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Lost at Sea: The Search for Longitude

Teacher's Ideas

Navigating Around the World by Observing the Sun
James I. Sammons, Jamestown School Rhode Island

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As you saw in the NOVA episode, Lost at Sea: the Search for Longitude, finding your position on the world oceans is very important. We take our location for granted today because we have modern electronics and radio. But what was it like when those inventions were not available? If you've looked out from the beach on a clear day, you know that it seems that you can see forever. Actually, depending on how high your eye is above the water, the horizon is only a few miles away. Even the highest mainmast lookout would lose sight of land at 25 to 35 miles on a good day! And that distance could be covered in hours.

Back when ships were made of wood and powered by wind, one of the most valuable members of the ship's company was the navigator. This was often the Captain, or the First Officer. That tradition continues today; that's why Star Trek's Captain Picard called his First Officer, Commander Riker, "Number One". To give you a feel for navigation at sea when it really counted, we're going to find the Latitude and longitude for your school as though it were a ship at sea back in the old days. Welcome aboard, First Office!

To begin, we have to introduce the concept of Horizon and Celestial Sphere. Whoa, don't panic. These are new terms, but the ideas are simple. We all know that the Earth is round, but when you look off in the distance, the horizon seems flat. That's because we're so tiny compared to the Earth, that we just can't see the curve. But compared to space, the Earth is also very tiny. We can make our navigation problem simpler if we assume the Earth is flat right where we're standing and draw a straight line to represent our horizon. Figure 1 Shows this idea. Remember, the observer figure is actually so close to the surface of the circle that he can't see around its sides.

Figure 1

Figure 1.

Now let's combine the Horizon and Celestial Sphere idea with measuring the angle between the Horizon and something in space. Figure 2 shows three observers; A, B, and C, at different Latitudes on the surface of the Earth. The Sun is very far away, but directly over the Equator. That is, an imaginary string stretched between the centers of the Earth and Sun would pass through the Earth's Equator.

Figure 2

Figure 2.

Remember that each observer is really a tiny point on the surface of the Earth. Above each point, the observers are shown in an enlarged Celestial Sphere so that we can see the detail. Check out observer B at Latitude 45 degrees North. That's written "L 45°N". If observer B measures the angle from the Horizon to the Sun at noon, the result will be 45 degrees as long as the Sun is over the Equator. This measurement is called the Altitude of the Sun.

If you're on your toes, you've probably figured out the whole Latitude thing for noon when the Sun is over the Equator. However, before you put to sea in an inner tube, check out the Altitude of the Sun (Angle between the Horizon and the Sun) for observers A and C. It's not so simple as reading the angle and that's your Latitude. Observer A at the North Pole is at Latitude 90°N, but observes a noon Altitude of the Sun of 0°. The Sun appears to him to be right on the horizon, just like at sunset. At the same time, Observer C at the Equator is at 0°, but observes a noon Altitude of the Sun of 90°, directly overhead. Can you figure the simple trick to convert the Altitude correctly?

Notice that as you move away from observer C on the Equator, towards observer A at the North Pole, the Latitude increases and the Altitude of the Sun decreases. This relationship is familiar to math types as complimentary angles and can be summarized as:
               L = 90° - Sun's Altitude
Let's assume that you observe the Altitude of the Sun to be 80° at noon and that the Sun is directly over the Equator. What is your Latitude?
               L = 90° - Sun's Altitude
               L = 90° - 80°
               Latitude = 10°N
Better get out your SPF 40 sunscreen, `cause it's going to get hot that close to the Equator!

This simple relationship works great when the noon Sun is directly over the Equator. How often is this true? Not very, I'm afraid. Fact is, only on two days of the year, March 21st. and September 21st. Not to panic, I told you this was simple and it is. After all, if those old timers could do it with a compass and a stick in the eye, called a cross staff, you can do it too.

In the practice example we subtracted the Altitude of the Sun from 90 degrees. Why? Because Latitude doesn't go higher than 90 degrees. If you had a Latitude greater than 90 degrees, you'd go over the Pole and start back toward the Equator. But what if the Sun were not on the Equator and was actually 5 degrees north of it? What would the observed Altitude of the Sun be then? Study figure 3 and see if you can figure out how you could correct the observed Altitude of the Sun for any Sun position.

Figure 3

Figure 3.

Our original formula works fine when the Sun is over the Equator. To make it work for any Sun position and therefore any date, we need to make a correction. We need to adjust the result for when the Sun is north (above), or south (below), of the Equator.

In figure 3, the Sun is shown 5 degrees north of the Equator. The observed Altitude of the Sun is 50 degrees and we know from the sketch that the correct Latitude is 45°N. But look what happens when we calculate Latitude:
               L = 90° - Sun's Altitude
               L = 90° - 50°
               Latitude = 40°N
Hmmm, the result should have been L 45°N. We can fix this if we add the number of degrees when the Sun is north of the Equator and subtract the number of degrees when the Sun is south of the Equator. This concept is called the Declination of the Sun and has to be included in our Latitude formula so that it will work at any date. Here's our final formula:
          L = 90° - Sun's Altitude + Declination of the Sun*
*Northerly Declination is added.
Southerly Declination is subtracted.

You may wonder how this new improved formula can work just like the first version when the Sun is over the Equator. That's easy, if the Sun is over the Equator, its Declination is zero. It's just like the new term at the end of the formula went away.

Try these examples to show your stuff. We'll assume that the observations were made at noon with the Sun to the south using your best cross staff and that the Declination of the Sun came from the Court Astronomer.

Problem Declination Altitude Latitude
1 15° North 42°

2 17° South 17°

3 23° North 68°

           1 = L 63°N     2 = L 56°N     3 = L 45°N
How did you do? Here's a little extra thought for you. Suppose you have the same information as in problem 1, but you had to face toward the north to measure the Altitude of the Sun. Your Latitude would be L 33°S. If you must face north to view the noon Sun, you're probably in the Southern Hemisphere. The same formula works for ships in the Southern Hemisphere, the correction for the Declination of the Sun is simply reversed. So in this case, 15° North Declination is subtracted from the observed Altitude of 48°.

OK, so now you're ready to find the Latitude of your school. If only you had a Court Astronomer to provide the Declination of the Sun. Hey - you can do that too! Figure 4 shows the Earth at four positions in its orbit around the Sun. The sizes are exaggerated to make it easier for you to see what's going on. The Earths are shown tipped 23.5 degrees relative to a vertical line straight through the Sun. Why the Earth is tipped like that is another story having to do with how planets are formed. Right now we have to figure a way of determining the Declination for any day.

Figure 4

Figure 4.

Note the Equator on the Earth at the June 21st. position. You can see that the direction to the Sun from that Earth shows the sun to be 23.5 degrees north (above) of the equator. A similar observation of the December 21st. Earth shows that the Sun direction is now 23.5 degrees south (below) of the equator. The remaining Earths at March 21st. and September 23rd. show the Sun to appear directly above the Equator. For Earth positions between the four shown, intermediate values are observed.

This concept was well known to ancient civilizations and was the main principle behind "primitive" solar observatories such as Stonehenge and Native American Medicine Wheels. You probably recognize it because it's studied in most middle level science classes as the cause of our seasons.

Chances are, your local Stonehenge is nonexistent. In lifeboat navigation, they teach you how to make a table of the Declination of the Sun by tracing something round like a life ring onto a piece of graph paper. But you can find the Declination of the Sun right in your classroom. Printed on most globes, usually off the Ecuadorian Coast but always on the Equator, is the Analemma. This figure eight is a table of the Declination of the Sun and the Equation of Time. Ignore the time bit for now and simply find the month you're looking for and pick off the number of degrees above or below the Equator where that month is printed. If the Sun is above the Equator, it's called North Declination, if below the Equator, South Declination. Now I'll bet you're the only kid in Kansas that knows what the Analemma is used for!

Although this explanation is wordy, it's a fairly simple concept. You'd be surprised at how accurately Latitude can be determined with very crude stuff. You've probably noticed that I have referred to Altitudes taken only at noon. The reason is that Latitude from the observed height of the Sun is simple, but only at Local Apparent Noon. Local Apparent Noon, or LAN, simply means noon at your exact location. Finding Latitude at other times involves some veeery heavy math (solid trigonometry) as referred to in the NOVA program.

This places a premium on knowing when Local Apparent Noon occurs as your ship surges along. How do you know? Remember, this skill was well developed long before Harrison's chronometer. This is one of those swell things that turns out to be easier than it looks. Local Apparent Noon corresponds to two events: when the Sun is exactly south of your ship and when the Sun is at it's highest point in the sky. Your ship's compass will tell you when the Sun is exactly south, and if you measure the Sun's height for a period of time beginning as the Sun is still climbing and ending when it's clearly sinking, you'll have the greatest height among your readings.

Congratulations, you are now a fully qualified First Officer, a Sun Navigator of the Days of Exploration.

Figure 5

And this is where the skill of marine navigation stood for a very long time. Ships could know their north-south line of Latitude, but they had no way of knowing how far east or west they were along that line. It's like two people trying to drive a car, but only one can see. One knows when to turn the wheel right, but the other can't see so doesn't know when to turn the wheel left. Smack up a lot of cars that way, and they smacked up a lot of ships for the same reason.

This is the whole point of the NOVA episode Lost at Sea: the Search for Longitude. If only you could know your longitude. Taken together with Latitude which is easy, these measurements allow plotting of the vessel's position on the surface of the Earth. With an accurate sense of time, you can find your east-west line of longitude. This is why Harrison sought the 20,000 pound prize by building an accurate clock.

Time and Rotation of the Earth
Just as we saw in finding Latitude, finding longitude is easy enough when we've learned three basic ideas. The first of these ideas is the relationship between time and the rotation of the Earth. It takes an average time of 24 hours for the Earth to rotate 360 degrees so that a spot on its surface will move from under the Sun and then just return to it's under-the-Sun starting position. In 12 hours, the Earth will turn half around. In 6 hours, a quarter. If you divide the number degrees in a circle by the number of hours in a day, we find that the Earth turns 15 degrees each hour.
          360°  ÷  24 hours  =  15° per hour
We can take this a step further and state that the Earth turns one degree in four minutes.
          1 hour = 60 minutes  ÷  15°  =  4 minutes per degree
The second idea is simple enough, but very confusing. Take some time here and be sure you understand it well. We have to distinguish between events and time. The events and time idea is what gives adults a headache when they try to figure out what to do to their clocks when Daylight Savings Time changes. That's why we have the no-brainer saying: "Spring forward, fall back."

To see how events and time fit together, imagine that you're standing beside an amusement ride that turns around and around. As you watch, two friends in separate cars pass you by. First one, then the other. Now imagine that the ride is stopped, you give each friend a stop watch, and tell them to start the watch just as they circle by you. You signal the operator, the ride makes one complete turn, stops, and you collect the stop watches.

Clearly the first rider went by you earlier than the second rider. For him, the event of passing you occurred earlier. For the second rider, the event, passing you, was later. But when you examine the stop watches, the time on the first rider's watch is later than the time on the second rider's watch. Why? Because the first rider's watch was started before the second's. Now imagine that you are really the Sun, the ride is really the surface of the Earth, and your two friends are really two different places on the Earth.

Figure 6

Figure 5.

Observer A in figure 5A observes Local Apparent Noon, which is simply noon for your exact location, and sets his watch to 12:00 based on Sun time. In figure 5B, we see that observer A has been moved eastward by the rotation of the Earth to be replaced by observer B. Observer B now observes Local Apparent Noon and sets his watch to 12:00 based on Sun time. Which observer experienced the event, Local Apparent Noon, first? Which observer's watch shows the earliest time? Observer C in figure 5B will experience Local Apparent Noon latest of all and yet observer C has the earliest time!

And so it is that we have to be careful about the difference between the events and time. Events like sunrise in the east always happen before the same event in the west. But time as shown on eastern clocks is later than on western clocks at the same instant. We can summarize this concept with our own no-brainer:
          Local time earlier, position is westward.
          Local time later, position is eastward.
To use our no-brainer, we have to be comparing our position to some other position. So if you knew, for example, that your time was 1 hour later than your pen pal's time, you would know that you were east of your friend. Your time earlier, you're west; your time later, you're east. It's that simple!

If we combine the first and second ideas, we can find any longitude. Every point around the Earth has its own unique Sun time. If you live one degree west (later events, earlier time) of me, your Sun time would be four minutes earlier than mine. Let's see how that works. The first idea, "Time and Rotation of the Earth" tells us that our time difference will be four minutes.
          1 hour = 60 minutes  ÷  15°  =  4 minutes per degree
The second idea, "Events and Time", tells us direction. Note that the example states that you are to the west and your time is earlier than mine. Turn that around before you use the no-brainer: I'm to the east, my time is later.

In the example above, we compared our position to some other position. As you learned in Lost at Sea: The Search for Longitude, Harrison was English and the astronomical data that he used to set his chronometer came from the Royal Observatory at Greenwich, England. So it's perfectly reasonable that the other position used by British ships should be Greenwich, England. To this day, all longitude is figured from the line of longitude that runs through Greenwich, England. This is the line of zero longitude and is called the Prime Meridian. Lines of longitude are measured in degrees east and west from the Prime Meridian.

If you think about this for a minute, you'll realize that if you sail westward from the Prime Meridian, your west longitude will increase until you reach the 180° line of longitude where east and west longitude meet on the opposite side of the Earth. As you cross this line, called the International Date Line, your longitude, now east longitude, will decrease until you return to the Prime Meridian or zero degrees longitude.

After Harrison developed his chronometer, British ships would find their position by observing the time and the height of the Sun at Local Apparent Noon. The height of the Sun would produce the ship's Latitude. The time of Local Apparent Noon, recorded as 12:00 local time, was compared to the time back in Greenwich as shown on Harrison's chronometer and the difference would produce the ship's longitude.

Well, almost. Back at the beginning of the Longitude section, I said that it takes an average time of 24 hours for the Earth to rotate 360 degrees. The third and last idea needed for longitude is the Equation of Time.

Whenever I've mentioned clock time, I've called it average time. That's because the time that everyone keeps for their daily affairs is an average value. But if you measure the length of the day by timing the exact amount of time that is required to go from LAN (Local Apparent Noon) on one day, to LAN on the next day, you discover a curious thing: the length of the day changes slowly. Starting with the clock day equal to the actual Sun day, the Sun day gets slightly longer, then slightly shorter until the clock day and the Sun day are of equal length again. The process continues, but this time the Sun day get shorter first, then longer. Don't confuse this changing day length with the seasonal change in daylight. I'm referring to changes in the whole day, light and dark together. Clock time and Sun time are equal in length only four times during the year, on the other days they are different by as much as 16.5 minutes.

It's not that the Earth rotates at different speeds; that stays the same. It has to do with the speed that the Earth revolves around the Sun which changes according to something called Keplerian Motion. The important thing is that if you're going to compare Sun time to Harrison's chronometer, you have to change the chronometer's clock time to Sun time so that you're comparing like terms. And that's what the Equation of Time does.

By applying the Equation of Time to the chronometer's clock time, we convert Greenwich Mean Time (Clock time.) to Greenwich Apparent Time (Sun time.) Greenwich Apparent Time, or GAT, is simply the Sun time back at Greenwich, England. Now we can observe Local Apparent Noon and do our simple subtraction of GAT to find our longitude.

By the way, I've used the old term, Greenwich Mean Time, so you can see the connection between the history of navigation and the terms used. Some years ago, most people who use time adopted a new name for GMT, Universal Coordinated Time or UTC (From the French Universal Time Coordinaire.) Today, you're more likely to run into the new UTC on the radio or TV.

Equation of Time
OK, last step. Where do you find the Equation of Time? The Analemma. The Analemma is shaped the way it is so that you can read both the Declination of the Sun and the Equation of Time from one cool shape. The Declination of the Sun is shown above and below the Equator, the Equation of Time is shown left and right. Different maps and globes have different systems of displaying the numbers, but if you look at it carefully, you'll be able to figure it out. Just remember that up and down, the Analemma gives you Sun's Declination, left and right gives you the Equation of Time.

Once you've got the Equation of Time, you'll have something like +15.5 minutes, or -3.0 minutes. You simply add, as in the first case, or subtract, as in the second case, the value from the chronometer's Greenwich Mean Time. The result is Greenwich Apparent Time which is then compared to Local Apparent Noon.

It goes like this: At sea, LAN is observed at 0832 hours, Greenwich Mean Time. (The time on the chronometer.) Remember, LAN means Local Apparent Noon, so it's lunch time as far as your stomach is concerned. First you have to convert the chronometer's clock time to Sun time by applying the Equation of Time. The Nautical Almanac tells us that on this date, the Equation of Time is 08 minutes. (Ships use the more accurate Nautical Almanac, you would use the Analemma.)
           8:32  GMT
         - 0:08  Equation of Time
           8:24   GAT
We recorded LAN at 12:00 local time and that's later than 0824 hours Greenwich Apparent Time, so we must be to the east of the Prime Meridian. Next we convert the time to degrees of east longitude by calculating the time difference. LAN can be written above or below GAT. Place the larger value over the smaller.
          12:00  LAN
        -  8:24  GAT
          3 hours  *  15°/hour  =  45°
          36 minutes  ÷  4 minutes/° =  9°
          [lambda]  =  54°E

Step by Step
We're now ready to find the position of any school on the face of the Earth.
  1. You must establish a shadow source of known height with a clear area to the north so that the shadow can be measured. It doesn't have to be a pole. For example, if a billboard has a pointed feature at its top that casts a sharp shadow, it'll do. You must know its height and the point exactly under it however. A six foot pole will do, but the taller the pole, the better the accuracy. Flagpoles work especially well. If you use a small pole, bury the end so that it doesn't wobble and true it up using a level.

  2. You must establish a true north line extending from the base of your pole, flagpole, etc. This is done by determining a magnetic north line with a compass and correcting that line for variation so that you can draw a true north line. West variation is subtracted from the compass reading, east variation is added. For example, Variation in the Boston area is about 15 degrees west, so the north line is plotted at 345 degrees according to the compass.

              True North = 360°  -  15°W = 345°

    Variation for your area can be found in the legend of any USGS maps or CGS charts. These steps represent the only setup for this activity. You may want to enlist the help of a surveyor if you have trouble setting the true north line.

  3. The purpose of the pole and north line is to allow you to determine the angle of the Sun at the moment that the Sun lies due south of your position. Determining the Sun's angle can be done two ways: you can measure the length of the shadow and together with the known height of the pole, and find the angle from the pole base to the tip of the shadow to the top of the pole using the Pythagorean Theorem. This is the best way if your shadow maker is tall. Ask your math teacher for help with this one if you aren't familiar with the Pythagorean Theorem. If you can reach the top of your shadow maker, you can use a string from the top to the tip of the shadow and measure the angle directly with a protractor. A word of caution, do not grovel on the ground and sight the top of the shadow maker with your eye. The extended exposure to the near sun image is an invitation to retinal burns. Makes the inside of your eyes feel like hot onion rings!

  4. To make an observation, start by setting a reliable watch with a second hand to a good time source. Cable TV is good and so are radio sources that include a tone. DJ voice announcements are often approximations. Get in position ten to fifteen minutes before LAN. Because this is not something that most people have done before, I recommend a dry run before you need good results. As the shadow moves toward true north line, take practice height readings. You should set up a graph with time on the baseline and observed altitude on the Y axis. If you plot your observations as time passes, you'll be able to see your consistency improve.

  5. You will be recording the time of LAN as shown on your watch and either the Sun's altitude angle or the shadow length. Whether you record the angle or length will depend on how you intend to calculate the Sun angle. (See three, above.) Begin recording your readings before the shadow crosses the north line. You can mark the readings when the shadow is centered on the north line, but continue recording times and shadow angles or lengths for another five minutes. Typically the height readings are a bit jumpy when plotted on the graph paper. With a straight edge, draw a line through the highest points on the graph and take that line as the value for the observed time and height of the Sun.

  6. If you were in the British Navy and had sailed from Portsmouth, your Chronometer would have been set to GMT before you left. You obviously set sail from your classroom and so your watch is set to your time zone, not Greenwich. Before you can do any longitude calculations, you must convert your Local Zone Time, as shown on your watch, to GMT. Each US time zone is earlier than GMT, so converting to GMT is simply a matter of adding some number of hours as shown in Table 1:

Time Zone

Standard Time

Daylight Time
Eastern 5 hours 4 hours
Central 6 hours 5 hours
Mountain 7 hours 6 hours
Pacific 8 hours 7 hours

Table 1.

OK First Officers, see if you can determine the position of my School. I'll tell you that the name of the School is Jamestown School, named after our East Coast town. There are lots of Jamestowns in the USA, so you'll need to calculate my position before an atlas will help you. Here are the data that you'll need:

On August 29, 1998, Local Apparent Noon was observed by my First Officer at 16 hours, 47 minutes, Greenwich Mean Time. The observed height of the Sun was 57.5 degrees above the horizon.
          Local Apparent Noon  =  16:47 GMT
          Height of the Sun = 57.5°
From the Nautical Almanac, the First Officer extracts the following. (As you might guess, the Nautical Almanac provides the same information found on the Analemma, but in more detail.)
          Declination of the Sun = 9° 10' N
          Equation of Time =  - 00' 49"
Before we get started, let's take a moment to think about accuracy. The values shown in the Nautical Almanac are much more accurate than your ability to measure Sun height and the time of LAN with a stick. Therefore, the work of calculating to the nearest minute of arc is really a waste of time. This is an important principle in mathematics, keeping your precision place values in line with your original information. To do this, we're going to round off numbers often. Good luck, the solution to the problem is shown below. (Unless Blackbeard the Teacher cut it off to make you think.)

          L  =  90° - Sun's Altitude + Declination of the Sun
          L  =  90° - 57.5° + 9°N
          L  =  41.5°N
So let's see what we have. A quick check of the map shows us that I'm not in Jamestown, Virginia. Can we use the clue that I'm in an East Coast town? Dang! The Coastline turns east and west at L 41.5°N. Could be several Jamestowns at that Latitude. Back to the chart table you go!

First we have to get that Greenwich Mean Time converted to Greenwich Apparent Time which is Greenwich time according to the Sun. So we apply the Equation of Time:
          16:47  GMT
         - 0:01  Equation of Time
          16:46  Greenwich Apparent Time
Next we find the time difference between our position and Greenwich, England. We know that Local Apparent Noon occurs at 12:00 local time and that that is earlier than 16:46 GAT. Our no-brainer tells us that our position is westward of the Prime Meridian:
          Local time earlier, position is westward.
To find how far westward, we'll subtract. To make the subtraction easier, we'll set it up with LAN on the bottom because it's smaller than GAT:
          16h:46'  GAT
        - 12h:00'  LAN
All that's left is to convert the time difference into degrees of longitude:
          4 hours  *  15°/hour  =  60°
          46 minutes  ÷  4 minutes/° =  11.5°
          Adding both gives  [lambda]  =  71.5°W
You have found my position to be:
          L  41.5°N,  [lambda]  71.5°W
which is slightly west of our school in Jamestown, Rhode Island.