2 May 1997 revision 11 whit
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Purpose
Determine the latitude and longitude of another site by measuring each site's sun angles at solar noon.
Overview
Your students make friends with students at another school at least 500 km distant. Both agree to make sun angle measurements on the same day at each school's local solar noon at a site of known latitude and longitude. Times and measurements are shared. Each school computes the latitude and longitude of the other school. Results are swapped and compared.
Time: 4 class periods
Overview, local solar noon estimation, UT, setting clocks, plan
Build clinometer, install vertical pole, check for horizontal surface
Perform measurement
Computations, discussion, comparison with other school
Level
Intermediate and Advanced
Prerequisites
Determine your site's latitude and longitude
GPS Activity: Working with Angles
Make arrangements with another school for same day measurements
Build and test the Landcover/Biology clinometer
Set a clock to local time
Estimate the approximate time of local solar noon
Concepts
Time and sun angle measurements can be used to determine the differences in latitude and longitude between two locations.
Skills
Measuring an angle:
(intermediate) using the clinometer
(advanced) trigonometric
Accurately setting a clock
Conversion between local and universal time
Using a compass to determine north and south
Addition and subtraction of angles
Multiplication and division
(Advanced) Application of tangent trigonometric function
Materials and Tools
An outdoors location with a flat surface and sunny view at solar noon
A pole which can be mounted vertically on a flat surface
A tape measure or ruler long enough to measure the pole's height
and its shadow's shortest length with millimeter resolution
A clock set to local time
A magnetic compass or a rough knowledge of the north/south directions
(intermediate) The Globe clinometer
(advanced) A trigonometric arc tangent table or scientific calculator
A globe or world map
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Background
In ancient times, Eratosthenes inferred Earth's circumference without having to walk all the way around our planet. He used geometry and a set of angle measurements of our sun taken in the two Egyptian cities of Syene and Alexandria which are separated by about 900 km. From this, he inferred Earth's circumference to be about 44,055 km. Although this is 15 percent larger than the actual 40,074 km, his result is an amazing demonstration of geometry and logic given the available measurement skills.
We now know Earth's dimensions quite well. Using our GPS receiver or a map, we can learn our own latitude and longitude. Can we use techniques similar to those of Eratosthenes to determine another school's latitude and longitude?
Yes. We can measure the angles of our sun at both our and another school to determine our differences in latitude. The difference between the two times of our sun's highest angle at each school tells us our difference in longitudes. The time when our sun is highest in the sky is called local solar noon
Intermediate students can directly measure angles by constructing a clinometer. Advanced students can infer angles by measuring the height of a pole and the length of its shadow using trigonometric techniques which tends to be more accurate than our clinometer measurement.
Before making any measurements, you need to become partners with another Globe school at least 500 km distant and plan a date on which you both can make outdoor measurements. You may use your Globe email facility to do this. You also need to determine your school's latitude and longitude, become familiar with the angles which describe latitude and longitude, and be able to accurately set a clock to your local time-of-day.
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Setting a Clock and Universal Time
You will need a clock which indicates time to the nearest minute and set it to the correct time-of-day. How do we learn the correct time-of-day? Easily available sources which typically can give you the time to better than one second accuracy include:
a GPS receiver
the hour tone on a local radio or television station
various shortwave radio stations
Internet based time transfer software for computers
Local Time versus Universal Time
In this activity, both you and your remote school are going to confirm your time of local solar noon by recording your local time-of-day when our sun's shadow becomes shortest. Because each school may be in a different time zone, you will convert your times to Universal Time before swapping data. You then will determine the time difference between local solar noon at both schools.
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What to Do and How to Do It
On the same calendar date, at each school's local solar noon, and from a site of known latitude and longitude, each school will record on the supplied measurement recording form:
height of the vertical pole
direction (north or south) of the pole's shadow
For twenty minutes before and after estimated local solar noon, each school will record at four minute intervals:
shadow length on flat ground from a vertical pole
(Intermediate only) angle of our sun
You can estimate a day's solar noon by following this procedure or by a rough performance of this experiment within the previous week.
Figure 641 Students Making Measurements
Intermediate Students
Use the clinometer to measure the angle from horizontal to our sun to the nearest degree recording at four minute intervals for twenty minutes before and after estimated local solar noon. DANGER: do not look through the clinometer at our sun! This will harm your eyes.
Figure 642 Students Using Clinometer
Figure 643 Sun Angle, Vertical Pole, Horizontal Surface
Measure to the nearest millimeter the horizontal lengths of the shadows from the vertical pole at four minute intervals for twenty minutes before and after estimated local solar noon.
Several people are needed to make these measurements:
(Intermediate only) One person should carefully hold and adjust the clinometer to keep it aligned with the sun by observing the spot of sunlight through the straw on his hand
(Intermediate only) Someone else should observe the angle indicated on the clinometer.
Another should be measure and mark the lengths of a shadow from a nearby vertical pole.
A timekeeper with the data recording form should observe the time of day. At four minute intervals during the experiment, he should ask the observer of the clinometer for his measured angle and the observer of the shadow for his measured length then record these values.
Questions
Does it matter if the pole is vertical? How can we confirm that the pole is vertical?
Does it matter if the ground is flat? How can we confirm that the ground is flat (horizontal)?
What if we get a fuzzy shadow?
How do we use the clinometer?
Should we see much east-west movement in the shadow? A little. During the 40 minute interval around noon, should we see much change in the shadow's length or sun's angle? Very little? But if you have time, measure and record what happens over several hours.
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Swap Measurements with Your Partner School
Be sure to confirm that your partner school indeed was able to make the measurement on the same calendar day. If weather or other events precluded a successful measurement at either school, the pair of measurements should be repeated on another day. Why? Because the angle of our sun changes each day as the seasons change.
While you are welcome to swap all of your data, exchange at least your measurements of:
universal time of shortest shadow
shortest shadow length
(intermediate) sun angle at the time of the shortest shadow
pole height
pole's shadow direction (north or south)
Then each school computes the latitude and longitude of the other.
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Find Their Latitude
If our sun were directly over the equator (spring and fall equinoxes on about 21 March and 21 September), then the it's rays would be parallel to the plane of our equator. Thus, knowing our sun angle at our and another school's locations would tell us something about our and their latitudes. If we subtract the two latitudes, we could find the difference in latitudes between schools.
However, on days other than the equinoxes, our sun is not directly over our equator, so it's rays come down at different angles. But, both schools experience the same parallel rays from our sun on the same day regardless of location. So if we subtract what would be our latitudes on the days of the equinoxes, we also shall be canceling out any offset due to our sun's seasonal movements . Thus, on any day we can determine the difference in latitudes between schools. Knowing this difference and knowing our latitude (from perhaps a GPS receiver measurement) allows us to infer the latitude of the other school.
To do this, copy your and your partner school's measurements at local solar noon onto the calculation form then perform the calculations as outlined:
For each school's data:
(Advanced) Compute sun angle
(Intermediate) Use the measured sun angle
Compute the zenith angle (90 - sun angle) [degrees]
Compute latitude difference (subtract zenith angles)
Compute the difference in your latitudes
Compute the partner school's latitude
Figure 644 Latitude Sun Angle Relationship
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Find Their Longitude
Because we are on a planet which rotates one degree every four minutes, knowing the difference between the times of our and another school's local solar noon tells us about our longitude differences . Universal Time should be used by both schools so that we are working in a common reference frame. Perform the computations as outlined.
For each school's data:
Use Universal Time for local solar noon
Convert times into minutes into the day (UT)
Find the time difference between schools' local solar noon
Convert to a longitude difference (one degree every four minutes)
Add or subtract from your school's measured longitude
Correct for hemisphere changes
Compute the partner school's longitude
Figure 645 Longitude Sun Angle Measurement
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Compare with Your Partner School
Communicate with your partner school and share results. What happened? Did you compute their location? If not, how far away were you? Do you know why? Can you compare any error in your answer to that of Eratosthenes'. Can you determine why?
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Questions and Further Study
Did our measured data make sense? Did the intermediate and final calculations make sense? If not, can we determine why?
Compute what the other school would believe to be our location.
Why not use a single sun angle measurement to determine our latitude?
(Our sun moves as seasons change)
How would shadow lengths behave at a planet's poles? (Same length all day.)
What days will have the shortest and longest shadows?
Where can errors be made which affect our computation of the other school's location? How will each error affect the resulting latitude and longitude?
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B&W figures to facilitate photocopies
B&W and color .gif file figures
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SETTING CLOCKS PAGE
How do we learn the correct time-of-day? Easily available sources which typically can give you the time to better than one second accuracy include:
the hour tone on a local radio or television station
various shortwave radio stations
Internet based time transfer software for computers
A GPS receiver
Broadcast Stations
Local television and radio stations need to coordinate their transmissions with other stations and their sources of information. Therefore, they are heavily motivated to know the time to better than the nearest second. Many commercial radio stations offer a tone at the "top of the hour" (zero minutes, zero seconds) to which one can set one's clock. Various international shortwave radio stations exist which broadcast only the time-of-day.
How long does it take for the signal to get from the radio station to you? If you are 100 km from the radio station when the tone occurs, the signal traveling at the speed of light (299,792,538 meters/second) will arrive at your radio one third of a millisecond (one millisecond = one thousandth of second) after it is sent. The sound from the radio will take 3 milliseconds to travel at the speed of sound (331 meters/second) from the radio to your ear if you are one meter from the radio. So any error in when you manually adjust the clock probably will be longer than the time it takes for the signal to get from the radio station to you.
Computer Networks
You can get the time from the US Navy via Internet. Your computer's communications software breaks its digital data into as packets which are shipped through the network via potentially different paths with unknown and varying delays. These packets also take some time to transmit. Therefore, we cannot easily know the time difference between when you see the time displayed and when the remote computer actually responded to your request.
Software exists for use via Internet which can transfer the time-of-day from another computer to yours. Some of this software is sufficiently sophisticated in that it can bounce messages between the two computers to measure then average the time delay between the two computers. Once this delay is estimated, it can be added to the time which was sent from the remote computer to attempt to correct for the various network travel delays.
Astronomers sometimes use a slightly different time (sidereal time) which is synchronized to the motion of the stars. This differs by about 4 minutes per day from our "Civil Time" (the time-of-day typically indicated on our clocks) which is referenced to our relationship to our sun. Other sources of time available on the web include:
http://www.greenwich2000.com/time.htm
http://www.bldrdoc.gov/doc-tour/atomic_clock.html
Global Positioning System
GPS is an inherently time based system. Because your locations are inferred from time signals sent from satellites with onboard atomic clocks which are accurately set, your GPS receiver can display the time-of-day. More elaborate GPS receivers even compensate for the time it takes for the signal to travel from the satellite to your GPS receiver because it knows the distance to the satellite and thus can infer the delay (which is about 67 milliseconds).
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In this activity, both your and the remote schools are going to find your time of local solar noon by recording your local time-of-day when our sun's shadow becomes shortest. You will then determine the time difference between local solar noon at both schools.
Because our sun appears overhead at different times at different longitudes, governing bodies decided to segregate our planet into 24 different time zones separated on average by 15 degrees in longitude which is the distance corresponding to one hour of Earth's rotation . Thus, it is quite possible that the time when our sun is highest in the sky (local solar noon ) near your school is quite different from that of a far away school. Also, the other school may be in another time zone where all their clocks may be different from yours by one or more whole hours. However, we can have both schools present their time measurements in a common time reference. We then can subtract the two times to determine a difference.
For historical reasons, the time along the meridian through Greenwich, England is defined. We change our local time to Universal Time by adding or subtracting a whole number of hours which depends on our location.
We can determine the number of hours to add or subtract for our conversion to Universal Time by looking at a map or globe which indicates time zones or asking someone who knows. This may change by the local application of Daylight Savings Time. The work of aviation and weather officials typically requires their knowledge of local time standards. Most GPS receivers can be set to display either local or universal time.
There exist web pages displaying Universal Time.
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Figure 660 = Figure 643 Sun Angle, Vertical Pole, Horizontal Surface
You can use a carpenter's level to confirm that your pole is vertical. It is possible for the pole to be vertical in a north-south plane but not in an east-west plane. So be sure to try the level on several sides of the pole to make sure that it is completely vertical.
A weight hung on a string will form a vertical line. If your pole happens to be a piece of pipe, the you can test your pipe by lowering a weight into the pipe. When you orient the pipe so that the string is centered along its entire length, then the pole is vertical. The weight is sometime called a "plumb bob" because the weights used to be made of lead. A former chemical name for lead was "plumbum" from which comes the English word "plumber".
Some folks do not even use a pole when performing this measurement. They just hang a string with a weight from some overhead object. The string must have a knot or some other object big enough to cast a visible shadow and should be attached about 1/2 to 1 meter above the flat surface. The distance from the surface to the knot is measured carefully and recorded as the vertical distance. However, this technique has problems if the wind blows the string and weight.
Errors will be introduced into the trigonometric angle measurement technique if the pole is not vertical and your surface is not flat. This is not a problem for the clinometer technique for measuring our sun angle, however, it increases the difficulty of determining the minimum shadow length.
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Figure 661 = Figure 643 Sun Angle, Vertical Pole, Horizontal Surface
If a soccer or other ball will roll away, your surface is not flat. More sensitive techniques use the tendency of liquids to move to their lowest possible point. You can use a carpenter's level to confirm that your surface is flat. Be sure to place the level so that the tubes containing the liquid are parallel to your surface.
A drop of water on a surface will form a ball and cling to the surface even if it is not exactly horizontal. Detergents are chemicals which reduce the surface tension of a liquid. Gently drop a slight amount of a domestic powered detergent into a small amount of water on a surface. This should reduce the surface tension of the water sufficiently so that it will flow downhill if the surface is not flat. This can assist you both in determining if your surface is flat and correcting it if you can move it.
A more sophisticated technique for determining a common horizontal plane uses a long flexible and transparent tube almost completely filled with a liquid. Two students, each holding one end of the tube, move far apart holding their ends of the tube up so not to spill the liquid. Regardless of their distance, the levels of the liquids at each end will be the same.
Geologists use a variation of this technique to detect slight upheavals or drops in our planet's surface. They bury or lay on the ground a horizontal pipe which may be hundreds of meters long and fill this pipe half full of water. The pipe is adjusted up and down until a half full pipe is observed at each end. Should the ground move slightly producing an angular change of even a fraction of a degree, the water will move to one end of the pipe. This is an example of building a sensitive instrument which indicates a slight difference with a dramatic change. You can do the same with a long glass tube of moderate diameter, however, be careful not to break the glass.
Errors will be introduced into the trigonometric angle measurement technique if the pole is not vertical and your surface is not flat. This is not a problem for the clinometer technique for measuring the sun angle, however, it increases the difficulty of determining the minimum shadow length.
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If you are on Earth using our sun, you should always get a fuzzy shadow. With a short pole, you might not notice the softness of the shadow's edge, but no shadows from our sun have sharp edges on Earth.
Why? Because the light from our sun does not appear to come from a point source. Instead, it appears to come almost uniformly from a circle about one half of a degree across (which interestingly is about the same angular size as our moon which makes our total solar eclipses so interesting).
Because it would harm your eyes to look directly at our sun, look at the full moon some night. Light radiates towards you from the center, top, bottom, right, and left sides of the moon. All this light is approaching you or any object which can cast a shadow from slightly different angles.
Figure 662 Fuzzy Shadows
For example, if a one meter tall vertical pole was located where our sun appears to be 45 degrees up in the sky, the light rays which pass immediately above the top of the pole are coming from slightly different parts of our sun. The rays from the center may be coming down at a 45 degree angle but the rays from the top will be coming down over the top of the pole at an angle one quarter of a degree steeper (45 + 0.25 = 45.25 degrees). Conversely, the rays from the bottom will be approach at a quarter degree more shallow angle (45 - 0.25 = 44.75 degrees). The shallower rays will land further away from the pole (1009 millimeters) while the steeper rays will be closer up to (911 millimeters). This is a 18 millimeter (almost 2 centimeter) difference between the edge of the full sunlight area and the full shadow area.
For measurement purposes, this is a problem. One could begin measuring at the edge of the full sunlight or the edge of complete shadow. But because the light gradually fades from full sunlight to full shadow, there is no distinct edge. Because it produces almost the desired angle, try to estimate the middle of the light to dark area and use this as your distance.
In the above example, if one were to use the lighter or the darker edge instead of the shadow's middle, this could induce an error of a quarter degree either way. This becomes an error of about 26 kilometers when converted to a difference in latitude.
Because of this problem, often stars are used when navigating by celestial bodies. Although they are quite large, they are so far away that they appear to be less than one arc second across. Thus for navigation purposes they appear to be points. Also, the top or bottom edges of our sun may be used. Instruments used for making these angle measurements with celestial bodies are called sextants.
Figure 663 Pinhole camera
One can see a picture of our sun by making a pinhole camera. Simply take some surface (a piece of aluminum foil or cardboard works well) and make a small hole in it (stick the pin through the foil). If you hold this up to our sun over a flat, light colored surface, you can see an inverted image of our sun projected on the surface. This is a good way to view sunspots or a solar eclipse.
Occasionally the arrangement of leaves in trees above us form small holes through which sunlight shines to cover the ground with circles of light. During a solar eclipse, you can see the ground covered with the projections of the arcs of the uncovered sun. If you are under heavy canopy at night so that your eyes have adapted to the dark when there is less than a full moon, you can see clear images of the partial moon projected on the ground through small gaps the leaves.
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The clinometer described in the Landcover/Biology uses a drinking straw for optical alignment.
Do not try to look at our sun through the straw on your clinometer! You will harm your eyes. Instead, hold the clinometer with one hand. Align it so that you cans see a spot of sunlight through the straw n your other hand as shown in the illustration.
Figure 664 = 642 Students Using Clinometer
Several people are needed to make these measurements:
(Intermediate only) One person should be carefully holding and adjusting the clinometer to keep it aligned with the sun by observing the spot of sunlight on his hand.
(Intermediate only) Someone else should be observing the angle indicated on the clinometer.
Another should be measuring and marking the lengths of a shadow from a nearby vertical pole.
A timekeeper with the data recording form should be observing the time of day. At four minute intervals during the experiment, he should ask the observer of the clinometer for his measured angle and the observer of the shadow for his measured length then record these values.
You can estimate a day's solar noon by following this procedure or by a rough performance of this experiment on the previous day.
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Determine your time of local solar noon from your measurements. This occurs when the shadow is the shortest. Because you may have several length measurements with the same short length, choose the time of the one closest to the center of your identically short length measurements. Convert this to universal time and record this on your calculation form in addition to the other measured data from you and your partner school.
Follow the computations shown on the worksheet. Advanced students will determine our sun angle from a trigonometric computation whereas others will use angles measured with the clinometer.
Figure 665 = 643 Sun Angle, Vertical Pole, Horizontal Surface
We really need the Zenith Angle. This is the angle between the top of the shadow pole and the incoming sun's ray. It is defined as being the difference between our angle to our sun and our angle to the zenith. The zenith is defined as always being directly over our heads wherever we are. The pole points to our zenith. Because the sum of all triangle internal angles is 180 degrees and we know that one of the angles is 90 degrees if the pole is vertical and the ground is horizontal (flat), then we can subtract 180 minus 90 minus our sun angle to get the zenith angle.
Why do we need the zenith angle? If our sun were directly over the equator (spring and fall equinoxes on about 21 March and 21 September), then our sun's rays would be parallel to our equator. Thus, the zenith angle would be the same as our latitude. Knowing our sun angle at another school would tell us their latitude. However, on days other then the equinoxes, our sun is not directly over our equator, so it's rays come down at different angles. But, both schools experience the same parallel rays from our sun on the same day regardless of location. So if we subtract what would be our latitudes on the days of the equinoxes, we also shall be canceling out the offset due to our sun's seasonal movements because this offset is experienced equally at both schools. Thus, on any day we can determine the difference in latitudes between schools. Knowing this difference and knowing our latitude (from perhaps a GPS receiver measurement) allows us to infer the latitude of the other school.
Figure 667 = Figure 644 Latitude Sun Angle Relationship
Corrections
Each school could be viewing our sun from different north-south directions. This can change throughout the year as our sun moves in its seasonal cycle. In this case, we should need to look for a sum instead of a difference between the school zenith angles. The relationships between the directions of the poles' shadows tells us whether we need a sum or difference between zenith angles. The worksheet has a table indicating conditions for summation or subtraction.
It is also possible that your partner school is in the hemisphere opposite yours. If so, you will get a negative latitude when you perform your final subtractions. In this case, just change hemispheres and make the result positive.
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All these phrases mean the same thing:
24 hours in a day
24 hours for Earth to rotate once relative to our sun
24 hours in a 360 degree rotation of Earth
1440 minutes in a 360 degree rotation of Earth
4 minutes in a 1 degree rotation of Earth
4 minutes in a 1 degree east-west movement of our sun
every 4 minutes, Earth rotates through 1 degree of longitude
So every 4 minutes, our sun moves 1 degree.
Because we are on a planet which rotates one degree in every four minutes, knowing the difference between the times of our and another school's local solar noon tells us about our longitude difference. Universal Time should be used by both schools so that we are working in a common reference frame. Perform the computations as outlined on the worksheet.
Figure 668 = Figure 645 Longitude Sun Angle Measurement
Time is important for the longitude computations. Contrast this to the latitude computations where angle measurement is important. Pendulum clocks were developed long before self-regulating spring driven clocks. But pendulums do not work well on a moving and tossing ship. Until the development of clocks which did not require pendulums, ships could determine their latitudes but not their longitudes. The epic struggle to develop a technology which solved this problem is outlined in the book Longitude (by Dava Sobel, 1995, Walker Publishing Company, NYC).
To facilitate not having to deal with fractions of hours, convert the UT time-of-day from hours and minutes into a number of minutes into the day for each school. Find the difference between these times to determine the amount of time between local solar noons at each site. Because Earth turns at a fixed one degree in every four minutes, divide the time difference by four to compute the angular longitude difference between schools.
Our planet rotates from west to east. You can remember this by recalling that the sun rises in the east which means that you must be traveling toward the sun and therefore moving to the east. Thus the school which experienced its local solar noon first is east of the other school. This tells you whether to add or subtract the differences in longitudes between schools to your longitude to get the other school's longitude.
If the other school's longitude value is negative, then they are across the Prime Meridian (0 degrees Longitude) from you. In this case, change east-west hemispheres and use a positive longitude value. Should the value be greater than 180 degrees, then they are across the International Date Line. In this case, switch hemispheres and subtract 180 degrees for their final longitude value.
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1. Data recording form (example and blank)
2. Computations form (example and blank)
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Earth orbits from west to east at an average distance of about 148 million kilometers from our sun. However, our planet's spin axis is tilted about 23.5 degrees out of the plane of its orbit. Therefore, on one side of our orbit, the northern hemisphere experiences sunlight closer to perpendicular over a larger area than does the southern hemisphere. This is the northern hemisphere's summer and the southern hemisphere's winter. The seasons swap for the hemispheres as our planet moves to the other side of our sun. A terrestrial observer sees the sun at a higher angle in his sky during his summer FIGURE 7.1 (P 7-7 ) IN GLOBE II TEACHER'S GUIDE.
Figure 669 Sun-Earth Seasonal Relationship
Several geographic and astronomical definitions are a consequence of this tilt. The Arctic and Antarctic Circles are defined as being about 23.5 degrees away from the north and south poles respectively. These are the lowest latitudes which can experience total darkness in their respective winters. The Tropic of Cancer and Tropic of Capricorn are defined as being 23.5 degrees north and south respectively of the equator. These are the latitudes farthest from the equator which ever experience our sun directly overhead. In the northern hemisphere, the yearly date of the highest sun angle usually occurs on 21 June. In the southern hemisphere, this occurs on 21 December. These days are defined as being the summer and winter solstices respectively in the northern hemisphere. Our sun appears to be directly overhead at the equator on about 21 March and 21 September. These are defined as being the vernal and autumnal equinoxes.
From any point on Earth, the angle of the sun appears to change daily with a year long cycle attached to the seasons. So if you were to make a sun angle measurement one day and your partner school were to do so on another, you would have an angle difference between your schools which is due to both different latitudes and different sun angles. However, in any 24 hour period, our sun's seasonal motion is less than a single degree.
How high would the sun appear in the sky during summer and winter for various places on Earth? On the equator? The north pole? Where you live? Astronomers developed equations which model the motion of celestial bodies. Computer programs exist which use these equations to compute the positions of our sun, moon, and other celestial bodies as seen from anywhere in the world at any time. If you have an IBM compatible personal computer, see http://fox.nstn.ca/~ecu/ecu.htm
Sunlight falling perpendicular to the ground presents about 1000 watts of solar radiation to every square meter of brightly sunlit ground. Intuitively, this is the equivalent to 10 medium sized incandescent light bulbs falling on every square meter under our sun. And there are a lot of square meters on our planet. Contrast this to sunlight falling at 45 degrees which presents only about 700 watts to every square meter on an otherwise equally clear day. This difference in incoming solar radiation accounts for the accumulated energy differences between seasonal extremes which are indirectly observed as temperature changes.
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How close (in degrees) are your computed latitude and longitude from those measured at the other school's site? What affects this difference?
Size of fuzzy shadow
Alignment of clinometer
Ground not flat or pole not vertical
Determination of time of local solar noon
How can you determine which error sources causes more trouble? Pretend to perform the experiment. Make up a set of measurement numbers that you would expect to get under ideal circumstances. Perform the calculations on your ideal numbers and make sure that you get the ideal answer that you expect. Then make up a set of numbers which are all ideal except for a single error. Choose the error value to be typical of what you might observe. For example, you might add a few millimeters to the shadow length, whereas, adding 100 meters clearly is too much. Perform the calculations on these numbers and compare to the ideal results.
When you do this, you are performing a simulation to test your experiment's sensitivity to each of the error sources. For complex experiments with a large number of measurements and a lot of equations, a computer program may be used to vary all of the identified sources of error to determine the various extremes in outcomes.
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Eratosthenes was a Greek mathematician and scientist who lived during the 3rd century BC. He noticed that on one particular day in the year, sunlight could be seen reflecting from the water of a deep well in the city of Syene, Egypt. This meant that our sun was directly overhead in Syene. On this same day, the angle of our sun's shadow from a vertical pole was measured carefully in the northern port city of Alexandria, Egypt and found to be one 50th of a circle which is about 7.2 degrees.
Eratosthenes paid someone to walk due north from Syene to Alexandria who measured this distance to be 500 stades. A stade is about 185 meters or 607 feet. Because he walked due north, he was walking along a line of constant longitude with only his latitude changing.
Figure 670 Eratosthenes' Experiment
Knowing the circumference distance and angular separation between two points on a circle allows us to infer the circumference of that circle. If we assume Earth is round (as did some ancients contrary to legend), we can deduce Earth's circumference from the above information. Eratosthenes did this resulting in a circumference estimate of 250,000 stades or 44,055km. Today, we estimate Earth's circumference to average about 40074 km. Thus, Eratosthenes was in error by about 15 percent. Given the technology and scientific knowledge of the time, this is a remarkable deduction.
Techniques similar to these are used have been developed over centuries for land, sea, air, and space navigation. A sextant is a hand-held theodolite which is used to make angle observations of celestial bodies for navigation purposes. It is just a higher accuracy version of our clinometer. When done correctly, a hand-held sextant, clock, and computation tables can be used to determine your location to within two kilometers worldwide. For details, see annually updated navigation book The American Practical Navigator: An Epitome of Navigation, Nathaniel Bowditch, US Defense Mapping Agency, Bethesda, Maryland, 1st Edition 1802.
Biographical and historical information about Eratosthenes
http://www.math.utah.edu/~alfeld/Eratosthenes.html
http://astronsun.tn.cornell.edu/courses/astro201/eratosthenes.htm
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Figure 641 Students Making Measurements
Figure 642 Students Using Clinometer
Figure 643 Sun Angle, Vertical Pole, Horizontal Surface
Figure 644 Latitude Sun Angle Relationship
Figure 645 Longitude Sun Angle Measurement
Figure 660 = 643 Sun Angle, Vertical Pole, Horizontal Surface
Figure 661 = 643 Sun Angle, Vertical Pole, Horizontal Surface
Figure 662 Fuzzy Shadows
Figure 663 Pinhole camera
Figure 664 = 642 Students Using Clinometer
Figure 665 = 643 Sun Angle, Vertical Pole, Horizontal Surface
Figure 667 = 644 Latitude Sun Angle Relationship
Figure 668 = 645 Longitude Sun Angle Measurement
Figure 669 Sun-Earth Seasonal Relationship
Figure 670 Eratosthenes' Experiment
HYPERLINK TO GPS INTRODUCTION.
HYPERLINK TO ATMOSPHERE'S LOCAL SOLAR NOON DISCUSSION (PAGE 2-14 IN GLOBE II TEACHER'S GUIDE).
HYPERLINK TO GPS ACTIVITY: WORKING WITH ANGLES
HYPERLINK TO DISCUSSION IN GPS ACTIVITY: WHAT...ANSWER:TIME STANDARDS
HYPERLINK TO WELCOME TO THE GPS INVESTIGATION:INTRODUCTION TO THE BIG PICTURE:GPS SATELLITES
HYPERLINK TO GLOBE SEASONS ACTIVITY: WHY ARE THERE SEASONS (P 7-7)
FIGURE 7.2 (P 7-25) IN GLOBE II TEACHER'S GUIDE.
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