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Traditionally, an orrery is a mechanical model of the solar system, or at least the major planets, a device driven by clockwork to show the motion of the planets (and, in some cases, major moons) around the Sun. Orreries are usually not to scale, partly because of the difficulty of mechanically modeling both Mercury and Neptune (and Pluto!) with a device that operates smoothly and in which the planets are clearly visible … the solar system is a very big place.
The Antikythera mechanism, which is dated to ~150-100 BC, may be considered the first orrery (that still exists, and which has been found). The name, orrery, comes from Charles Boyle, the 4th Earl of Orrery, who in 1713 commissioned instrument maker John Rowley to make a copy of a device he’d built, based on a design by George Graham.
Many modern planetaria (or planetariums) have a projection orrery, as do some older ones. These project onto the dome a point of light (representing the Sun), with planets going round it (usually only Mercury to Saturn).
Today, with so much computing power available for so little cost, ‘orrery’ often means ‘digital orrery’, a software program for calculating the relative positions, and motions, of solar system bodies (and displaying them). There are many available, such as this java applet from the Department of Physics, in the University of Texas at Austin, and Orrery, a Solar System Visualizer from the Geometry Center, part of the University of Minnesota (runs under Unix).
There is, however, a Digital Orrery (capital D, capital O); it is the name of a special purpose computer, designed specifically to model the long term motions of the outer planets (which then, 1985, included Pluto). It was built to study a long-standing question about the solar system – is it stable, over billions of years? (The answer? No; Pluto’s orbit is chaotic). This device is now at the Smithsonian Institution in Washington DC.
Perhaps the most fun orrery is the Human Orrery, at Armagh Observatory, in Northern Ireland , which allows people to play the part of the planets Mercury, Venus, Earth, Mars, Jupiter, and Saturn, as well as Ceres and two comets … all to scale! That website also has a good, short, history on the orrery, as well as an explanation of how it works.