In celestial mechanics, a horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating (instantaneous) orbital period of the smaller body remains very near that of the larger body, and if its orbit is a little more eccentric than that of the larger body, during every period it appears to trace an ellipse around a point on the larger object's orbit. However, the loop is not closed but drifts forward or backward 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, , , and possibly . A broader definition includes 3753 Cruithne, which can be said to be in a compound and/or transition orbit, or and . By 2016, 12 horseshoe librators of Earth have been discovered. 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: A body closer to the Sun completes an orbit more quickly than a body farther away. 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