# How do horseshoe orbits work?

An asteroid was recently discovered that is in a horseshoe orbit with respect to the earth. Is there an intuitive explanation for these orbits? It seems that the earth acts as a repulsive force where one would expect gravity to be attractive. From the above linked article:

Accepted in MNRAS, Apostolos A. Christou, David J. Asher, 1104.0036, A long-lived horseshoe companion to the Earth

A better description of the horseshoe orbits, along with the tadpole orbits and Lagrange points:

• Is the earth inside the orbit? Meaning is the orbit approximately parabolic near the earth with the earth being close to the focal point?
– user1567
Apr 9, 2011 at 20:31
• The earth goes around the sun once per year. The asteroid does too, but slowly speeds up and slows down so it follows the horseshoe shown, with a period of 350 years. That is, after 350 years, the average time the asteroid takes to go around the earth is exactly a year. Apr 9, 2011 at 20:33
• Very funky phenomenon, +1 Apr 9, 2011 at 20:58
• there's a lot of weirdness embedded in the phase space of the n-body problem: blogs.discovermagazine.com/cosmicvariance/2006/07/23/n-bodies Apr 9, 2011 at 21:06
• @Jerry, great luck that Kepler did not know this orbit :=) Apr 10, 2011 at 10:17

From the perspective of an inertial frame in heliocentric coordinates this asteroid is in a circular orbit (topologically a circle) around the sun. When the orbit is closer to the sun than the Earth’s orbit the asteroid has a smaller orbital period, or equivalently it has a larger velocity. It will eventually catch up with the Earth, but it is not necessarily gravitationally drawn into the Earth. It interacts with the Earth’s gravity field in its frame with an effective and repulsive potential $L^2/2mr$, $L$ = angular momentum $r$ = distance from Earth. The gravitational potential plus this effective potential pushes the asteroid into a higher orbital radius. The Lagrange points $L_4$ and $L_5$ act then as attractor points in the rotational frame of the asteroid. Its orbital velocity is now smaller and recedes away from the Earth. Eventually the Earth approaches the asteroid and the process is repeated.