# Lagrange points tutorial was: Fly Me to L 1

Joseph S. Barrera III joe at barrera.org
Sun Dec 7 06:19:30 PST 2003

You should visit the link to see the nice pictures and links to follow.
But I've included the text here in case the link fails one day.

http://www.physics.montana.edu/faculty/cornish/lagrange.html

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
http://www.physics.montana.edu/faculty/cornish/lagrange.pdf 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.

[PICTURE] 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.

[PICTURE] 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.

David N. Spergel / dns at astro.princeton.edu
Gary Hinshaw / hinshaw at stars.gsfc.nasa.gov
Charles L. Bennett / bennett at stars.gsfc.nasa.gov