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.
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
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.
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.
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Last updated: Friday, 05-21-1999