Are the Lagrangian points fixed with respect to the Earth?

Question posed by dthomson

What are Lagrangian points?
Put as simply as possible, Lagrangian points are places near to two orbiting bodies in which a third body with relatively little mass can remain 'stationary' relative to the two larger masses.

In slightly more detail, the Lagrangian points describe positions in a three body system where the combined gravitational pull of the two larger bodies provides exactly the force required for a third, much less massive body to rotate at the same rate.

These points were originally theorised by the Italian-French mathematician Joseph Louis Lagrange when studying the three-body system in 1772. He found five points at which the net force from two bodies acting upon the third was zero. These points are known as L1, L2, L3, L4 and L5 (as shown in the image borrowed from http://www.jwst.nasa.gov/, using the Sun- Earth system (not to scale!) The third body may be a man-made satellite, for example).

Points L4 and L5 are stable points. This means that if, for some reason, an object at one of these points is nudged away from it, forces will conspire to push it back towards the stable point*- imagine a marble in the bottom of a curved bowl. Whichever way you push it, when you let go it will eventually settle back into a rest position at the bottom.

The first three points, L1, L2 and L3 are only partly stable: It depends which direction you nudge the third object in. If, for example, you were to give it a push in the plane perpendicular to an imaginary line joining the larger two objects, it would be drawn back to the Lagrangian point it came from. If on the other hand you give it a nudge directly towards either of the two large bodies, it will continue to be drawn away from its starting point and towards one of the bodies. If you're having trouble getting to grips with this, imagine a marble balancing on a pringle, and what would happen if you pushed it in different directions.


But the Solar System isn't a 'three body problem'!
The original questioner also said "I wonder if the one between Earth and Mars' orbit might fluctuate, since the outer planets don't stay put," which is a good point. The thing is, multiple-body problems increase dramatically in complexity with the number of bodies: A one body system is simple as it's always stationary relative to itself; A two body problem is just a matter of the bodies orbiting around a central point; a third body introduces various difficulties; many more and it all goes a bit bonkers.

Naturally the planets and other bodies in our solar system will have an effect on the positioning and stability of these points. In the case of points L1, L2, L4 and L5, that effect is negligible due to the distances involved**. At L3, however, there is a different story and other planets get more of a say in matters: every 20 months, for example, Venus finds itself only around 0.3 AU away from the Earth-Sun L3 point; Earth itself is, by definition, around 2 AU away at all times.

It's worth noting that L4 and L5 are also by definition around 1AU from the Earth at all times, so other planets do have more of an effect on these than Earth now and then. This increased effect is, however, short lived, and Earth's dominance reigns. Any disturbance is automatically righted as they are stable points.


Why the interest in Lagrangian points?
Stable points allow the positioning of man-made satellites for various purposes. Arguably most famous is the SOHO mission at the Sun-Earth L1 point, which is a Sun observation platform. Orbiting at L1 means that the satellite's view of the Sun is never obstructed by the Earth. There are other research missions currently being carried out at L1, including ACE and WIND. L2 is currently host to WMAP among other space observatories and probes. Other missions have been proposed, and thoughts for the far future include L4 and L5 based manufacturing plants as stepping stones further out into the solar system.

The larger planets Neptune, Saturn and Jupiter, have collections of captured asteroids orbiting at their respective L4 and L5 points. These are known as Trojan asteroids, and much speculation has been made (particularly in works of science fiction) regarding these as potential sources of raw materials for missions, observatories and waypoints even further away from home.

The animation to the left (from Petr Scheirich) shows the simulated orbits of a number of Jovian Trojans in green (the red ones are the Hilda asteroids, which look good in this animation but are largely irrelevant to this post).




* Or, more accurately, it will start to orbit around the stable point itself; much like what you would actually observe when you first try the marble-bowl experiment described above.
** And the fact that the effect of gravity reduces with the square of the distance

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