A conundrum in writing a discussion like this is that it is hard to know where to start. In explaining one thing, you want to talk about something you have yet to define or to use terms you have yet to describe. I have done my best to avoid this, though I may have erred in places. Sometimes I use a term before defining it, and quickly follow up with some clarification.
Almost every topic being discussed has fine points and nuances that are not being discussed in depth. Some limitation of this sort is essential to allow the reader to get an overall perspective of the topics, which is the purpose of this discourse.
The line of zero longitude passes through both poles and through the Royal Observatory at Greenwich, England. In general, longitude is given as positive or negative values with magnitudes less than 180 degrees. Positive values are east of Greenwich, and negative values are west of Greenwich. Longitude in North America is measured from east to west in units of degrees. Since North Americans often think as if the rest of the world does not exist, they often will speak of a "longitude of 119 degrees, while they really should say either 119 degrees west, or negative 119 degrees longitude.
Latitude is measured from zero at the equator to 90 degrees at either pole. Latitudes north of the equator are positive or "north" latitudes; latitudes south of the equator are negative or "south" latitudes.
The pair of numbers, longitude and latitude, specify the location of a site on the surface of the earth. The coordinates of Tucson, Arizona are (longitude, latitude) -110.93 and +32.12.
Radians are also used to measure angles, but always in hidden places within software. There are 2 pi radians in a circle. Nobody who doesn't deserve to be severely punished ever lets radians escape from inside a computer program.
RA and dec serve to specify the position of objects on the celestial sphere. The celestial poles are projections of the earths axes of rotation. The celestial equator is the projection of the plane perpendicular to that rotation axis. The zero point for RA is known as "the first point in Aries", which is the place in the Sky where the sun crosses the celestial equator during the Spring Equinox. If you are new to all of this, it is perhaps best to accept this as some arbitrarily chosen point in the sky and to ignore the details for a while.
It is worth noting briefly a couple of fine points with regard to stellar coordinates. The first is that the stars themselves are in motion. For many purposes (and many stars) this can be ignored, but this motion is significant for stars fairly close to the earth. The change of the coordinates of a star with respect to time is called "proper motion" and has been determined and recorded for stars for which it is important.
The second point worth noting is that the reference points for the entire system of stellar coordinates is in motion (relative to the earth). This is because the earths rotation axis is moving relative to a system of coordinates fixed among the stars. This motion is called precession. The rotation axis of the earth sweeps out a cone with angular radius of nearly 23.5 degrees every 26,000 years. Precise astronomical calculations must account for precession, and stellar coordinates are given for a specific moment in time (most commonly the year 2000, specified as J2000), and may be adjusted, if desired, for the current time.
Polaris has an RA of 2:31:49.09456 hours and a declination of +89:15:50.7923 degrees in the J2000 Epoch. Notice the sexigesimal notation where RA is given in hours, minutes, and seconds and declination is given in degrees, minutes, and seconds. Also notice that RA minutes and seconds are units of time, different from the angular minutes and seconds given for declination.
It is convenient to think of sidereal time as tracking the motion of the stars overhead. The local sidereal time gives the hour angle (see below) of the zero point for RA (the infamous "first point in Aries"). Of course what sidereal time really does is to track the rotation of the earth with respect to a reference frame fixed with respect to the stars (insofar as we can talk about such a thing). I say this because I expect some dunce to send me an email and point out that it is the earth that is rotating, and not the stars. It is really a question of choosing our frame of reference, and it is usually most convenient to fix our coordinate frame at the current location on earth and talk about the stars rotating, so I go ahead and do just that.
A sidereal day is shorter than a solar day (a solar day being what civil time deals with). This is rooted in the fact that the earth orbits the sun. Each night at the same solar time, we find ourselves looking at a slightly different view of the sky (being that much further along on our trip around the sun). If we were to go outside and carefully view the sky the same time on two consecutive nights, we would find that the stars had moved by about 4 minutes. After a year had gone by, we see the same view we had begun with. 4 minutes is 24 hours divided by 365. A sidereal day is 23.93447 solar hours.
HA = LST - RANotice that hour angle and RA have opposite sign. HA increases for objects further west, whereas RA increases for objects further east.
Given the RA and Dec coordinates of an object, the observers location in longitude and latitude,
and the local sidereal time, it is possible to calculate the altitude and azimuth of an object.
This link gives details on the math required to do this.
Some telescope are built with altazimuth mounts. They have two axes of motion (azimuth and altitude) and these axes can be moved in such a way as to follow objects moving in the sky.
Care must be taken not to confuse this with the position angle.
To describe the parallactic angle we need to discuss two lines (actually great circles) in the sky. One is the line of constant RA which passes through the celestial pole and the object. The other is the line that passes through the zenith and the object. Notice that the second line is dynamic - as the object moves from east to west across the sky, this line constantly moves.
The parallactic angle is the angle between the constant RA line (circle) containing the object and the line (circle) containing both the object and the zenith. Notice that there is a mathematical singularity if the object passes through the zenith. This is resolved by defining the parallactic angle as zero when the object crosses the meridian. The parallactic angle of an object is constantly changing as the object moves from east to west.
The parallactic angle is important for imaging objects with a telescope with an altazimuth mount. If the telescope rotates the image it produces in synchronization with the parallactic angle, the image will be stationary on a detector.
Care must be taken not to confuse this with the parallactic angle. Position angle was first defined to specify the orientation of visual binary stars. Again, two lines (circles) on the sky must be defined. The first is the line of constant RA from the primary star to the north celestial pole. The second is the line from the primary star to the secondary star. It is positive from north through the east. It is the angle between the line from the primary star If we consider an extended object such as a planet or galaxy (as described above), we can call the
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