Saturday, September 25, 2010

Latitude & Longitude

In all of the courses I'm teaching this semester, students have to be familiar with the concept of latitude and longitude. Latitude and longitude allow you to specify an exact location on the Earth’s spherical surface.  In geology, it's important because we work with maps and in astronomy it's important because it helps to understand right ascension and declination - a coordinate system used for celestial objects (subject of tomorrow's post).

Let’s start with a digression… In mathematics, a circle is traditionally divided into 360 equal parts called degrees (º). Blame the ancient Babylonians for this – they used a base 60 number system (same reason we have 60 seconds in a minute and 60 minutes in an hour). The origin of 360º arose because it was 6 x 60º and these numbers came up in some geometrical problems they were working on at the time. Still around over 3,500 years later, go figure!

Sometimes, degrees are a little large for your measurement purposes so you need a smaller subdivision of a degree. Once again, taking our cue from the Babylonians, we can subdivide a degree into 60 smaller units called arcminutes (sometimes they’re just called “minutes” but people confuse that with time on a clock which it has nothing to do with so I’ll use the term arcminutes). An arcminute can be similarly subdivided into 60 arcseconds (also often just called seconds). This means there are 60 x 60 or 3,600 arcseconds in 1º. The symbol for arcminutes is the single quotation (’) and the symbol for arcseconds is the double quotation (”).

We can therefore express an angular measure as 14º 15’ 18” which is far more precise than saying it’s “around” 14º. We can also specify accurate angular measurements in decimal notation instead of degree-minute-second notation. Since there are 60 minutes in a degree and 3,600 seconds in a degree, we can rewrite 14º 15” 18” as:

14º + (15” / 60”/º) + (18” / 3,600”/º) = 14º + (0.25)º + (0.005)º = 14.255º

Similarly, if we were given an angular measurement of 72.765º, how do we convert it to degree-minute-second notation?

Start with 72º

Then multiply 0.765º by (60’ / 1º) = 45.9’. Keep the 45’ and carry the 0.9’

Take that remainder of 0.9’ and multiply it by (60” / 1’) = 54”

Put them together to get 72º 45’ 54”

Digression over, now let’s look at latitude.

The Earth is a sphere (roughly, but for our purposes we can assume it’s exactly that). The Earth also rotates on its axis once every 24 hours. This gives two poles of rotation which we call the North Pole and the South Pole. We can draw a line on the Earth halfway between the North and South Poles and call it the Equator. It divides the Earth into a Northern and Southern Hemisphere (“hemisphere” literally means “half a sphere”).

The Equator is defined as the reference line for latitude and we say that any point on the Equator lies at 0º of latitude. Since traveling from the Equator to the North Pole means you have traveled ¼ of the way around the sphere of the Earth, we then define the North Pole as 90º North latitude (¼ of the way around 360º is 360 º/4 = 90º). Similarly, the South Pole is now defined as 90º South latitude. Between the Equator and the Poles, we can draw 90 equally-spaced lines of latitude each a degree apart.

The college where I teach lies close to the 42º N line of latitude. We’re roughly 42º north of the Equator (almost half way to the North Pole). While we’re close to this line of latitude, we’re not exactly on it and herein lies a problem. If you go to Wikipedia and look up “Earth,” you will see that the meridional circumference (meridians are north-south lines so another way of saying this is the polar circumference) of the Earth is 40,008 km. Divide this by 4 and you’ll have the distance from the Equator to the Pole which is 10,002 km. Divide this by the 90 lines of latitude between the Equator and the Pole and you’ll see that each line of latitude covers 111 km (rounded off). For those who are metrically-impaired, 111 km is about 69 miles. So when we say that the my college lies close to the 42º N line of latitude, it means we could be as far as 34.5 miles away from it (69 mi / 2).

That’s why we have to worry about arcminutes and arcseconds. They locate our position on the Earth’s surface far more accurately. Since each degree line of latitude covers about 111 km of distance, we can see that each arcsecond of latitude covers 0.0308 km (111 km / 3,600). That’s 30.8 meters (there are 1,000 meters in a kilometer). So if we specify our location to the nearest arcsecond, that will get us within ± 15.4 meters (30.8 m / 2) of the exact spot. That’s about 50 feet – close enough for many purposes (including ours). If you want to even be more exact, you can specify tenths or hundredths of an arcsecond (the U.S. military likes this degree of precision when launching things like cruise missiles).

So, to be more precise, my office at the college is best specified with a latitude of 41º 51’ 02.50” N. That kind of precision is ± 15.4 cm (± 6 inches)! Pretty damn good.

Now let’s look at longitude. Longitude lines, called meridians, run from pole to pole and are a bit more problematic because we can place the 0º longitude reference line anywhere. In the past, people have tried to locate it in places like Jerusalem, Paris, Copenhagen, Rome, St. Petersburg, and even Philadelphia. It ended up, however, in England running through the Royal Observatory in the area of greater London called Greenwich.

Why England? Because back in the 1700s the English needed, and were developing, accurate maps for seafaring and England was the dominant naval power of the day (a 1740 patriotic song still sung today has the line “"Rule, Britannia! rule the waves!”).  They made a lot of maps placing 0º longitude in Greenwich.  After much arguing among countries, an international conference in 1884 made it official, the Prime Meridian, or zero line of longitude, was defined to run through Greenwich, England.

The Prime Meridian now divides the Earth into an Eastern and Western Hemisphere (we’re in the Western Hemisphere here in the U.S.). Longitude lines go east or west of the Prime Meridian to the opposite side of the globe, in the middle of the Pacific Ocean, where they meet at the International Date Line which has a longitude of 180º E or W (doesn’t matter). Why 180º? Because you’ve gone halfway around the globe and 360º / 2 = 180º.

My office at the college has a longitude of 074º 07’ 42.88” W.

Remember that latitudes will always have N or S designations and longitudes will always have E or W designations (unless it’s assumed – for example, a location in the U.S. will always be N latitude and W longitude).

One other note, many computer programs require you to enter latitude and longitude as plus (+) or minus (-) instead of N/S or E/W. In this system, N latitudes are positive and S latitudes are negative. E longitudes are positive and W longitudes are negative. In this system, my office's location would be:

+41º 51’ 02.50” for the latitude and -074º 07’ 42.88” for the longitude.


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