Comets
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There are over 5000 known comets in our Solar System. However, there are probably very very many cometary bodies hanging out around the Oort cloud — since comets are loosely defined by their volatile composition and highly elliptical orbits, any given comet spends a majority of its life near their aphelia in the icy outer reaches of the Solar System.

Comets are generally given designations based on the names of their discoverers or orbit calculators. I am not interested in giving a distinctive symbol for every known comet, because their names aren't very distinctive. However, there are several comets which are notable enough that I believe they garner a symbol. I list them here, in addition to comets which are also categorized as minor planets.

C/-43 K1

Also called Caesar's Comet or The Great Comet of 43 BC.

C/1995 O1

Also called Comet Hale-Bopp or The Great Comet of 1997.

D/1993 F2

Also called Comet Shoemaker-Levy 9.

C/2006 P1

Also called Comet McNaught or The Great Comet of 2007.

1P/Halley

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This comet, with a period of about 75 years, has been observed many times throughout human history, according to historical and archaeological record. In 1705, 2688 Edmond Halley (no, we don't know if his name is supposed to rhyme with "daily" or "valley", though I tend towards the former), using the laws of motion and gravity outlined by his colleague Isaac Newton, first discovered that the multiple appearances of this comet in the recent astronomical record were in fact sightings of the same object that was orbiting the Sun. Thus began the study of periodic comets.

The orbit of Halley's comet is unstable on the scale of hundreds of years. It's period and orbit have been changing ever since humans began observing it thousands of years ago, and it will continue to change, and we will continue to observe it.

The symbol for this comet is a capital Latin letter H in a cometary symbol.

3D/Biela

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Also called Biela's Comet.

9P/Tempel
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19P/Borrelly

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Also called Borrelly's Comet, this comet with a period of 6.8 years was discovered in 1904 by Alphonse Borrelly. In 2001, it was imaged by the NASA spacecraft Deep Space 1.

The symbol of this comet is a capital Latin letter B in a cometary symbol.

26P/Grigg–Skjellerupr

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This comet with a period of 5.31 years was discovered in 1902 by John Grigg and John Francis Skjellerup. It is the source of the small Pi Puppis meteor shower. Its orbit is often the victim of Jupiter's gravitational influence; it has a rather uncertain future. The ESA Giotto mission did a close flyby of this comet in 1992.

The symbol of this comet is a cometary symbol, with the circle replaced by a small version of the symbol for Puppis.

67P/Churyumov–Gerasimenko
81P/Wild

95P/Chiron

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This is the simultaneous cometary designation for the minor planet 2060 Chiron.

103P/Hartley

107P/Wilson-Harrington

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This is the simultaneous cometary designation for the minor planet 4015 Wilson-Harrington.

174P/Echeclus

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This is the simultaneous cometary designation for the minor planet 60558 Echeclus.

176P/LINEAR

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This is the simultaneous cometary designation for the minor planet 118401 LINEAR.

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