Coordinate Systems in Astronomy

We need a systematic and universal method for finding any point on the celestial sphere. With such a method, observers all over the world can find objects easily. The system we will use is the Equatorial Coordinate System.

Navigators use a similar coordinate system for finding places (points) on the Earth's surface. The two terrestrial coordinates are latitude and longitude. Both of these are angles with some arbitrary point on the Earth's surface. Latitude measures the angle between a location and the Earth's equator while longitude measures the angle between a location and the Prime Meridian.

The Equatorial Coordinate System is very similar. It uses two reference points - the celestial equator and the celestial poles. These are shown in Figure 1. The axis of the Earth's spin runs directly through the celestial poles while the celestial equator is above the Earth's own equator. The celestial poles are exactly 90° away from the celestial equator.

There are two coordinates in the system - right ascension and declination. The first - declination or DEC - is the angular distance from the celestial equator. It is measured along a great circle through the celestial poles (see figure 1). Declination is postive in the northern half of the sphere, negative in the southern half. A point on the equator has a declination of zero.

The second equatorial coordinate - right ascension or RA - is the angle measured eastward¹along the equator from the vernal equinox to the great circle through the point in question and both celestial poles (this great circle is called the hour circle of the point) (see figure 2). The vernal equinox and the autumnal equinox are the points where the ecliptic - the path of the Sun and the planets, crosses the celestial equator. The ecliptic is tilted 23.5° with respect to the equator. Right ascension may be measured in hours from 0-24 or in degrees from 0-360.

Figure 1: Right ascension $\alpha$ and declination $\delta$ as seen from the outside of the celestial sphere, in the right-hand drawing. The left-hand drawing locates the vernal equinox (V.E.) and the autumnal equinox (A.E.).

These coordinates are absolute. That is - they will not change over the course of the year. Planets move rapidly through these coordinate systems and stars move very slowly through them, mainly due to precession. But for the kind of observing you will do, the coordinates will not change. Star Atlases and other references usually contain the RA and DEC of any objects you wish to observe.

The problem is that the sky will change over the course of the night and the semester. As the Earth spins, objects rise and set. They are constantly changing location relative to the observer. For this reason, you need to be able to use a relative coordinate system.

The reference points for the relative coordinate system are the meridian and the horizon. The point directly overhead is known as the zenith (see figure 2). The great circle through the celestial poles and the zenith is the observer's meridian. When a star crosses the observer's meridian or transits, its distance from the horizon is greatest²and it is well placed for observation is the sky is dark. The horizon is just what it sounds like - the great circle created by the Earth itself.

Figure 2: Hour angle H. The arc length on the equator and the angle formed at the pole have equal angular measure

The two coordinates of the relative system are altidude and the hour angle. The hour angle or HA is the angle measured along the equator between the hour circle of the star and the observer's meridian (see figure 2). Hour angle is positive to the west, i.e. after transit. It is often measured in hours (1 hour = 15 degrees). A star with an hour angle of 23 hours would be 1 hour before transit, and might be referred to as having an hour angle of -1 hour. Sidereal time is the hour angle of the vernal equinox. In simple terms, the Sidereal time represents the right ascension of the object passing directly overhead. Thus, the hour angle represents how long it will be before an object is overhead. On object at an HA of -1 will be overhead in one hour while one with an HA of 2 was overhead two hours ago.

The hour angle of a known object can be calculated quite easily. Hour angle is simply the sideral time minus the right ascension:

HA = S - $$\alpha$$

Sidereal time runs faster than the mean solar time that our watches use by 3 minutes and 56 seconds per day, a result of the Earth's orbital motion around the sun. This means that the sidereal time at a given standard time changes continuously by 2 hours per month. The following table gives the sidereal time for 9PM EST for selected dates throughout the year.

DATE	ST @ 9PM EST
Jan. 21	5
Feb. 21	7
Mar. 21	9
Apr. 21	11
Sep. 21	21
Oct. 21	23
Nov. 21	1

The second relative coordinate - altitude or ALT - measures the angle between the star and the nearest point on the horizon. Because of lights, haze, seeing, or local obstacles, it is often difficult to observe objects at altitudes less than 30°. The altitude may be calculated from an object's hour angle, its declination, and the latitude from which it is observed; but this involves trigonometry and will not be covered here.

It is easy to calculate an object's altitude when it crosses the meridian (i.e. has H = 0), however. Consult figure 3. There we see that the altitude above the southern horizon for an object on the meridian is SALT = $\delta$ + (90° - LAT) where LAT is the latitude of the observing site. The altitude above the northern horizon is NALT = (LAT + 90°) - $\delta$. Objects with $\delta$ = LAT evidently cross through the zenith ( SALT = NALT = 90°); the celestial equator (where $\delta$ = 0°) has SALT = 90° - LAT on the meridian.

Solar system objects, which lie near the ecliptic, have altitudes on the meridian which vary with the time of year. The maximum altitude of the sun occurs on June 21 when its $\delta$ = 23°.5 and SALT = 113°.5 - LAT. Since for Charlottesville, LAT = 38°, the maximum SALT for the sun is 75°.5. Its minimum SALT on the meridian (Dec. 21, $\delta$ = - 23°.5) is 28°.5. The Moon's orbit is inclined 5° to the ecliptic, so its maximum and minimum SALT's are 80°.5 and 23°.5. The full moon is directly opposite the sun in the sky, so the SALT of the full moon is large when the SALT of the sun is small (December) and vice versa. Of the bright planets, only Mercury has an orbital inclination greater than 5° from the ecliptic.

Figure 3: A view of a slice through the observer's meridian showing the relationship of a star's declination $\delta$ and the observer's latitude lat, to the altitude of the star above the Southern horizon SALT when it is on the meridian.