic orbits: an example

The most
classical example of chaotic orbits is the trajectory of
a light body such as an asteroid or a comet. In fact,
there are many physical mechanisms (such as the
fly-by of a planet, or a resonance phenomena) that act on
asteroids and comets, perturbing their initial orbits and
making them become NEOs (click here to
know more about these mechanisms). An example of these mechanisms, is the close approach of a planet. As shown in the picture, when the asteroid passes near a planet, even for very similar initial conditions, the orbits diverge. |