# Horseshoe orbit

A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body (such as Earth). The orbital period of the smaller body is very nearly the same as for the larger body, and its path appears to have a horseshoe shape in a rotating reference frame as viewed from the larger object.

The loop is not closed but will drift forward or backward slightly each time, so that the point it circles will appear to move smoothly along the larger body's orbit over a long period of time. When the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, with the larger body between the 'horns'.

Asteroids in horseshoe orbits with respect to Earth include 54509 YORP, 2002 AA29, and 2010 SO16, and possibly 2001 GO2. A broader definition includes 3753 Cruithne, which can be said to be in a compound and/or transition orbit,[1] or (85770) 1998 UP1 and 2003 YN107.

Saturn's moons Epimetheus and Janus occupy horseshoe orbits with respect to each other (in their case, there is no repeated looping: each one traces a full horseshoe with respect to the other).

## Explanation of horseshoe orbital cycle

### Background

The following explanation relates to an asteroid which is in such an orbit around the Sun, and is also affected by the Earth.

The asteroid is in almost the same solar orbit as Earth. Both take approximately one year to orbit the Sun.

It is also necessary to grasp two rules of orbit dynamics:

1. A body closer to the Sun completes an orbit more quickly than a body further away.
2. If a body accelerates along its orbit, its orbit moves outwards from the Sun. If it decelerates, the orbital radius decreases.

The horseshoe orbit arises because the gravitational attraction of the Earth changes the shape of the elliptical orbit of the asteroid. The shape changes are very small but result in significant changes relative to the Earth.

The horseshoe becomes apparent only when mapping the movement of the asteroid relative to both the Sun and the Earth. The asteroid always orbits the Sun in the same direction. However, it goes through a cycle of catching up with the Earth and falling behind, so that its movement relative to both the Sun and the Earth traces a shape like the outline of a horseshoe.

### Stages of the orbit

Figure 1. Plan showing possible orbits along gravitational contours. In this image, the Earth (and the whole image with it) is rotating counterclockwise around the Sun.
Figure 2. Thin horseshoe orbit

Starting out at point A on the inner ring between L5 and Earth, the satellite is orbiting faster than the Earth. It's on its way toward passing between the Earth and the Sun. But Earth's gravity exerts an outward accelerating force, pulling the satellite into a higher orbit which (per Kepler's third law) decreases its angular speed.

When the satellite gets to point B, it is traveling at the same speed as Earth. Earth's gravity is still accelerating the satellite along the orbital path, and continues to pull the satellite into a higher orbit. Eventually, at C, the satellite reaches a high enough, slow enough orbit and starts to lag behind Earth. It then spends the next century or more appearing to drift 'backwards' around the orbit when viewed relative to the Earth. Its orbit around the Sun still takes only slightly more than one Earth year.

Eventually the satellite comes around to point D. Earth's gravity is now reducing the satellite's orbital velocity, causing it to fall into a lower orbit, which actually increases the angular speed of the satellite. This continues until the satellite's orbit is lower and faster than Earth's orbit. It begins moving out ahead of the earth. Over the next few centuries it completes its journey back to point A.

### Energy viewpoint

A somewhat different, but equivalent, view of the situation may be noted by considering conservation of energy. It is a theorem of classical mechanics that a body moving in a time-independent potential field will have its total energy, E = T + V, conserved, where E is total energy, T is kinetic energy (always non-negative) and V is potential energy, which is negative. It is apparent then, since V = -GM/R near a gravitating body of mass M, that seen from a stationary frame, V will be increasing for the region behind M, and decreasing for the region in front of it. However, orbits with lower total energy have shorter periods, and so a body moving slowly on the forward side of a planet will lose energy, fall into a shorter-period orbit, and thus slowly move away, or be "repelled" from it. Bodies moving slowly on the trailing side of the planet will gain energy, rise to a higher, slower, orbit, and thereby fall behind, similarly repelled. Thus a small body can move back and forth between a leading and a trailing position, never approaching too close to the planet that dominates the region.