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Is there name for the line that goes from celestial pole to pole at RA 0 degrees 0 minutes 0 seconds? On Earth we would call it the Prime Meridian. Is it called the "Celestial Meridian"?

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That is correct, the line going through 0RA (and 12, actually) is the Celestial Meridian.

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This is incorrect! The celestial meridian is the arc that connects the north celestial pole, the zenith, and the south celestial pole. This has nothing to do with passing through 0h RA. –  user11266 Dec 30 '12 at 5:32
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The zero right ascension point (the celestial equivalent of the "zero longitude" meridian) is the vernal equinox point - the place where the Sun is during the spring equinox.

http://csep10.phys.utk.edu/astr161/lect/time/coordinates.html

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The question asks about the line that runs through the point this answer discusses. –  Andrew Oct 5 '11 at 22:14
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The word you are looking for is "colure." There are two colures on the celestial sphere. One passes through the two celestial poles and the two equinoxes and is called the equinoctial colure. The other one passes through the the two celestial poles and the two solstices and is called the solstitial colure. Note that the colures are also great circles and go all the way around the celestial sphere.

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