Microwave Anisotropy Probe

The Lagrange Points

The Italian-French mathematician Josef Lagrange discovered five special points in the vicinity of two orbiting masses where a third, smaller mass can orbit at a fixed distance from the larger masses. More precisely, the Lagrange Points mark positions where the gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them. Those with a mathematical flair can follow this link to a derivation of Lagrange's result.

Of the five Lagrange points, three are unstable and two are stable. The unstable Lagrange points - labelled L1, L2 and L3 - lie along the line connecting the two large masses. The stable Lagrange points - labelled L4 and L5 - form the apex of two equilateral triangles that have the large masses at their vertices.

Lagrange Points

Lagrange Points of the Earth-Sun system (not drawn to scale!).

The L1 point of the Earth-Sun system affords an uninterrupted view of the sun and is currently home to the Solar and Heliospheric Observatory Satellite SOHO. The L2 point of the Earth-Sun system will soon be home to the MAP Satellite and (perhaps) the Next Generation Space Telescope. The L1 and L2 points are unstable on a time scale of approximately 23 days, which requiress satellites parked at these positions to undergo regular course and attitude corrections.

NASA is unlikely to find any use for the L3 point since it remains hidden behind the Sun at all times. The idea of a hidden "Planet-X" at the L3 point has been a popular topic in science fiction writing. The instability of Planet X's orbit (on a timescale of 150 days) didn't stop Hollywood from turning out classics like The Man from Planet X.

The L4 and L5 points are home to stable orbits so long as the mass ratio between the two large masses exceeds 24.96. This condition is satisfied for both the Earth-Sun and Earth-Moon systems, and for many other pairs of bodies in the solar system. Objects found orbiting at the L4 and L5 points are often called Trojans after the three large asteroids Agamemnon, Achilles and Hector that orbit in the L4 and L5 points of the Jupiter-Sun system. (According to Homer, Hector was the Trojan champion slain by Achilles during King Agamemnon's siege of Troy). There are hundreds of Trojan Asteroids in the solar system. Most orbit with Jupiter, but others orbit with Mars. In addition, several of Saturn's moons have Trojan companions. No large asteroids have been found at the Trojan points of the Earth-Moon or Earth-Sun systems. However, in 1956 the Polish astronomer Kordylewski discovered large concentrations of dust at the Trojan points of the Earth-Moon system. Recently, the DIRBE instrument on the COBE satellite confirmed earlier IRAS observations of a dust ring following the Earth's orbit around the Sun. The existence of this ring is closely related to the Trojan points, but the story is complicated by the effects of radiation pressure on the dust grains.

Finding the Lagrange Points

The easiest way to see how Lagrange made his discovery is to adopt a frame of reference that rotates with the system. The forces exerted on a body at rest in this frame can be derived from an effective potential in much the same way that wind speeds can be infered from a weather map. The forces are strongest when the contours of the effective potential are closest together and weakest when the contours are far apart.

Effective Potential

A contour plot of the effective potential.

In the above contour plot highs are colored yellow and lows are colored purple. We see that L4 and L5 correspond to hilltops and L1, L2 and L3 correspond to saddles (i.e. points where the potential is curving up in one direction and down in the other). This suggests that satellites placed at the Lagrange points will have a tendency to wander off (try sitting a marble on top of a watermelon or on top of a real saddle and you get the idea). A detailed analyis confirms our expectations for L1, L2 and L3, but not for L4 and L5. When a satellite parked at L4 or L5 starts to roll off the hill it picks up speed. At this point the Coriolis force comes into play - the same force that causes hurricanes to spin up on the earth - and sends the satellite into a stable orbit around the Lagrange point.

This page was written by Neil J. Cornish as part of MAP's education and outreach program.

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Last updated: Friday, 05-21-1999