Coordinate System of the Earth and Celestial Sphere.

Latitude and Longitude together are used as a geometric coordinates system to locate places on the surface of the earth. It is like the Cartesian coordinate system, which is used to locate points on a plane where the X-axis (Horizontal lines) and the Y-axis (Vertical lines) cross. In a Cartesian coordinates, a point on a plane whose coordinates are (3, 4), is 3 units to the right of the Y-axis and 4 units up from the X-axis. Latitude and longitude coordinate system is similar to the Cartesian coordinate system, except that the earth, which the latitude and longitude system is used to locate places on its surface, is spherical in shape. However, the principles are the same for the most part.

Latitude lines run horizontally east-west, parallel to one another. Latitude gives location to places north or south of the equator, which is an imaginary line that divides the earth into northern and southern hemispheres. Latitude is measured in degrees from 0°, at the equator, to 90° at the north and south poles. Each degree of latitude is approximately 111km (69 mi) apart, but the slight flattening at the poles causes the length of a degree of latitude to vary from the equator to the poles.

On the other hand, Longitude lines, which are also called the meridian, run vertically from north to south on the sphere. They converge at the poles and are widest at the equator, about 111 km (69 miles) apart. Just like the length of a degree of latitude decreases from the equator to the poles, the length between the lines of longitude also decreases from the equator to the poles. Zero degrees longitude is located at Greenwich, England. The degrees of longitude continue 180 degrees east and 180 degrees west of Greenwich and they meet and form the International Date Line in the Pacific Ocean. Greenwich, the site of the British Royal Greenwich Observatory, was established as the site of the Prime Meridian by an international conference in 1884. Therefore the meridian is an artificial line and any government or region can create their own meridian and hence a different time keeping.

This coordinate system can then be used to precisely locate places on the surface of the earth. For example, Banjul, the capital of the Gambia is located a few degrees north of the equator and about fifteen degrees west of the longitude zero degrees (that is about one hour behind, and about 111km due west of, Greenwich in London). Each degree of latitude and longitude are divided into minutes and second for more precise location of places and time zones.
Furthermore, this coordinate system of latitude and longitude is used to find great circle distances between any two places on the surface of the earth. Like flat plane geometry, the great circle distance is calculated using spherical geometry because the earth is spherical in shape. In flat plane geometry, the shortest distance between any two points is a straight line and in spherical geometry, the shortest possible distance between any two points is the great circle distance. A great circle of the earth is any line that passes through its center, dividing it into two equal hemispheres. This means that a great circle is equal to the circumference of the earth. The equator of the earth is the only great circle among the lines of latitude and all the lines of longitude, including the meridian are all great circles of the earth. Other than locations on the equator or on due north or south, the great circle distances on a flat map seem to be curved. The great circle distance between places located on the northern or southern hemispheres seem to curve towards the north or south poles, respectively.

Therefore, to find the shortest possible distance between two locations on the surface of the earth must follow the great circle linking the two points. For example, to find the distance between Banjul International Airport in The Gambia and John F Kenney International Airport in New York, I would need a set of coordinates (latitude and longitude) for the two places to calculate the smaller arc of the great circle. Banjul Airport is located at 13° 20’ 19” N and 16° 39’ 17” W; JFK is located at 40° 38’ 36” N and 73° 47’ 22” W. Approximately, Banjul is located at 13.34° North (0.2328 rad) and 16.66° west (0.2908 rad); and JFK is located at 40.64° North (0.7093 rad) and 73.64° west (1.2853 rad). Now I can use the great circle formula to find the shortest distance between Banjul and New York. From this calculation, I found out that the shortest distance between Banjul Airport and JFK about 6280 Kilometers. The calculation is as follows:
Great Circle Distance between Banjul International Airport and JFK in New York City is as follows using the great circle distant formulae.

Distance=0.9852929*6,372.795km

Distance=6279.069km.

This is the shortest possible distance between Banjul International Airport and JFK in New York City. Great circle distance is useful for airlines, shipping routes, and other purposes for navigation on the surface of the earth.

Celestial coordinate system is similar to the spherical geometry. This system is based on the motions of the earth around the sun, our star. The days, months and years on the earth are base on the activities of the earth (rotating and orbiting) around the sun. 24-hour day is called the solar day because it’s based on the time it takes for the sun to make one complete circuit around the local sky. This is slightly (about 4 minutes) longer than a sidereal day which is based on the time it takes a distant start to complete one circuit around the local sky. This is possible because, while the earth rotates on its axis at an average speed of about 1,000 km per hour, it is at the same orbiting around the sun at an average speed of about 100,000 km per hour. Therefore, for the original opposite sides of the earth to meet the sun again, the earth must complete slightly more than one full rotation on its axis. On the other hand, stars in our local sky are too far away for us to notice their movement; as a result, they seem to be fixed at one position. Solar day is not exactly 24 hours; it varies slightly between 25 seconds longer or shorter. The average day for the year is about 24 hours.

A Sidereal month is about 27 days while synodic month, the meeting of the sun and moon in the sky, occurs about every 29 days. The difference is still due to the motions of the earth around the sun; the moon must complete more than one full rotation around the earth before it can meet the sun in our local sky. In the same way, our year, which is the tropical year, is also longer than the sidereal year. The tropical year or calendar year is based on seasons, starting with spring equinox of one year to next spring equinox. Our year is marked with four distinct seasons in most parts of the world due to the sun’s movement in our local sky, which is in an ecliptic to the celestial sphere. The sun crosses the celestial equator on the southern side at spring equinox, which means spring on the northern hemisphere and fall in the southern hemisphere.

The illusionary celestial sphere is very important in locating objects in the sky. The celestial coordinate systems works like the latitude and longitude, and Cartesian coordinate systems. The north and south celestial poles represent the geological north and south poles, the celestial equator represent the equator of the earth, and Declination (dec) and Right Ascension (RA) represent latitude and longitude respectively. Therefore, the lines of declination are parallel to the equator and the celestial equator has a declination of zero degree, just like the equator of the earth has zero degree of latitude. While latitude is measured in positive numbers from zero to 90 degrees, north or south from the equator, declination is measured differently. The north celestial has a declination of +90 degrees, the celestial equator has a declination of zero degrees, and the south celestial pole has a declination of -90 degrees.

Right Ascension, on the other hand is also closely related to longitude in its applications. Like longitude, Right Ascension lines run from the north celestial pole to the south celestial pole. It is measured in hours, minutes and seconds east of the spring equinox. Greenwich is the similitude of Spring Equinox—zero degrees of Longitude and Right Ascension. Let’s say an object in our celestial sphere has a declination of 60 degrees and Right Ascension of 5 hours, its means that the object is located 60 degrees north of the celestial equator and each day the object crosses the meridian 5 hours of sidereal time after spring equinox. Another example, if a star has a declination of zero degrees and Right Ascension of zero hours, the star would rise due east, reach the meridian at spring equinox, and set due west. Stars with declination of more than zero degrees rises north of due east and sent south of due west, and the stars with less than zero degrees rise south of due east and sets north of due west. Stars with a declination of more than +90 degrees are circumpolar, meaning they do not rise or set, and stars with lest than -90 degrees never rise above the horizon.

Unlike the stars, which appear fixed at one place, the sun’s motion in our sky is apparent. Due to the tilting of the earth on its axis during its orbit around the sun, the sun appears to move in an ecliptic in the celestial sphere. The sun completes a full revolution around the celestial sphere in one year. At the spring equinox, the sun has a declination of zero degrees and RA of zero hours. For the next three months, the declination steadily increases to 23.5 degrees and Right Ascension steadily increase to 6 hours on Summer Solstice (June 21). The next three months, the declination of the sun decrease to zero and Right Ascension increase to 12 hours on Fall Equinox (Sept 21). This means that the sun rises due east and set due west at spring equinox, rises north of due east and sets south of due west at Summer Solstice, rises due east and sets due west at fall equinox, and rises south of due east and sets north of due west at winter solstice. It also means that day light is longer at spring and fall equinoxes and shorter during the solstices.

Now how can I determine my latitude and longitude in the celestial sphere? The secret is finding either the celestial north or south poles. An observer’s latitude is equal to the altitude of the celestial pole in the sky. If you know a star’s declination to be 20 degrees north and it appears in your sky at altitude of 70 degrees south, it means the celestial equator crosses your meridian at an altitude of 50 degrees south (70°-20°=50°), and your altitude is 40 degrees north (90° minus 50°). During the day, altitude can be found from the altitude of the sun, by knowing the date and the sun’s declination on that date. For example, on December 21, if the sun crosses the meridian in the south at altitude 75 degrees, therefore, the celestial equator crosses the meridian at an altitude of 51.5 degrees (75° minus 23.5°) and the latitude of the observer is 38.5 degrees north. To find your longitude using the position of an object in the sky is to compare the objects current position to its known longitude to determine how far it has moved and compare this time with the Greenwich Time to know the local time.

By ML Touray