Horseshoe orbit

# Horseshoe orbit

Overview
A horseshoe orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

is a type of co-orbital motion
Co-orbital configuration
In astronomy, a co-orbital configuration refers to two or more celestial objects that orbit at the same, or very similar, distance from their parent object as each other, i.e. they are in a 1:1 mean motion resonance....

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
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...

as viewed from the larger object.
Discussion

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Encyclopedia
A horseshoe orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

is a type of co-orbital motion
Co-orbital configuration
In astronomy, a co-orbital configuration refers to two or more celestial objects that orbit at the same, or very similar, distance from their parent object as each other, i.e. they are in a 1:1 mean motion resonance....

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
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...

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 Earth's orbit over a long period of time. When the object approaches Earth closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe
Horseshoe
A horseshoe, is a fabricated product, normally made of metal, although sometimes made partially or wholly of modern synthetic materials, designed to protect a horse's hoof from wear and tear. Shoes are attached on the palmar surface of the hooves, usually nailed through the insensitive hoof wall...

, with the Earth between the 'horns'.

Asteroids in horseshoe orbits with respect to Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

include 54509 YORP
54509 YORP
54509 YORP is an Apollo Near-Earth Object discovered on August 3, 2000 by the Lincoln Laboratory Near-Earth Asteroid Research Team at Socorro. Measurements of the rotation rate of this object provided the first observational evidence of the YORP effect, hence the name of the asteroid...

, , and , and possibly . A broader definition includes 3753 Cruithne
3753 Cruithne
3753 Cruithne is an asteroid in orbit around the Sun in approximate 1:1 orbital resonance with the Earth. It is a periodic inclusion planetoid orbiting the Sun in an apparent horseshoe orbit. It has been incorrectly called "Earth's second moon", but it is only a quasi-satellite. Cruithne never...

, which can be said to be in a compound and/or transition orbit, or and .

Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

's moons Epimetheus
Epimetheus (moon)
Epimetheus is an inner satellite of Saturn. It is also known as Saturn XI. It is named after the mythological Epimetheus, brother of Prometheus.-Discovery:Epimetheus occupies essentially the same orbit as the moon Janus...

and Janus
Janus (moon)
Janus is an inner satellite of Saturn. It is also known as Saturn X . It is named after the mythological Janus.-Discovery and orbit:Janus occupies practically the same orbit as the moon Epimetheus...

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).

### Background

The following explanation relates to an asteroid which is in such an orbit around the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, 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 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 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
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

. 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.

Figure 1 above shows shorter orbits around the Lagrangian point
Lagrangian point
The Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...

s and (e.g. the lines close to the blue triangles). These are called tadpole
A tadpole or polliwog is the wholly aquatic larval stage in the life cycle of an amphibian, particularly that of a frog or toad.- Appellation :...

orbits
and can be explained in a similar way, except that the asteroid's distance from the Earth does not oscillate as far as the point on the other side of the Sun. As it moves closer to or farther 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
Libration
In astronomy, libration is an oscillating motion of orbiting bodies relative to each other, notably including the motion of the Moon relative to Earth, or of Trojan asteroids relative to planets.-Lunar libration:...

.

An example of a body in a tadpole orbit is Polydeuces
Polydeuces (moon)
Polydeuces is a very small natural satellite of Saturn that is co-orbital with Dione and librates around the trailing Lagrangian point . Its diameter is estimated to be about 3.5 km....

, a small moon of Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...

which librates around the trailing point relative to a larger moon, Dione
Dione (moon)
Dione is a moon of Saturn discovered by Cassini in 1684. It is named after the titan Dione of Greek mythology. It is also designated Saturn IV.- Name :...

.

• Box orbit
Box orbit
In stellar dynamics a box orbit refers to a particular type of orbit which can be seen in triaxial systems, that is, systems which do not possess a symmetry around any of its axes...

• Co-orbital moon
Co-orbital moon
In astronomy, a co-orbital configuration refers to two or more celestial objects that orbit at the same, or very similar, distance from their parent object as each other, i.e. they are in a 1:1 mean motion resonance....

• Interplanetary Transport Network
Interplanetary Transport Network
The Interplanetary Transport Network is a collection of gravitationally determined pathways through the solar system that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little...

• Natural satellite
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....

• Quasi-satellite
Quasi-satellite
A quasi-satellite is an object in a 1:1 orbital resonance with its planet that stays close to the planet over many orbital periods.A quasi-satellite's orbit around the Sun takes exactly the same time as the planet's, but has a different eccentricity , as shown in the diagram on the right...