Libration point trajectories

These are natural trajectories of a spacecraft in a gravitational vector-field governed mainly by two primaries (e.g. Sun and Earth, Earth and Moon, Sun and Jupier,...) where the relative position of the spacecraft with respect to the primaries is almost constant.

These trajectories are located around five fixed points with respect to the primaries. Three of them are placed in the line joining the primaries, and the remaining two form an equilateral triangle with them. In a rotating reference frame which keeps the primaries in fixed positions, libration points are seen as places where the gravitational forces of the primaries compensates the centrifugal force of the non-inertial frame.

Due to its privileged location, libration point trajectories, among other applications, are suitable places to put satellites devoted to solar science, space telescopes and possible future gateway stations to the Solar System. See http://sohowww.estec.esa.nl/ for a current libration point mission.

In order to illustrate libration point orbits, consider the following figure.

The white sphere on the left represents the Moon, and the blue sphere on the right represents the Earth. The moving spheres (click here for an animation of the previous figure) stand for a cluster of artificial satellites, flying in formation. Their trajectories describe a geometrical object known as "invariant torus" in Dynamical Systems theory. The trajectories of these satellites are examples of quasi-periodic motion with two frequencies: the one corresponding to the rotation of the cluster, and the other corresponding to the translation of the cluster along the invariant torus.

If we plot a dot every time a satellite crosses the horizontal plane (which is the plane of rotation of the Moon around the Earth) we get a closed curve, which is the intersection of the invariant torus with the horizontal plane. This curve is surrounded by the green curve, which is a closed trajectory, known as Lyapunov planar periodic orbit. For a given energy level, the traces on the horizontal plane of all the bounded trajectories of the vicinity of the libration point L1 (which is the point where the gravitational forces of the Earth and the Moon compensate) are inside the Lyapunov periodic orbit of this level of energy.

These trajectories cannot be obtained by direct numerical integration because they are highly unstable. One approach to compute them is to use a technique known as "reduction to the central manifold", which consists in computing, using symbolic manipulation, the expansions of a change of coordinates that allows to remove the unstable directions. The resulting system of differential equations can be integrated numerically for long time spans. In the third demo, we do this for a representative set of all the trajectories of a fixed energy level, and plot their intersections with the horizontal plane in the figure below. The trajectories are integrated in parallel.

The drawback of the above approach is limited to the regions where the expansions computed are accurate. Out of this regions, one has to use a different approach, that is more expensive in terms of computing time. It consists in computing these trajectories by direct numerical methods (in opposition to the above approach, which is semi-analytical). In the fourth demo, we start the parallel computation of several of these trajectories. We do not actually compute the trajectories, but the invariant tori in which they are contained.

In the above figure, each '+' symbol represents an invariant torus, similar to the one represented in the animation above. The horizontal coordinate is the energy level, and the vertical coordinate is the quotient of the frequencies of the trajectories on each invariant torus (this is known as "rotation number").