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A term used to describe the direction of a secondary body's orbital motion around its primary. Specifically, a retrograde orbit describes a situation in which the orbiting object follows a path opposite to the rotation of its primary. The opposite term - and the much more usual situation - is a prograde orbit, in which a body orbits in the same direction as its primary rotates.
When a star forms a planetary system, that process begins with matter around a protostar being drawn (by conservation of angular momentum) into a protoplanetary disc, and that disc follows* the same direction of rotation as the young star. As the matter in the disc then condenses to form planets, those planets will in turn rotate around the star in the same prograde direction. The equivalent pattern is also seen as planets form systems of moons, and so most major moons follow prograde orbits around their primary planets, just as planets do around their star.
As might be expected, then, all of the major planets in the Solar System, as well as the vast majority of the asteroids in the Asteroid Belt, follow prograde orbits - that is, they rotate around the Sun in the same direction that the Sun turns on its axis. There are, however, a small number of exceptions within the system, bodies that follow retrograde orbits, with the most common examples being the irregular moons of the gas giants.
The major moons of Jupiter formed alongside that planet and follow prograde orbits, but many of the giant planet's smaller outer moons do not. These are moons that did not form at the same time as Jupiter, but were originally errant asteroidal bodies that strayed into Jupiter's gravity well and were captured. Commonly this effect had violent repercussions for the newly-captured moon, causing it to break apart to form a family of smaller moons, all following similar orbits. In cases like these, there was no constraint on the direction of the orbit of the new moons, and in many cases they settled into retrograde orbits, travelling around the planet in the opposite direction to its rotation.
The Solar System is a relatively well-ordered stellar system in these terms. Observations of planets orbiting other stars show that retrograde planetary orbits can be rather more common. This is especially true for hot Jupiters - gas giants following very close orbits around their stars - which implies that many of these planets had their orbits disrupted early in their existence (a factor that can also help to explain their unusual proximity to their parent stars).
On a wider scale, stars orbiting within the disc of the Galaxy tend to follow a standard prograde orbital direction around its core. There are notable exceptions, such as the nearby Kapteyn's Star in Pictor, which traverses the Milky Way in a retrograde direction. In these cases some violent event in the star's past tends to be responsible for its unusual behaviour (for Kapteyn's Star, it is hypothesised to have been captured by the Milky Way during an ancient merger with a dwarf galaxy). While objects within the galactic disc are typically uniform in their motion around the Galaxy, this is much less true of the Galaxy's halo, and many of the Milky Way's attendant globular clusters pursue retrograde orbits around their host galaxy.
The term retrograde (which simply means 'moving backward') is used with slightly different meanings in various astronomical contexts. Retrograde motion is distinct from a retrograde orbit, and describes a body appearing to move backward on its own orbital track (an illusory motion that can be seen in various objects within the Solar System, because Earth itself is also moving relative to the other planets). Retrograde rotation is not directly related to an object's orbital path, but instead describes the rotation of an object on its axis (within the Solar System, all the major planets have prograde orbits, but two - Venus and Uranus - rotate on their axes in a retrograde direction).
* Strictly, a protoplanetary disc will usually follow the same rotation as its primary, but there are a small number of exceptions. In these highly unusual cases distortions or perturbations caused by the host protostar, or by a binary companion, cause parts of the disc to change direction and take up a retrograde orbital course. (An example of such a system is the binary protostar IRAS 16293-2422 in Ophiuchus.)
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