The mathematician Joseph Lagrange studied the so-called restricted three-body systems, where one

of the three masses is much smaller than the other two. In this case, to a very good approximation,

the two large masses (M1 and M2, with M1 < M2) follow a standard two-body Keplerian orbit,

while the small mass (m) moves in the gravitational potential generated by M1 and M2. Lagrange

examined quasicircular orbits: he determined that, in the corotating frame where M1 and M2 are

at rest, there are ve points of equilibrium for m, now known as the Lagrange points L1{L5. In

the inertial frame, the Lagrange points correspond to stationary orbits, which have the property

that the distance between m and M1 and M2 is constant at all times.

In the corotating frame, the Lagrange points L1{L3 sit along the line that joins M1 and M2;

these points are unstable, meaning that a small mass initially placed there will gradually veer

o (this did not keep NASA from placing the SOHO solar observatory at L1 in the Earth{Sun

system, and from planning the WMAP microwave-radiation anisotropy probe for L2, because

unstable orbits can still be corrected with thrusters). On the contrary, L4 and L5 are stable; they

sit roughly along the orbit of M1, respectively sixty degrees ahead and behind this mass. Apart

from mathematical arguments, we can nd evidence of their stability by looking up at the sky:

the L4 and L5 points of the Jupiter{Sun system are home to the two families of Trojan asteroids

(so called because they were given names associated with the Iliad ). Trojan asteroids have been

found also at the Lagrange points of Mars and Neptune.

When this assignment was originally written, no trojan asteroids of earth had been found.

However, a trojan asteroid of earth has now been discovered:

http://news.nationalgeographic.com/news/2011/07/1107128-trojan-asteroid-earth-planet-orbit-nasa-

space-science/

Caltech and students from around the globe met at Caltech in the summer of 2011 to compete

in the Caltech Space Challenge to design a mission to a near earth asteroid. They considered this

new trojan asteroid, 2010 TK7, but as the article explains, it is more di cult to reach than some

other known near earth asteroids. Once you are done with this assignment, which considers trojan

asteroids around jupiter, you may wish to model the 2010 TK7 orbit.