Horseshoe orbit Totally Explained

Everything about Horseshoe Orbit totally explained

A horseshoe orbit appears when a viewer on an orbiting body (like Earth) watches the movement of another orbiting body, whose orbit is skinnier (more eccentric), but has about the same period. As a result, the path appears to have the shape of a kidney bean. The loop isn't closed but will drift forward or backward slightly each time, so that the point it circles will appear to move smoothly along Earth's orbit over a long period of time. When the object approaches Earth closely at either end of its trajectory, gravitational exchange of energy changes the object's apparent direction. Over an entire cycle the center traces the outline of a horseshoe, with the Earth between the 'horns'.
Several asteroids (such as 3753 Cruithne and ) are known to occupy horseshoe orbits with respect to Earth. Saturn's moons Epimetheus and Janus occupy horseshoe orbits with respect to each other (in their case, there's 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 further 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 only becomes apparent 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

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 and – counter-intuitively – slowing it down.
When the satellite gets to point B, it's traveling at the same speed as Earth. Earth's gravity is still accelerating the satellite along the orbital path. The twisted rule above continues to pull the satellite into a higher orbit. Eventually, at C, the satellite reaches a high enough, slow enough orbit and starts to fall behind Earth. It then spends the next century or more drifting 'backwards' around the orbit. Remember that its orbit around the Sun still takes only slightly more than one Earth year. The apparent backwards motion is only relative to the Earth.
Eventually the satellite comes around to point D. Earth's gravity is now reducing the satellite's orbital velocity. The counter-intuitive result is a lower, faster orbit. This continues until the satellite's orbit is lower & 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.

Tadpole orbit

Figure 1 above shows shorter orbits around the Lagrangian points L4 and L5 (for example the lines close to the blue triangles). These are called tadpole orbits and can be explained in a similar way, except that the asteroid's distance from the Earth doesn't oscillate as far as the L3 point on the other side of the Sun. As it moves closer to or further from the Earth, the changing pull of Earth's gravitational field causes it to accelerate or decelerate, causing a change in its orbit known as libration.
An example of a body in a tadpole orbit is Polydeuces, a small moon of Saturn which librates around the trailing L5 point relative to a larger moon, Dione.

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