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Coordinate Systems

The Geographical System of Coordinates

Latitude and longitude are the coordinates used to specify a location on the Earth’s surface. Latitude indicates the number of degrees that a place is north or south of the equator. For example, a city on the equator has a latitude of 0 degrees, the North Pole has a latitude of 90 degrees north, and a point midway between the North Pole and the equator has a latitude of 45 degrees north. The longitude of a place is the number of degrees that it is east or west of Greenwich, England. Longitude is always between 0 and 180 degrees west or 0 and 180 degrees east. To uniquely specify a location, both the latitude and longitude must be given. Greenwich has a latitude of 52 degrees north and a longitude of 0 degrees. Detroit, Michigan, has a latitude of 42 degrees north and a longitude of about 83 degrees west.

Longitudes are sometimes expressed in units of time rather than in angular units. The connection between time and angle comes from the fact that the Earth rotates through 360 degrees in 24 hours and hence through 15 degrees in 1 hour. Thus, 1 hour is equivalent to 15 degrees. Likewise, in 1 minute the Earth turns through 15 minutes of arc, and in 1 second of time it turns through 15 seconds of arc. Since the terms second and minute can refer to both time and angular measurement, the phrase "of arc" is used to indicate angular units.

Recall that there are 360 degrees in a circle, 60 minutes of arc in 1 degree, and 60 seconds of arc in 1 minute of arc. Since 15 degrees is equivalent to 1 hour, 90 degrees is equivalent to 6 hours, 180 degrees is equivalent to 12 hours, and so on. Expressed in time units the longitude of Greenwich, England, is 0 hours 0 minutes and that of Detroit, Michigan, is 5 hours 35 minutes west of Greenwich.

Altitude and Azimuth

If you go out on a clear evening and look up at the sky it seems as though you are at the center of a large dome that comes down and meets the Earth in a big circle. This circle is called the astronomical horizon.

Figure 4.1 The Celestial Dome

Around the horizon are the four cardinal points; north, east, south, and west. Directly overhead, 90 degrees from the horizon, is the zenith. The paths that objects trace out in one day are known as diurnal circles (Figure 4.1). The daily east to west motion of celestial objects is caused by the Earth’s rotation and is known as diurnal motion.

Centuries ago Greek astronomers imagined that there was another part to the sky below the astronomical horizon. The entire sky, they suggested, was a sphere with half of the sphere hidden by the Earth. Today, the sky sphere is often called the celestial sphere (Figure 4.2).

The point underfoot, 90 degrees below the

Figure 4.2 The Celestial Sphere

horizon, is the nadir. Circles passing through the zenith and nadir are known as vertical circles while the circles parallel to the horizon are termed parallels of altitude.

Longitude and latitude locate a place on the surface of the spherical Earth. Astronomers use similar coordinate systems to locate objects on the celestial sphere. One such set of coordinates is the horizon system (see Figure 4.2). The coordinates in this system are called altitude and azimuth.

Altitude is the angular distance measured from the horizon along the vertical circle through the object.

Altitude ranges from 0 to +90 degrees for objects on or above the horizon and from 0 to -90 degrees for objects below the horizon. For example, any object on the horizon has an altitude of 0 degrees, the zenith, directly overhead, has an altitude of 90 degrees, and a star midway between the horizon and zenith has an altitude of 45 degrees.

Azimuth is the angular distance measured from the north point eastward around the horizon to the vertical circle through the object.

The following table lists some altitudes and azimuths. The nn for the zenith’s azimuth indicates that azimuth is not needed since the zenith is the only point with an altitude of 90 degrees.

    Object             Altitude            Azimuth

    Name              (degrees)          (degrees)

   N                         0                       0

   E                          0                     90

   S                          0                    180

   W                        0                    270

Zenith                    90                     nn

NCP                 latitude                    0

Star rise                   0               0 to 180

Star set                    0           180 to 360

Upper transit      max              0 or 180

The coordinates of the cardinal points, the NCP, and the zenith are always the same, but this is definitely not true for celestial objects. An object’s altitude and azimuth change continually as it moves from east to west (diurnal motion). When an object is rising its altitude is 0 and its azimuth is a minimum value somewhere between 0 and 180 degrees. When it is setting its altitude is again 0 degrees but its azimuth lies somewhere between 180 and 360 degrees. An object is said to be at upper transit when it has its maximum possible altitude. This usually occurs when the object lies directly above the south point and thus has an azimuth of 180 degrees. However, circumpolar objects have an azimuth of 0 degrees when they are at either upper or lower transit.

Figure 4.3 shows the sky viewed from particular latitude at a certain time of the year and time of day.

Figure 4.3 The Local Horizon System

THE EQUATORIAL SYSTEM OF COORDINATES

One of the potential disadvantages of the horizon system is that diurnal motion causes altitude and azimuth to change with time. Another is that the coordinates depend on an observer’s latitude. This is because observers at different latitudes on a spherical Earth have different zeniths and horizons (see Figure 4.4). At any instant a star above one observer's horizon need not be above the other’s horizon.

Figure 4.4 TheZenith and Horizon

The disadvantages of the horizon system are eliminated in the equatorial system of coordinates. While the horizon system is a local system depending on time and location, the equatorial system is more general. It can be used by any observer at any time. Latitude and longitude are independent of time because the coordinate grid is marked out on the rotating Earth and moves with it. In a similar fashion, the coordinate grid of the equatorial system is marked out on the rotating celestial sphere.

The equatorial system is in many respects similar to the geographical coordinate system of latitude and longitude. Extending the Earth’s equator out to the sky produces the celestial equator while the extension of the North and South Poles results in the corresponding celestial poles (see Figure 4.5). Circles through the North and South Poles are called circles of longitude while circles through the north and south celestial poles are called hour circles.

Figure 4.5 Extending the Earth’s Coordinate System

In Figure 4.6 a star is shown on an arc passing through the NCP. This arc is part of a circle that also passes through the SCP. Any circle that passes through the celestial poles is called an hour circle, and there are an infinite number of these circles. Each celestial object lies on an hour circle and is located on the celestial sphere by the coordinates of declination and right ascension.

 Figure 4.6 The Equatorial System of Coordinates

Declination (Dec), is an object’s angular distance north (+) or south (-) of the celestial equator. It is measured along the hour circle through the object and is expressed in angular units.

Declination is similar to latitude on the Earth’s surface. Latitude is measured north and south of the Earth’s equator while declination is measured north and south of the celestial equator.

Right ascension (RA) is an object’s angular distance from the vernal equinox. It is measured from the equinox eastward (counterclockwise) around the celestial equator to the hour circle containing the object. Right ascension is expressed in time units.

Right ascension is similar to longitude on the Earth’s surface. Longitude is measured around the Earth’s equator, and right ascension is measured around the celestial equator.

The following table shows the right ascensions and declinations of a few celestial objects.

                             Object                                 Declination               Right Ascension          Sun
                             Name                                     (degrees)                       (hours)                    Date

    
                             Vernal equinox                          0                                    0                         Mar 21
                             Summer solstice                      23.5                                 6                         Jun 21
                             Autumnal equinox                      0                                 12                          Sept 21
                             Winter solstice                      -23.5                               18                          Dec 21
                             NCP                                        90.0                               nn
                             SCP                                       -90.0                                nn                                 

One of the major advantages of the equatorial system is that the coordinates of stars are essentially independent of time and the observer’s latitude. This is not true, however, for solar system objects. For example, the right ascension and declination of the Sun vary continuously as it circles the ecliptic once each year. The Moon cycles through its coordinates in about one month (it's period of revolution about the Earth). In a similar way  the time required for a planet to cycle through its coordinates depends on its period of revolution about the Sun.

Figure 4.6  shows the celestial sphere without either the Earth or the observer in the picture. The Earth’s revolution about the Sun makes the Sun appear to move eastward among the 12 constellations known as the zodiacal constellations (see Chapter 2). Each month the Sun is in a different zodiacal constellation. The circle labeled ecliptic represents the Sun’s yearly path through the zodiacal constellations. It is actually the Earth’s orbital plane extended out to the sky. The ecliptic is inclined to the celestial equator by a 23.5 degree angle because the Earth’s equator is tipped 23.5 degrees relative to its orbital plane (the ecliptic plane).

The Sun moves counterclockwise (eastward) along the ecliptic about one degree per day. On March 21 it is at the point marked V.E. On June 21 it is at point S.S. On September 21 the Sun is at point A.E. and on December 21 it is at point W.S. These points are called the vernal equinox, the summer solstice, the autumnal equinox, and the winter solstice, respectively.
While we often think of the equinoxes and solstices as dates, they are also points on the ecliptic. The vernal equinox, for example, is in the zodiacal constellation of Pisces. These points, like celestial objects, have a diurnal motion. The vernal equinox rises at the east point and sets at the west point on the horizon every day in the year. However, the Sun is at the vernal equinox (in Pisces) only on March 21. On this date the Sun, along with the equinox, rises at the east point and sets at the west point. The daily motion of the Sun is discussed in more detail in the next section.

Each year as the Sun moves around the ecliptic its right ascension and declination vary as shown in the above table. The Sun’s declination ranges from north (+) 23.5 degrees in the summer to south (-) 23.5 degrees in the winter as its right ascension varies from 0 hours through 24 hours. You might think of the Sun as a person who goes north for the summer months and south for the winter months!

The Diurnal Motion of the Sun

Figures 4.7, 4.8, and 4.9 illustrate the daily motion of the Sun on the dates of the equinoxes and solstices as seen from latitudes 90 degrees north, 45 degrees north, and 0 degrees, respectively. In each instance the altitude of the NCP has been drawn equal to the latitude since the two are always numerically equal.

On the dates of the equinoxes, when the Sun’s declination is 0 degrees, its daily path is along the celestial equator. At the North Pole (Figure 4.7) where the horizon and celestial equator are one and the same circle, the Sun circles around the horizon and never sets. At the Earth's equator (Figure 4.9) and at middle latitudes (Figure 4.8) the Sun rises at the east point and sets at the west point, remaining above the horizon for 12 hours and below the horizon for 12 hours. However,

Figure 4.7 Latitude 90 Degrees North

On the summer solstice when the Sun's declination is +23.5 degrees, the Sun is 23.5 degrees north of the celestial equator. At the North Pole (Figure 4.7) the Sun circles parallel to the horizon at an altitude of 23.5 degrees and never sets. Daylight lasts 24 hours. At middle latitudes (Figure 4.8) the Sun rises north of east, sets north of west, and is above the horizon longer than it is below the horizon and hence days are longer than nights. At the equator (Figure 4,9) the Sun is once again above the horizon 12 hours.

Figure 4.8 Latitude 45 Degrees North

On the winter solstice the Sun’s declination is -23.5 degrees and the Sun is south of the celestial equator by that amount.   At the North Pole (Figure 4.7) the Sun is 23.5 degrees below the horizon and never rises. The night is 24 hours long. At middle latitudes (Figure 4.8) the Sun rises south of east, sets south of west, and is below the horizon longer than it is above the horizon. Days are shorter than nights. At the equator (Figure 4,9) the Sun is once again above the horizon 12 hours.

Figure 4.9 Latitude 0 Degrees, Equator

Variations on the Theme of Day and Night

At the Earth’s poles and equator there are some interesting variations on the theme of night and day.

At the North Pole (Figure 4.7), between March 21 and September 21, when the Sun’s declination is positive, the Sun is always above the horizon. But between September 21 and March 21, when the Sun's declination is negative, it is always below the horizon. In other words day and night are both six months long. At the North Pole, where the horizon and celestial equator are one and the same circle, any object with a positive declination always remains above the horizon (circumpolar) and any object having a negative declination always remains below the horizon. Thus, half the sky above the horizon is always visible and the other half below the horizon is never seen.

At the equator (Figure 4.9) the Sun is above the horizon 12 hours and below the horizon 12 hours every day of the year. Also, the equator is the only place where the Sun is at the zenith at noon on the dates of the equinoxes. At the equator, not only the Sun but all celestial objects rise and set vertically to the horizon. This means that each day every object is above the horizon for 12 hours and below it for 12 hours and there are no circumpolar objects. 

Summary

The coordinates systems discussed above are summarized in the following table using the general terminology of spherical coordinate systems (see Figure 4.10).

                                                                     Figure 4.10  Spherical Coordinate Systems


Definitions

Great circle: The intersection between a sphere and a plane passing through the center of the sphere.

Small circle: The intersection between a sphere and a plane not passing through the center of the sphere.

Fundamental circle: A great circle chosen as a reference circle.

Poles: The two points 90 degrees in either direction from the fundamental circle.

Secondary circle: Any great circle through the poles of a sphere.

Origin: A chosen point on the fundamental circle.

First coordinate: The angular distance above or below the fundamental circle measured along the secondary circle through the object.

Second coordinate: The angular distance measured from the origin in a specified direction along the fundamental circle to the secondary circle through the object.

Spherical Coordinate Systems

                    General                          Geographic                          Horizon                 Equatorial    

 
                Fundamental circle        Earth’s equator                      Horizon                  Celestial equator
               Poles                              North & South                       Zenith & Nadir       NCP and SCP
                Secondary circles          Meridians of   Longitude       Vertical                   Hour circles
                Origin                            Greenwich England               North Point             Vernal equinox
               First coordinate             Latitude                                 Altitude                   Declination
                Second coordinate         Longitude                              Azimuth                   Right ascension