----- The following copyright 1991 by Dirk Terrell
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 Lecture #17    "The Equatorial Coordinate System-"

   Continuing with our discussion of astronomical coordinate systems, let's 
now look at the one that is perhaps the most commonly used -- the 
Equatorial (or Q) system. This system has certain advantages over the 
Horizon system that we defined last time. The primary advantage is that an 
object's position in this system changes only very slowly with time, whereas 
its position in the H system changes very rapidly due to the rotation of the 
Earth. Another advantage is that the Q system is geocentric which means that 
the origin of the coordinate system is at the Earth's center. The H system, 
on the other hand, is a topocentric system -- one whose origin lies on the 
surface of the Earth, making an object's coordinates dependent on the 
location of the observer. Of course, here I am saying "advantage" in the 
context of making a map of the sky or a star catalog. The Q system is not as 
useful, if you want to walk outside and know where to look for an object. In 
that case the H system is more advantageous.

   First we need to define a rather commonly used term -- the ecliptic. Most 
of the time we "define" the ecliptic as the plane of the Earth's orbit, or 
the path of the Sun against the background stars. Most of the time that is 
sufficient for our purposes. However, let me give, for completeness, a more 
correct definition of the ecliptic. Define a plane by the center of the Sun, 
the barycenter of the Earth-moon system, and the heliocentric velocity 
vector of the Earth-moon system. Over time, this plane's orientation will 
change with respect to an inertial frame, albeit very slowly. The changes 
can be mathematically broken down into periodic and secular parts. If you 
remove the periodic parts (which, incidently, arise mainly from the 
gravitational tugs of Venus and Jupiter), the plane that remains is the 
ecliptic.

   Because the Earth's rotation axis is tilted by some 23.5 degrees with 
respect to a line vertical to the plane of the ecliptic, the plane defined 
by the equator is not the same as the ecliptic. Therefore, these two planes 
intersect in a line and this line defines the direction of the equinoxes. 
The Sun is near one of these is late March, and we call this one the vernal 
equinox. Since the two planes (equator and ecliptic) change their 
orientations, this point also changes, going through a complete rotation in 
about 26,000 years.

   Now that we have those terms defined, we can define the Q system. It is a 
right-handed system, with the x-direction in the direction of the vernal 
equinox, and the z-direction in the direction of the north celestial pole. 
The longitude angle in this system is called the right ascension and is 
measured in hours, minutes, and seconds rather than the usual degrees, 
minutes, and seconds. RA is usually represented by the Greek letter alpha. 
The latitude angle in this system is called the declination and is measured 
in degrees, minutes, and seconds. Dec is usually represented by the Greek 
letter delta, lowercase, that is.

   Have you ever wondered why we're always talking about the epoch of 
coordinates (i.e., 1950 vs. 2000)? The reason is because of the changes in 
the ecliptic and equator over time. Today's equinox is different from 
yesterday's. Therefore, when we give an object's coordinates (RA and dec) we 
have to specify which equinox our positions are measured from, so that 
everyone can measure them the same way. If there is interest, we will 
discuss how to convert positions from one epoch to another in a subsequent 
lecture.

   The program that I posted earlier converts an object's RA and dec into 
altitude and azimuth (a Q to H transformation). These transformations are 
accomplished by rotating a coordinate system around it's axes so that they 
correspond to the desired system. Try to imagine the axes for the Q system 
and those for the H system (Tinkertoys are great for this). What are the 
rotations you need to perform? That is, what are the angles that you need to 
rotate through and which axes do you rotate about? (Hint -- the program has 
subroutines R1, R2, and R3. R1 is a rotation about the x-axis, R2 about y, 
and R3 about z.).

 Dirk