Horseshoe orbit

An example of horseshoe orbit

Orbits of the co-ordinate asteroid 2002 AA ₂₉ and the earth around the sun in the perpendicular view of the ecliptic ; Image: JPL

Horseshoe orbit from 2002 AA ₂₉ along the earth's orbit over 95 years viewed from the reference system moving with the earth's orbital motion; Image: JPL

The horseshoe orbit , the horseshoe orbit or the horseshoe orbit is a special orbit of a co-orbital object , which, together with a second (mostly much larger) body in the same or a very similar orbit, orbits a central star. In the normal resting frame of reference of the central star, the orbit of the co-orbital companion looks like a normal Kepler elliptical orbit. From the reference system that moves along with the movement of the larger object around the central star (in which the larger celestial body seems to rest) one then only sees the relative movement of the co-orbital companion. From this reference system, the co-ordinate companion describes a large arc along the orbit of the larger body, which it periodically swings back and forth. The shape of the arch is reminiscent of the closed contour of a horseshoe , hence the name horseshoe orbit.

stability

Because of their very similar orbits, co-ordinate objects have the same mean period around the central star as the larger celestial body. They are in gravitational interaction with the larger celestial body and are in a so-called 1: 1 orbit resonance due to the same mean period of rotation . Such orbits are only stable under certain conditions, of which normally the most important is that the co-ordinate companion has a vanishingly small mass in relation to the larger body (so-called restricted three-body problem ).

However, bodies in a horseshoe orbit do not have to have negligible mass in order to occupy a stable orbit (see section Examples ).

Explanation

Figure 1 : Horseshoe orbit of an object (turquoise) in the sun-earth system. The reference system of this diagram (not to scale) rotates with the rotation of the earth around the sun, so that the sun and earth are fixed in the diagram.

The Figure 1 shows a possible horseshoe orbit an object in the Sun-Earth system, in which both the Earth and the object rotate together counterclockwise around the sun and the object changes regularly around the sun between a sun near and sun distant orbit, especially but, from the point of view of the earth, it is sometimes more "in front of", sometimes more "behind" it, so that its relative position to the sun and earth describes the shown horseshoe over the course of time.

Assuming that the object is at the beginning in point A of a circular orbit around the sun, so to speak "behind" the earth orbiting the sun: Since this orbit is closer to the sun, the object has a slightly higher orbital speed than the earth, so it moves, like a celestial body “overtaking” the earth on the left, slowly towards it.

Due to the increasing gravitational force of the earth, it is additionally accelerated again along the orbit, i.e. counterclockwise, with the result that the object begins to drift outwards due to the increased centrifugal force into an orbit further away from the sun, in which its orbital velocity from point B gradually begins to decrease again than that of the earth.

Now circling more slowly around the sun than the earth, from point C the object gradually begins to fall further and further behind the position of the earth and thus to move further and further away from the sphere of influence of its gravity, in order to then "fly behind" the earth at an ever greater distance , until the object appears again at point D, now apparently on the right "in front of" it from the perspective of the earth, and again comes under its gravitational influence.

This time, however, the earth pulls the object from “behind”, which causes it to be braked again and, as a result of dwindling centrifugal force, changes back to the orbit closer to the sun, where its orbital speed increases again so far that the object moves with it from point E. now again higher orbital speed than the earth, one more time away from it, to finally reappear at point A, apparently "behind" the earth, whereby everything can repeat itself as described.

Transition to Trojans

The transition from a Trojan horse to a horseshoe track is fluid: If the distance of a Trojan horse to the Lagrangian point L ₄ - or L _{5 is} too great, then once on the orbit it will cross the point opposite the larger celestial body and then in the direction of the other Lagrangian Wander point and thus swing back and forth in a large arc.

Examples

So far, only a few objects on horseshoe tracks are known. Noteworthy are the two co-ordinate companions known to date on Earth , the asteroid 2002 AA ₂₉ (an object less than 100 m in diameter) and the approximately 300 m large (419624) 2010 SO ₁₆ . Between 1996 and 2006, a quasi-satellite of the earth was the small asteroid 2003 YN ₁₀₇ , which since then has described a horseshoe orbit along the earth's orbit as before. Two other co-ordinate objects on unusual horseshoe orbits are the small, almost equally large Saturn moons Janus and Epimetheus , which orbit Saturn in very similar orbits, come very close every four years and swap their orbits.

Planetoids are also known that are currently in a quasi-orbit around the planet Neptune, that is, a horseshoe-shaped orbit around Neptune, whose closest point to the Sun is on Uranus and whose point furthest from the Sun is about twice as far away: (309239) 2007 RW ₁₀ , (316179) 2010 EN ₆₅ .

Web links

Wiktionary: Horseshoe orbit  - explanations of meanings, word origins, synonyms, translations

• Article about horseshoe orbits ( using the example of the first co-orbital companion discovered on Earth, the asteroid (3753) Cruithne ; English)
• Articles about 2002 AA ₂₉ , a small revolving on a horseshoe path with Earth asteroids (English)
• GIF animations of the orbital movements of the earth and the asteroid 2002 AA ₂₉ (English)

Individual evidence

1. ↑ This relationship results from the formula for the first cosmic circular velocity . That is the speed at which centrifugal force and gravitational force cancel each other out. Then d. that is, an object in a smaller radius orbit has a greater velocity. ${\displaystyle v^{2}\sim {\frac {1}{r}}}$
2. ↑ Asteroid 2010 SO16 is following Earth in its orbit around sun . earthsky.org. April 6, 2011. Retrieved April 10, 2011.
3. ↑ C. de la Fuente Marcos, R. de la Fuente Marcos: (309239) 2007 RW10: a large temporary quasi-satellite of Neptune . In: Astronomy & Astrophysics Letters . 545, 2012, p. L9. arxiv : 1209.1577 . bibcode : 2012A & A ... 545L ... 9D . doi : 10.1051 / 0004-6361 / 201219931 .
4. ↑ C. de la Fuente Marcos, R. de la Fuente Marcos: Four temporary Neptune co-orbitals: (148975) 2001 XA255, (310071) 2010 KR59, (316179) 2010 EN65, and 2012 GX17 . In: Astronomy and Astrophysics . 547, November 2012. arxiv : 1210.3466 . bibcode : 2012A & A ... 547L ... 2D . doi : 10.1051 / 0004-6361 / 201220377 .