Coordinate Systems used in Astronomy
With so many stars in the sky, it's important to have a system to locate any given star by position, so we can find it again, or tell other people where the star is located. Several coordinate systems are used to describe the positions of objects in the sky. Most of them are based on the celestial sphere (an imaginary huge sphere of unspecified radius, concentric and coaxial with the Earth), but non-Earth-centric (galactocentric) coordinate systems are also used for describing more distant objects.
The Altitude-Azimuth System
This is the simplest system, based on local coordinates. It uses two numbers: the altitude and the azimuth, to describe the star's position in the sky. This system is purely local, meaning that the Altitude and Azimuth for a given star depend on your specific location and time. Since the earth rotates and stars appear to move across the sky, the position of any given star will appear to change over time. Also, depending upon where you are on the earth's surface, the star will have a different location in the sky. Therefore, Altitude and Azimuth numbers are only useful when combined with location (latitude + longitude) and time of observation. However, since most people stay put in one location while observing (as do fixed observatories), this is still a useful system.
Since the movement of stars is predictable, charts or astronomical software can be used to generate ephemerides, or tables which describe the positions of objects in a given location at a given time. Many observers use such charts or software to generate an ephemeris for the objects they are interested in observing, and then use the ephemeris (which often describes the positions in the Altitude-Azimuth system) to help point their telescopes in the proper direction.
The system works as follows. At any observation site on the surface of the earth, there is a horizon, where the earth appears to meet the sky. The true horizon may be obscured by trees or buildings, but it can easily be determined by drawing a line parallel to the ground. For a telescope, a bubble-level can be used. When the bubble-level is perfectly horizontal, it is parallel to the true horizon. The altitude, then, is the angle between the horizon and the star. The altitude can be anywhere from 0 (star at horizon) to 90 degrees (star directly overhead). The true altitude is the same angle measured from the center of the Earth instead of from the local horizon (the center of the Earth representing the true horizon). See the figure below.
The Azimuth is the angle of the star from true north. True north is based on the earth's geographical poles, passing through the earth's axis of rotation. This is not the same as the magnetic north pole, which is a few hundred kilometers away from the true north pole. Unless one is located at extremely high latitudes, true north and magnetic north are close enough for most purposes.
Azimuth is also measured as an angle, with 0 degrees being true north, 90 degrees being East, 180 degrees being South, and 270 degrees being West. In practice, many observers (at least in low latitudes) use a compass to locate north. Alternatively, if the pole star (Polaris) is visible, it can be used (along with an ephemeris) to locate north. Since Polaris is almost (but not quite) true north, an ephemeris can be used to correct for its deviation from true north. In principle, any bright and recognizable star can be used with an ephemeris to set both altitude and azimuth settings for a given location.
The Equatorial Coordinate System
The problem with the altitude-azimuth system is that the coordinates of a star are based on the location of the observer and the local time, and are therefore constantly changing as the Earth rotates on its axis and revolves around the Sun. The Equatorial Coordinate System avoids these problems, and is popular among telescope users.
The Equatorial Coordinate System is basically the same as the earth's own lines of latitude and longitude projected to the sky, with a couple of differences. The dome of the sky can be thought of as a sphere. Projecting the earth's equator to the celestial dome gives us a "horizon" plane, which is tilted at the same angle (23.5 degrees) to the ecliptic as the earth's axis. This plane is called the Celestial Plane.
Altitude (the height of a star above the horizon) can be obtained by projecting the earth's latitude lines to the sky, in the same way as the equator. The equator is simply latitude zero. The earth's axis would "intersect" the sky dome at 90 degrees N (if you're in the northern hemisphere), which corresponds to zenith. Nadir is the earth's axis similarly projected southwards. In the Equatorial Coordinate System, the altitude is known as Declination (abbreviated Dec), and its value ranges from +90 (zenith), through 0 (horizon, celestial plane), to -90 (nadir).
Setting the longitude is slightly more complicated. We could project earth's longitude lines to the sky as we did for latitude, but there is a problem in that the earth constantly spins on its axis. So the earth's prime meridian sweeps a path across the sky and is not stationary. Therefore, instead of projecting the earth's prime meridian to the sky, we pick a different point as prime.
The earth's equator, being inclined to the ecliptic by 23.5 degrees, intersects the plane of the ecliptic at two points in space. These are the points in the earth's orbit when the earth is at equinox. One of these, the spring or vernal equinox, has been chosen as the location of the prime meridian. See the diagram below.
The measure in this case is not degrees, like lines of longitude around the earth. Instead, it's hours, minutes and seconds. Earth's full orbit around the Sun (360 degrees) is divided into 24 hours. Each hour has 60 minutes, and each minutes has 60 seconds. The measure is named Right Ascension (RA for short), and the direction of measurement is the same as the direction of movement of the earth in orbit. That is, vernal equinox (March 21) is 0 Hours RA. Then as the earth moves to the summer solstice (June 21), the position then is 90 degrees or 6 Hours RA. When it's opposite the starting point at autumn equinox (Sept 23), it's at 180 degrees, or 12 Hours RA, and so on. This system is obviously based on the perspective from the northern hemisphere, but it has stuck.
With these two coordinates (Dec and RA), the position of any star can be precisely described independent of the rotation of the earth. Each hour of RA is 15 degrees, each minute is a quarter of a degree, and each second is 1/240 of a degree. Conversely, each degree is 4 minutes RA, and each arc-minute is 4 seconds RA.
The obvious problem with the Equatorial Coordinate System is that the earth's axis does not maintain a constant 23.5 degree angle to the ecliptic; in fact it precesses over a 25765 year cycle. Therefore the inclination of the celestial plane with respect to the ecliptic plane constantly changes, which changes the position of the equinoxes (precession of equinoxes). Therefore, while the Equatorial Coordinate System works reasonably well for short durations, there will be errors when considering a long time span. For this reason, a standard year is picked, and the location of the vernal equinox in that year becomes the standard for the coordinate system.
The J2000 coordinates are based on the position of the vernal equinox in the year 2000 AD. The location of Earth on March 21, 2000 sets the position of the prime meridian. These are referred to as J2000 coordinates. Any other year may also be picked as the standard. The choice of the year 2000 AD is arbitrary.
In practice, the Equatorial Coordinate System is used with some reference year in mind, so equatorial coordinates are often actually described as J2000 coordinates.
The Ecliptic Coordinate System
This is simply a variation of the Equatorial Coordinate System. The plane of the horizon used in this system is the Ecliptic Plane instead of the Celestial or Equatorial Plane. Altitude is measured as before from the horizon, but the horizon is now the Ecliptic Plane. Altitude is still called Declination, and it has similar values from -90 to 90, with 0 being the ecliptic plane. RA is still measured from vernal equinox, also in the same eastward direction.
The Ecliptic Coordinate System is often used for bodies in the solar system, such as the paths of comets and planets and asteroids. The Ecliptic Coordinate System can actually have two frames of reference, one centered on the Sun (Heliocentric System), and the other centered on the Earth (Geocentric System). It's a trivial matter to translate between these two.
The Galactic Coordinate System
This is also simply a modification of the Equatorial Coordinate System. Here, the plane of the horizon is not the celestial or ecliptic planes, but the galactic plane, which is the plane of the Milky Way. The Milky Way is a spiral which is flattened into a disc shape. The Galactic Plane is the transverse plane of the disc.
Declination is measured as height above galactic equator. RA is measured differently though - the zero line or prime meridian is the position of the Sun with respect to galactic center. Since the Sun revolves westwards around the center of the Milky Way, the north and south galactic poles are switched compared to Earth.
This system is most often used for extragalactic objects.
Since nothing is stationary in the universe, all these systems must also be referenced by a date. The Ecliptic Coordinate System is also usually zero'd at year 2000, that is, the position of vernal equinox in that year. The Galactic Coordinate System can similarly be zero'd to the year 2000, with prime meridian corresponding the position of the Sun relative to galactic center in that year.