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Equatorial coordinates are the main coordinate system used in astronomy. Equatorial coordinates are spherical coordinates which use as their reference origin the projection of the Earth's equator onto the celestial sphere, and a meridian passing through the First Point of Aries. The north-south position -- the declination -- is measured in degrees of arc, and ranges from +90° (north celestial pole) to -90° (south celestial pole). The east-west position -- the right ascension -- is measured in time units of hours, minutes, and seconds, and the values increase eastward.

#### The coordinate system

Equatorial coordinates are fixed on the celestial sphere, following the daily rotation of the Earth and its yearly revolution about the Sun. Because of this, the main requirement for locating an object upon the sky is simply the local mean sidereal time, and the direction of the north celestial pole (approximately coincident with Polaris). I said main requirement because one also must know the epoch of the coordinate system. Because of the Earth's precession, the right ascension and declination change by several arcseconds per year. Common epochs you'll find in modern catalogs are B1950 and J2000, which give the equatorial coordinate positions measured in the years 1950 and 2000. (The "B" and "J" stand for Besselian and Julian.) One should also realize that the "celestial sphere" is not truly a fixed sphere of stars, and that stars themselves have proper motion through space. Some stars have very large proper motions -- nearby stars can move several arcseconds per year -- so one may also need to account for this.

#### Getting there

To determine the apparent position of an object on the sky given its equatorial coordinates, first determine the local mean sidereal time, which is the right ascension of the meridian -- the north-south arc in the sky that runs directly overhead. From this, you can obtain the hour angle, which is the number of hours east or west of the meridian the object is. The hour angle, HA, is defined by

HA = LMST - α

where LMST is the local mean sidereal time. As an example, suppose you have a sidereal clock which keeps track of your local mean sidereal time, and it says the time is 02:00:00. The object you're interested in has a right ascension of 04:30:00. Therefore, the hour angle is -02:30:00, or two and a half hours east of the meridian. (This means the star will cross your local meridian in about two and a half hours.)

The declination is simply the distance in degrees due north or south of the celestial equator. Again, assume the object has a declination of +16° 00' 00". If you draw an hour circle -- a circle at the hour angle passing through the north and south celestial poles -- you will find the object 16 degrees north of the intersection of this hour circle and the celestial equator.

The practical (if tedious) method for locating the apparent point on the sky of a set of equatorial coordinates is to use set of trigonometric equations which convert equatorial coordinates to altazimuth coordinates. You need to know the hour angle, your latitude, and the equatorial coordinates of the object. These equations are:

Altitude: sin(altitude) = sin(δ) sin(latitude) + cos(δ) cos(latitude) cos(HA)

Azimuth: ( sin(δ) - sin(altitude) sin(latitude) ) / ( cos(altitude) cos(latitude) )

In practice, the chances are that you will have a star chart that will easily help you find your way to the object you're interested in based on some easily spotted reference stars, or else you're lucky/well-to-do enough to have a telescope with an equatorial mount, which makes computing the altaz coordinates unnecessary. However, the equations given above are used frequently in research astronomy for (at least) two reasons. First, they're a convenient means of figuring out when your object of research interest will be visible in the sky. Second, most modern, large telescopes do not use equatorial mounts, but instead use altazimuth mounts to save on cost and size. The telescope then tracks positions on the sky based on these equations, rather than simply fixing the declination and allowing the telescope to follow the right ascension.

#### Origins of the system

The time-frame of the invention of equatorial coordinates is apparently in some dispute. Some suggest that equatorial coordinates were known and used as early as the time of Hipparchus. In his Commentary on Aratus (In Arati et Eudoxi phaenomena commentariorium), Hipparchus apparently used what appear to be the right ascension and declination of several bright stars whose positions are correct if the modern coordinates are precessed back to 150 BC. Regardless of whether this is the case, equatorial coordinates clearly go back before the origins of the telescope around the turn of the 17th century. The system was known and used by Arabic astronomers and was certainly well-known and used by the time of Jamshid al-Kashi and Ulugh Beg. Equatorial mounts and alignments were used even for pre-telescopic astronomical instruments, and equatorially-mounted telescopes (such as the English or Yoke mount) were likely invented before the mid 18th century.

#### Sources

Kraus, J.D., Radio Astronomy, Cygnus-Quasar Books, 1986
Burnham, R., Burnham's Celestial Handbook, vol. 1, Dover Books, 1978
Duke, D.W., "Hipparchus' coordinate system," Archive for History of Exact Sci ences, 56, 423 (available from http://www.csit.fsu.edu/~dduke/coordinates4.p df)
http://home.att.net/~srschmitt/celestial2horizon.html (for equations)
There's also a convenient visibility calculator called Skycalc, written by Dartmouth astronomer John Thorstensen, available from http://imagiware.com/astro/skycalc_notes.html

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