3753 Cruithne

3753 Cruithne is an asteroid.
Discovery
3753 Cruithne was discovered on October 10, 1986, by Duncan Waldron on the UK Schmidt Telescope at Siding Spring Observatory, Coonabarabran, Australia. The 1983 appearance is given to Giovanni de Sanctis and Richard M. West of the European Southern Observatory in Chile. It was not until 1997 that its unusual orbit was found by Paul Wiegert and Kimmo Innanen, working at York University in Toronto, and Seppo Mikkola, working at the University of Turku in Finland. The asteroid is named after the Cruithne, a people of early medieval Ireland.
Measurements
3753 Cruithne is about 5 kilometres (3.1 mi) in diameter. From 1994 through 2015, 3753 Cruithne makes its annual closest approach to Earth every November. Although Cruithne's orbit is not thought to be stable over the long term, calculations showed that it has probably been synchronized with Earth's orbit for a long time. There is no danger of a collision with Earth for millions of years, if ever. Its orbital path and Earth's do not cross.
Orbit
Cruithne is in a normal elliptic orbit around the Sun. Its period of revolution around the Sun, about 364 days at present, is almost equal to that of the Earth. Because of this, Cruithne and Earth appear to follow each other in their paths around the Sun. This is why Cruithne is sometimes called "Earth's second moon". However, it does not orbit the Earth and is not a moon. In 2058, Cruithne will come within 13.6 million kilometres of Mars. Cruithne's distance from the Sun and orbital speed vary a lot more than the Earth's, so from the Earth's point of view Cruithne actually follows a kidney bean-shaped horseshoe orbit ahead of the Earth, taking slightly less than one year to complete a circuit of the "bean". Because it takes slightly less than a year, the Earth "falls behind" the bean a little more each year, and so from our point of view, the circuit is not quite closed, but rather like a spiral loop that moves slowly away from the Earth.
After many years, the Earth will have fallen so far behind that Cruithne will then actually be "catching up" on the Earth from "behind". When it eventually does catch up, Cruithne will make a series of annual close approaches to the Earth and gravitationally exchange orbital energy with Earth; this will alter Cruithne's orbit by a little over half a million kilometres so that its period of revolution around the Sun will then become slightly more than a year. The kidney bean will then start to migrate away from the Earth again in the opposite direction -- instead of the Earth "falling behind" the bean, the Earth is "pulling away from" the bean. The next such series of close approaches will be centred on the year 2292 -- in July of that year, Cruithne will approach Earth to about 12,500,000 kilometres.
After 380 to 390 years or so, the kidney-bean-shaped orbit approaches Earth again from the other side, and the Earth, once more, alters the orbit of Cruithne so that its period of revolution around the Sun is again slightly less than a year (this last happened with a series of close approaches centred on 1902, and will next happen with a series centered on 2676). The pattern then repeats itself.
More resonant near-Earth objects (NEOs) have since been discovered. These include 54509 YORP, (85770) 1998 UP1, 2002 AA29, and 2009BD which exist in resonant orbits similar to Cruithne's.
Other examples of natural bodies known to be in horseshoe orbits include Janus and Epimetheus, natural satellites of Saturn. The orbits these two moons follow around Saturn are much simpler than the one Cruithne follows, but operate along the same general principles.
Mars has four known co-orbital asteroids (5261 Eureka, 1999 UJ7, 1998 VF31, and 2007 NS2, all at the Lagrangian points), and Jupiter has many (more than 1000 known objects, the Trojan asteroids); there are also other small co-orbital moons in the Saturnian system: Telesto and Calypso with Tethys, and Helene and Polydeuces with Dione. However, none of these follow horseshoe orbits.