Coordinating the Heavens Help (page 3)
Coordinating the Heavens
A thousand generations ago, people had no quantitative concept of the sky. In the past few millennia, we have refined astronomical measurement as a science and an art. Mathematics, and geometry in particular, has made this possible.
Points On A Sphere
It is natural to imagine the sky as a dome or sphere at the center of which we, the observers, are situated. This notion has always been, and still is, used by astronomers to define the positions of objects in the heavens. It’s not easy to specify the locations of points on a sphere by mathematical means. We can’t wrap a piece of quadrille paper around a globe and make a rectangular coordinate scheme work neatly with a sphere. However, there are ways to uniquely define points on a sphere and, by extension, points in the sky.
Meridians And Parallels
You’ve seen globes that show lines of longitude and latitude on Earth. Every point has a unique latitude and a unique longitude. These lines are actually half circles or full circles that run around Earth.
The lines of longitude, also called meridians , are half circles with centers that coincide with the physical center of Earth (Fig. 1-1 A ). The ends of these arcs all come together at two points, one at the north geographic pole and the other at the south geographic pole. Every point on Earth’s surface, except for the north pole and the south pole, can be assigned a unique longitude.
The lines of latitude, also called parallels , are all full circles, with two exceptions: the north and south poles. All the parallels have centers that lie somewhere along Earth’s axis of rotation (Fig. 1-1 B ), the line connecting the north and south poles. The equator is the largest parallel; above and below it, the parallels get smaller and smaller. Near the north and south poles, the circles of latitude are tiny. At the poles, the circles vanish to points.
All the meridians and parallels are defined in units called degrees and are assigned values with strict upper and lower limits.
Degrees, Minutes, Seconds
There are 360 degrees in a complete circle. Why 360 and not 100 or 1000, which are “rounder” numbers, or 256 or 512, which can be divided repeatedly in half all the way down to 1?
No doubt ancient people noticed that there are about 360 days in a year and that the stellar patterns in the sky are repeated every year. A year is like a circle. Various familiar patterns repeat from year to year: the general nature of the weather, the Sun’s way of moving across the sky, the lengths of the days, the positions of the stars at sunset. Maybe some guru decided that 360, being close to the number of days in a year, was a natural number to use when dividing up a circle into units for angular measurement. Then people could say that the stars shift in the sky by 1 degree, more or less, every night. Whether this story is true or not doesn’t matter; different cultures came up with different ideas anyway. The fact is that we’re stuck with degrees that represent 1/360 of a circle (Fig. 1-2), whether we like it or not.
For astronomical measurements, the degree is not always exact enough. The same is true in geography. On Earth’s surface, 1 degree of latitude represents about 112 kilometers or 70 miles. This is okay for locating general regions but not for pinpointing small towns or city blocks or individual houses. In astronomy, the degree may be good enough for locating the Sun or the Moon or a particular bright star, but for dim stars, distant galaxies, nebulae, and quasars, smaller units are needed. Degrees are broken into minutes of arc or arc minutes , where 1 minute is equal to of a degree. Minutes, in turn, are broken into seconds of arc or arc seconds , where 1 second is equal to of a minute. When units smaller than 1 second of arc are needed, decimal fractions are used.
Let’s take a close look at how latitude and longitude coordinates are defined on the surface of Earth. It will help if you use a globe as a visual aid.
In geography classes you were taught that latitude can range from 90 degrees south to 90 degrees north. The north geographic pole is at 90 degrees north, and the south geographic pole is at 90 degrees south. Both the poles lie on the Earth’s axis. The equator is halfway between the poles and is assigned 0 degrees latitude. The northern hemisphere contains all the north-latitude circles, and the southern hemisphere contains all the south-latitude circles.
As the latitude increases toward the north or south, the circumferences of the latitude circles get smaller and smaller. Earth is about 40,000 kilometers (25,000 miles) in circumference, so the equator measures about 40,000 kilometers around. The 45-degree-latitude circle measures about 28,000 kilometers (17,700 miles) in circumference. The 60-degree-latitude circle is half the size of the equator, or 20,000 kilometers (12,500 miles) around. The 90-degree-latitude “circles” are points with zero circumference. Every latitude circle lies in a geometric plane that slices through Earth. All these planes are parallel; this is why latitude circles are called parallels . Every parallel, except for the poles, consists of infinitely many points, all of which lie on a circle and all of which have the same latitude.
There is no such thing as a latitude coordinate greater than 90 degrees, either north or south. If there were such points, the result would be a redundant set of coordinates. The circle representing “100 degrees north latitude” would correspond to the 80-degree north-latitude circle, and the circle representing “120 degrees south latitude” would correspond to the 60-degree south-latitude circle. This would be confusing at best because every point on Earth’s surface could be assigned more than one latitude coordinate. At worst, navigators could end up plotting courses the wrong way around the world; people might mistakenly call 3:00 the “wee hours of the morning”!
An ideal coordinate system is such that there is a one-to-one correspondence between the defined points and the coordinate numbers. Every point on Earth should have one, and only one, ordered pair of latitude-longitude numbers. And every ordered pair of latitude/longitude numbers, within the accepted range of values, should correspond to one and only one point on the surface of Earth. Mathematicians are fond of this sort of neatness and, with the exception of paradox lovers, dislike redundancy and confusion.
Latitude coordinates often are designated by abbreviations. Forty-five degrees north latitude, for example, is written “45 deg N lat” or “45°N.” Sixty-three degrees south latitude is written as “63 deg S lat” or “63°S.” Minutes of arc are abbreviated “min” or symbolized by a prime sign (′). Seconds of arc are abbreviated “sec” or symbolized by a double prime sign (″). So you might see 33 degrees, 12 minutes, 48 seconds north latitude denoted as “33 deg 12 min 48 sec N lat” or as “33°12′48″N.”
As an exercise, try locating the above-described latitude circles on a globe. Then find the town where you live and figure out your approximate latitude. Compare this with other towns around the world. You might be surprised at what you find when you do this. The French Riviera, for example, lies at about the same latitude as Portland, Maine.
Longitude coordinates can range from 180 degrees west, down through zero, and then back up to 180 degrees east. The zero-degree longitude line, also called the prime meridian , passes through Greenwich, England, which is near London. (Centuries ago, when geographers, lexicographers, astronomers, priests, and the other “powers that were” decided on the town through which the prime meridian should pass, they almost chose Paris, France.) The prime meridian is also known as the Greenwich meridian . All the other longitude coordinates are measured with respect to the prime meridian. Every half-circle representing a line of longitude is the same length, namely, half the circumference of Earth, or about 20,000 kilometers (12,500 miles), running from pole to pole. The eastern hemisphere contains all the east-longitude half circles, and the western hemisphere contains all the west-longitude half circles.
There is no such thing as a longitude coordinate greater than 180 degrees, either east or west. The reason for this is the same as the reason there are no latitude coordinates larger than 90 degrees. If there were such points, the result would be a redundant set of coordinates. For example, “200 degrees west longitude” would be the same as 160 degrees east longitude, and “270 degrees east longitude” would be the same as 90 degrees west longitude. One longitude coordinate for any point is enough; more than one is too many. The 180-degree west longitude arc, which might also be called the 180-degree east-longitude arc, is simply called “180 degrees longitude.” A crooked line, corresponding approximately to 180 degrees longitude, is designated as the divider between dates on the calendar. This so-called International Date Line meanders through the western Pacific Ocean, avoiding major population centers.
Longitude coordinates, like their latitude counterparts, can be abbreviated. One hundred degrees west longitude, for example, is written “100 deg W long” or “100°W.” Fifteen degrees east longitude is written “15 deg E long” or “15°E.” Minutes and seconds of arc are used for greater precision; you might see a place at 103 degrees, 33 minutes, 7 seconds west longitude described as being at “103 deg 33 min 7 sec W long” or “103°33′07′″W.”
Find the aforementioned longitude half circles on a globe. Then find the town where you live, and figure out your longitude. Compare this with other towns around the world. As with latitude, you might be in for a shock. For example, if you live in Chicago, Illinois, you are further west in longitude than every spot in the whole continent of South America.
Practice problems of this concept can be found at: Coordinating the Heavens Practice Problems
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