To locate an object on the sky, astronomers use several different coordinate systems which are similar in concept to the system for plotting a point in a plane. However, the celestial sphere, which is the apparent sphere of the sky, is not a plane: It is the inside of a sphere, and one that rotates relative to Earth's surface. The two most frequently used coordinate systems are the horizon system and the equatorial system, and they are analogous in principle to the grid lines of geographic longitude and latitude. On the celestial sphere, distances are measured in degrees along the arc of a great circle. A great circle, formed by passing a plane through the center of a sphere, divides the sphere into two equal halves. Several features are common to all astronomical coordinate systems. Each has a principal axis, or polar axis, about which the system rotates. The points of intersection of this axis and the celestial sphere are the poles of the system. Perpendicular to the principal axis is a great circle, which is the principal reference circle along which one coordinate is measured. Finally, there is an infinite number of secondary reference circles that are great circles perpendicular to the principal reference circle, and which meet at the poles of the principal axis, as shown in Figure A2.1. The most natural and the easiest coordinate system to visualize is the equatorial system.
The principal axis of this system is defined so it is parallel locally to the direction of gravity. Extended upward, the principle axis intersects the celestial sphere at a point known as the zenith (Figure A2.2). The principal reference circle, the astronomical horizon, is the great circle marked on the celestial sphere by a plane perpendicular to the zenith-nadir axis and tangent to the Earth at the point of the observer. There are four reference points on the astronomical horizon: north, east, south, and west, which are 90o from each other.
If we project onto the celestial sphere the terrestrial meridian of the longitude passing through the observer's position, it defines the great circle called the celestial meridian. The celestial meridian passes through the north and south points of the horizon as well as the zenith. It also contains the two points that are the intersection of the Earth's axis of rotation extended to the celestial sphere: the north and south celestial poles. The position of the celestial poles relative to the north and south points of the horizon depends upon the observer's latitude. For an observer in the northern hemisphere the angular distance of the north celestial pole above the north point of the horizon is equal to the observer's latitude.
Coordinates in the horizon system are known as azimuth and altitude. Azimuth is the angular distance measured along the horizon from the north point. The secondary reference circles in this system are known as vertical circles. Thus azimuth is measured to the foot of the vertical circle passing through the object of interest. The azimuth of the north point is 0o; the east point, 90o; the south point, 180o; and the west point, 270o. Altitude is the angular distance of the object above or below the horizon measured along a vertical circle. The altitude of the zenith is +90o, and the nadir, -90o. In Figure A2.2, the altitude of the star is about 60o, and azimuth is about 220o.
The major disadvantage of the horizon system is that it is peculiar to the observer and not at all a general system. Also, since the altitude and the azimuth of an object continually change as Earth rotates, one must know the exact location and time at which an altitude and azimuth are measured in order for them to have any meaning to anyone besides the observer. A more general system is the equatorial coordinate system.
The principal axis of the equatorial system coincides with Earth's axis of rotation. Its poles are the north and south celestial poles, as defined in the preceding section. The principal reference circle is the celestial equator, whereas secondary reference circles are called hour circles (Figure A2.3). In the horizon system, the horizon and vertical circles remain fixed relative to an observer with the celestial sphere moving relative to them, whereas in this system, the celestial meridian and hour circles are on the celestial sphere and thus rotate with it relative to an observer. Figure A2.4 shows the relationship between the equatorial and horizon systems at one particular time.
As Earth rotates from west to east, stars trace paths from east to west across the sky called diurnal (daily) circles. For an observer located (Figure A2.3) at intermediate latitudes, the sky rotates at an oblique angle, the value depending on the latitude. And the celestial equator passes through the east and west points of the horizon regardless of an observer's latitude.
Coordinates in the equatorial system are called right ascension and declination. Declination is the angular distance of an object north or south of the celestial equator measured along an hour circle. Thus the declination of the north celestial pole is +90o, and that of the south celestial pole is -90o. The other coordinate is called right ascension and is measured eastward along the celestial equator from the vernal equinox to the foot of the hour circle passing through the object. The vernal equinox, so named because the Sun moves through this point on or about March 21, is the point of intersection of the celestial equator with the annual apparent path of the Sun on the celestial sphere, the ecliptic. Right ascension is measured in units of time rather than degrees of arc. Since a 360o rotation by the Earth corresponds to 24h, then 1h equals 15o of arc, or 1o equals 4m, and so on. Finally, the angular distance of the hour circle from the observer's celestial meridian is called the hour angle of the star. As an example, the star's declination in Figure A2.4 is about +60o, its right ascension about 6h, or 90o, and its hour angle about 1.5h, or 22.5o of arc.
It is obvious that the right ascension and declination of a celestial object remain fixed as the sky rotates. This makes it possible to catalog astronomical objects by right ascensions and declinations in the same way in which places on Earth are cataloged by longitudes and latitudes. Because of precession (Section 1.3), the vernal equinox slides westward along the ecliptic by about 50" per year. Thus over a number of years, a star's right ascension and declination change so that cataloged positions are not accurate indefinitely and must be updated periodically to correct for precession.
In addition to the horizon and equatorial systems, there are two less frequently used systems of astronomical coordinates, the ecliptic and the Galactic systems. The principal reference circle for the ecliptic system is the ecliptic, whereas in the Galactic system it is the central plane of our Galaxy. The four systems are summarized in Table A2.1. Although astronomical coordinate systems are necessary in order to make precision observations, a general knowledge of the sky can be obtained with the aid of star charts such as those inside the front and back covers.
| Coordinate System | Principal Axis | Principal Reference Circle | Coordinates (units) | Secondary Reference Circles | Coordinate (units) |
|---|---|---|---|---|---|
| Horizon | zenith-nadir | astronomical horizon | azimuth (0o-360o) | vertical circle | altitude (+0o-90o) |
| Equatorial | north-south celestial pole | celestial equator | right ascension (0h-24h) | hour circle | declination (+0o-90o) |
| Ecliptic | north-south ecliptic pole | ecliptic | celestial longitude (0o-360o) | no name | celestial latitude (+0o-90o) |
| Galactic | north-south Galactic pole | plane of the Galaxy | galactic longitude (0o-360o) | no name | galactic latitude (+0o-90o) |
Relating coordinate systems to each other can be done through
time systems. Astronomical time systems are constructed on the
principle of tracking a reference point on the celestial sphere
relative to an observer's celestial meridian. Thus time is measured
basically by Earth's rotation. A convenient reference position
from which to mark the passage of time is an observer's celestial
meridian. Since both the vernal equinox ( ) and the Sun are carried
westward by the rotation of the sky, each can serve as an "hour
hand" on the celestial clock.
1B.3.1. Sidereal Time
One sidereal day is equal to the interval between two successive crossings of an observer's celestial meridian by the vernal equinox. This corresponds to one complete rotation of Earth on its axis. The local sidereal time (LST) is given by the local hour angle of the vernal equinox (LHA ), which is equal to the sum of any object's right ascension and hour angle (Figure A2.4). Local sidereal time progresses uniformly at the rate of 15o to the hour and can be read from a sidereal clock whose rate is that of Earth's rotation. The clock is set to read 0h0m0s at the instant the vernal equinox crosses the celestial meridian, 1h0m0s when the vernal equinox has moved 15o west of the meridian, corresponding to a local hour angle of 1h, and so on around the sky through 24h, when a new sidereal day begins.
Apparent solar time is given by the local hour angle of the Sun, (LHAsun) + 12h, to start the solar day at midnight. One apparent solar day is equal to the interval between two consecutive passages over an observer's meridian by the Sun. Time measured by the apparent or real Sun is slightly variable for two reasons: First, the Sun's annual motion along the ecliptic varies a little because of Earth's orbital eccentricity (recall Kepler's law of areas, Section 2.4). Second, the projection of the real Sun's motion along the ecliptic onto the celestial equator, where hour angle is measured, varies slightly over the course of a year. A more suitable clock time is one that does not change in an irregular manner from day to day.
This is accomplished by introducing an imaginary sun called the mean sun. It moves uniformly eastward along the celestial equator at a daily rate that is equal to the average daily rate of the real Sun along the ecliptic, so that both arrive at the vernal equinox simultaneously 1 year later, which is where they started out together. Mean solar time is equal to the local hour angle of the mean sun, (LHAsun) + 12h, again starting the mean solar day at midnight. Since mean solar time progresses uniformly, mechanical and electronic clocks can be made to keep this kind of time, which is what we use in everyday life. The greatest difference between apparent and mean solar time, which varies during the year, is +15m.
A tropical year, a year of seasons, is equal to 365.2422 mean solar days. It represents the time it takes the Sun to complete one revolution around the ecliptic with respect to the vernal equinox. The sidereal year, on the other hand, is the period of the Sun's revolution with respect to the stars, and it is equivalent to the period of Earth's orbital revolution. Because of Earth's annual motion around the Sun, a mean solar day is longer than a sidereal day by 3m56s of solar time (Figure A2.5).
Obviously from the way mean solar time is kept, it is different at places separated by differences in longitude. To get around this inconvenience, the standard time system was introduced in 1884. There are 24 time zones, each theoretically 15o wide, with the zero time zone centered on Greenwich, England, the 0o meridian of longitude, and proceeding east or west of Greenwich. Through the approximate center of each zone runs the standard meridian, whose local mean solar time is the standard time within the zone. In the United States and Canada, the standard meridians are at 60o, Atlantic standard time (AST), 75o, eastern standard time (EST), 90o, central standard time (CST), 105o, mountain standard time (MST), 120o, Pacific standard time (PST), 135o, Yukon standard time (YST), and 150o, Alaska-Hawaii standard time (AHST). These standard time zones are respectively 4h, 5h, 6h, 7h, 8h, 9h, and 10h behind Greenwich mean time (GMT), also known as universal time (UT). Time changes by 1h as the traveler crosses a zone boundary. The zones frequently have irregular boundaries to suit local conditions. Also there are countries that actually span several standard time zones, in which the entire country keeps the same zone time.
The successive time zones run either west or east of Greenwich until they meet halfway around the world at 180o longitude at the international date line, which passes through the center of the 12h zone. The half portion of the zone west of the line is 1d ahead of the other half east of the line even though the standard time is the same in each half. A traveler crossing the date line from Tokyo to San Francisco, for example, gains a day, whereas one traveling in the opposite direction loses a day.
The common calendar of ancient peoples was based on the lunar cycle of 29.5d, because changes in the Moon's phase are so readily apparent. Since 12 lunar months cannot be contained in a tropical year a whole number of times, an extra month was added from time to time to bring the seasons back on schedule. Attempts to synchronize the lunar month with the year never proved satisfactory. The Egyptians were the first to base their calendar on a tropical year of 365.25d. Their year consisted of 12 months of 30d each, with 5d extra set aside at the end of the year for celebrations. However, by the time the Julian calendar was adopted on January 1, 45 B.C., it had to be made 445d long in order to bring the seasons back on schedule. The Julian calendar consisted of 365d with every fourth year being 366d.
The true length of the tropical year is 365d5h48m46s, or about 11m14s shorter than 365.25d. Hence, the longer year used in the Julian calendar results in a discrepancy of 3d in 400 y. By 1582 A.D., the accumulated difference amounted to 10d, so the date of the vernal equinox had retreated from March 21, at the time of the Council of Nicaea convened in 325 A.D., to March 11. Accordingly, Pope Gregory XIII called upon the astronomer Clavius to revise the Julian calendar. His solution was to drop 10d from the calendar, so that the day following Thursday, October 4, 1582, thereby became Friday, October 15, 1582. And to avoid future calendar discrepancies, only century years divisible by 400 were leap years, such as 1600 and 2000. This new calendar became known as the Gregorian calendar, and it was readily adopted by Catholic countries. The Lutherans and Protestants finally made the adoption in 1700. When Great Britain and the American colonies changed to the Gregorian calendar in 1752, September 2 had to be followed by September 14, the discrepancy having grown to 12d. Early in this century other countries in Europe and Asia finally adopted the Gregorian calendar or one very close to it. The present Gregorian calendar is accurate to about 1d in 3300 y. However, its accuracy is improved beyond that by interspersing an extra second between days every few years to compensate for a number of irregularities in the celestial clock.
Plan of the various astronomical coordinate systems. Beginning with the principal axis, the principal reference circle is everywhere 90o from the poles of the system. An infinity of secondary reference circles is perpendicular to the principal reference circle and passed through the poles.
Horizon system of coordinates. For a star in the southwestern part of the sky, the coordinates locating its position on the sky at one instant of time are called azimuth and altitude. Azimuth is measured clockwise in degrees from the north point along the astronomical horizon to the foot of the vertical circle passing through the star. Altitude is measured in degrees from the horizon along the vertical circle to the star.
Relationship between equatorial and horizon coordinate systems. The relationship is shown for three different geographical latitudes: (a) the north geographic pole, (b) the geographic equator, and (c) 30oN geographic latitude.
Equatorial system of coordinates. The horizon system is shown for orientation. For a star in the southwestern part of the sky, the coordinates locating its position are called right ascension and declination. Right ascension is measure counterclockwise in units of sidereal time from the vernal equinox along the celestial equator to the foot of the hour circle passing through the star. Declination is measured in degrees from the celestial equator along the hour circle to the star. As the Earth rotates, the hour angle of the vernal equinox continually increases, repeating after a 360o rotation of the Earth or 24 sidereal hours. Thus the local hour angle of the star, which continuously changes over time, can be found from the sidereal time and the star's right ascension.
Why the solar day is longer than the sidereal day. With an observer at position 1, the Sun is on the observer's celestial meridian. After a 360o rotation relative to the stars, Earth has moved about 1o along its orbit and the observer will be at position 2, where the Sun is now 1o east of his or her celestial meridian. Consequently, Earth must rotate one additional degree, which is done in approximately 4m, in order to bring the Sun to the observer's celestial meridian in position 3.