Polaris on the size of the Earth...

This classroom guide begins with a fleshing out of the superb ideas outlined on pages 215-217 of the National Research Council's National Science Education Standards. The purpose in writing it out in this format is to help bridge the gap between having brilliant ideas for the classroom and implementing an actual lesson so that students can achieve the desired goals. The lesson plan included below is rather structured, largely for the purpose of helping us at Scopes for Schools (who have limited educational experience) keep organized in the classroom. Naturally, this plan can be altered to better meet our goals according to the instincts of more experienced educators (we always appreciate feedback from teachers on this issue), but you should expect it to take about five class periods.

Ultimately, these lessons will give students an idea of how distances and sizes are measured in the solar system. Students will actually learn the basic principles required to build a scale model of the solar system! Along the way, they will work intensively with some basic mechanics of the solar system, with proportionality, and with geometry.

The core activity will take about five class days, and it very nicely sets up a few one-day offshoot activities such as the scale model of the solar system or measuring the diameter of a sunspot. This series of lessons can either be taught as a coordinated effort between math and science teachers or by a single teacher working alone. It starts with (for example) a science teacher showing students a couple of photographs of the sky taken an hour or so apart. The students will observe that there appears to be one star (Polaris, the North Star) that doesn't move, while all others do. Students are challenged to figure out from this simple observation where Polaris would appear in the sky if you were standing on the North Pole (straight up). Where would it appear to someone standing on the equator? The first major goal of this activity is for students to come up with a correct model (or diagram) describing how the Polaris lies in line with the Earth's axis, but many Earth diameters away, and to use this model to relate the angle of elevation of the North Star to latitude on Earth.

With this relationship known, we can take two locations on Earth, one due south of the other and a known distance apart, and by knowing the angle of elevation of Polaris viewed from each location, the kids can figure out the latitude of each place, and with a little thought they can figure out the circumference of the Earth from this information. Once they know how big the Earth is, they can determine how far a point on its surface moves in the course of an hour due to rotation (1/24^th of the Earth's circumference if the point is on the equator).

With precise enough methods, we could measure the angle of elevation of a celestial body (e.g., the Moon) at a couple of times or from a couple of different locations. A careful diagram and a little mathematics (similar triangles, in particular) can then yield the distance to the object (scientists call this method and its variations parallax). Once the distance to, say, the sun is known, the diameter can be determined with a simple experiment and another application of proportionality. This key observation sets up other nice activities. Interestingly, scientists built up their precise methods using cellestial bodies themselves: they figured out the distance to the Moon using the size of the Earth, then they figured figured out the distance to the Sun using the size of the Earth-Moon system, and then they figured out the distance to other stars and planets using the size of the Earth-Sun system!

Connections:

• After this activity, the students can see how simple it is in principle for scientists to measure the planets' sizes and distances to Earth -- designing a scale model of the solar system requires only ideas the students themselves have developed! (though it might take very precise instruments)

• Since the Polaris can tell us our latitude, it's no great mystery how sailors in previous centuries were able to sail all the way across the Atlantic Ocean and land at the exact port they wanted.

• Exercises in proportionality: Since we now know how big the sun and the planets are, we can get pictures from the web (try the Big Bear Solar Observatory or the Hubble Space Telescope Gallery) and figure out the size of a sunspot or a solar flare or Jupiter's Great Red Spot!

The importance of the relationship between the procedures outlined in this guide and the NRC's National Science Education Standards cannot be over-stated. In fact, the goals of this activity consist of standards we hope will be realized for the students. For a discussion of the relevant standards and the goals and rationale underlying this activity, please follow this link.

• Either a few pictures of the night sky taken facing north from an unchanging spot and at one-hour intervals (it helps to get a tree or a building in the shot for reference), or a picture that shows time-lapse star trails, like this one,
• Graph paper (one sheet per three students should be plenty),
• Protractors (again, one per three students),
• Overhead Projector, or some other way to plot some points for the class.

Five to ten minutes to get things started:

Once you've got the class warmed up, explain that over the next couple of days they'll be using the stars to see how astronomers in Columbus' day estimated the size of the Earth and how more recent astronomers estimated the distance to the Sun, the size of the Sun, and the size of- and distance to the planets. Show them the image(s) of the northern sky at night. The first major goal is for the kids to explain the movement the image reveals.

Ten to fifteen minutes to analyze the stars' motion:

Break the class up into groups to answer the question, "There is one star which doesn't appear to move over time. This tells us that the this star must have a very special location relative to the Earth. Where is it?" The groups that finish early can move on to some other questions: Where would this star appear if you were right at the equator? At the north pole? Do they know this star's name? (the North Star, also called "Polaris")

Ten Minutes for reporting out with brief discussion:

Bringing the class back together for discussion (but keeping everyone seated with their groups), sketch the Earth on the chalkboard and ask where the odd star is. If there are discrepencies, ask for explanations and see if the kids can reach a consensus. Once the class can agree on the correct model, draw a dot on the north pole and ask where the star would be if you were standing at the north pole (answer: straight up). (What is this star called?) Do the same for a point on the equator (this time, the star would appear on the horizon).

Last fifteen minutes to explore the model further:

Working in groups with protractors and graph paper, have the students figure out where the star would appear to an observer standing on a point half way between the point on the equator and the north pole. Tell them it is very important that they write out their reasoning and use careful diagrams so they can use today's work tomorrow. Challenge groups that finish early to work on points with other latitudes, such as 1/3, 2/3, 1/6 or 5/6 of the way up from the equator.

The questions on this homework worksheet should help the students solidify some of the concepts encountered today.

About twenty-five minutes for some discussion

Let's see, where were we? We were trying to figure out what the North Star's angle of elevation would be if were standing at various places on Earth. It might be helpful to make a chart on the board, with two columns, Location (fraction) and Elevation (degrees). What angle of elevation did we come up with for the equator? (Zero degrees) What about at the north pole? (Ninety degrees) Half-way in between? (45 degrees, though they may need to discuss a bit to come to agreement) Does anyone else have other points to put up? Based on this information, does anyone have an idea of how location and elevation are related? How about ideas for expressing this information differently so that we might better see a relationship? (plotting the Elevation vs Location might be a good idea!)

Plot the information, with location on the x-axis and elevation on the y-axis. Can anyone suggest a relationship now (that is, where would other points go on the graph)? If no ideas spring forth, it might help to get more points on the graph (you can either break into groups for this, or do it as a class). With enough points on the graph, someone will undoubtably suggest that the graph should be a straight line. (Be on the look-out now for students who try to express what this really says: Viewed some fraction of the way from the equator to the north pole, the star's angle of elevation is that same fraction of 90°!)

This is a great time to make sure everyone understands what latidude means. It is the measure of the angle made between some point A on Earth, the center of the Earth B, and a point C on the equator due south (or north) of A. So all points on the equator have zero degrees latitude, the north pole is 90° north, and a point half-way in between is 45° north. In terms of latitude, what does our graph say? (Answer: Latitude equals angle of elevation!)

At this point, we haven't really proved that latitude equals angle of elevation, but we have strong reasons to believe this is the case. If you have the time to prove it or are working with a math teacher who is interested, then we encourage you to present a proof.

Last twenty minutes to estimate the size of the Earth:

Now we know that the North Star's angle of elevation tells us our latitude. Can anyone think of how someone five hundred years ago could use this information to figure out how big around the Earth is? Hint: They had reasonably accurate estimates of the distances between cities back then. In fact, Eratosthenes used observations like these to come up with a fairly accurate measure of the circumference of the Earth over 2000 years ago! (Carl Sagan's Cosmos, Episode 1: The Shores of the Cosmos has more on this.)

Let's look at a specific example (you may want to have the kids come up with their own more precise examples using a map): According to my map, Milwaukee is about 100 miles due south of Green Bay. In Green Bay, the North Star appears about 1.5° higher in the sky than in Milwaukee. How could we use this information to figure out how big around the Earth is? This is worth a few minutes' open-ended discussion. If the kids need help, you might show them how you can illustrate the situation by drawing a quarter circle with a protractor, with dots at 1.5° apart; the distance between the points is 100 miles.

This should be enough of a hint to set the students onto the task of determining the circumference of the Earth in their groups for the remainder of the hour. Remind them to keep careful notes of their reasoning for use tomorrow. If they finish early, have them consult a map for more precise estimates of the latitudes and distances. If a student's group does not finish finding an estimate, it should be finished for homework.

About fifteen minutes for discussing yesterday's results.

Last time, the kids finished the hour using the difference in the North Star's angle of elevation in Green Bay and Milwaukee (1.5°), along with the fact that Green Bay is about 100 miles due north of Milwaukee, to figure out the distance around the Earth. Let's see what they came up with. How many of the groups completed the task? Before getting numerical answers, try to get an explanation or two of how the groups approached the problem. How do these estimates compare with the known circumference of the Earth (about 24,812 miles around the poles)?

About five minutes to introduce the next task:

Now that we have an estimate of the size of the Earth, can anyone think of a way to estimate the distance to the moon? What observations would we need in order to estimate this distance? It might help to draw a diagram on the board:

If they draw blanks for too long, you might ask how knowing the size of the Earth can help. Do they see any connection? More help: If the Earth is about 24,000 miles in circumference, how far do we travel due to rotation in one hour? (answer: about a thousand miles). How can we use this information?

About fifteen minutes to think about an approach:

Break the kids up into groups to discuss the question, "How can we use the known circumference of the Earth to determine the distance to the Moon? Encourage them to draw diagrams and communicate their methods to their group partners.

[It should be noted that while there are several ways to measure the distance to the moon, I outline only one here. If the students come up with other reasonable methods, tell them to go for it! Also, this method doesn't really fit into the historical progression of such measurements very neatly - the ancients compensated for a lack in precise instrumentation with very elaborate reasoning, which is hard to motivate and in any event not appropriate for eighth graders.

This activity might be a bit too hard for eighth graders, so you may need to help steer them towards a solution such as this: We know we move about a thousand (more precisely, 1034) miles in an hour. If we measure the angle of elevation of the Moon at a couple of times an hour apart, we can figure out "Angle a" in this diagram:

Then we could just draw a triangle similar to the one made by the Moon and the two points we observed from. Now, 1000/(distance to Moon) = (measure of small side of triangle)/(measure of larger side), so we can figure out the distance to the moon!

About ten minutes for discussion:

Hopefully, every group will now have an idea of how the distance to the Moon can be measured, so let's see what they came up with. Here are a couple issues that may arise during this discussion:

1. The moon is orbiting around the Earth at a rate of 360° per 29.5 days, or about 0.5° per hour, so we would have to add a half a degree to our measure of "Angle a" to account for this. This optional worksheet guides the students in figuring this out for themselves. The worksheet may be too hard (you be the judge!) for a homework, but might make a nice group assignment in either a math or science class. Alternatively, the first two questions could be homework, with the third worked through in the next class.
   
   
2. Since the Earth rotates 15° in an hour, if we were actually standing outside measuring the angle of elevation of the Moon at two different times and subtracting, we would have to add or subtract 15 degrees to account for the Earth's rotation.

It turns out that to actually measure the distance to the moon, the angle we'd need to measure is quite small (about 0.25°) and consequently very difficult to measure with enough precision (though it can be done with care and patience!). It's probably best to circumvent this precision problem, along with problems 1 and 2 (above) by simply giving the students the measure of the angle: in terms of the diagram above, Angle a is about 0.25°. (This is too small to draw with a protractor, but we can theoretically construct it anyway: Think of the triangle as a wedge cut out of a giant circle centered at the moon, with the Earth on its edge. This circle is so big that an angle of 0.25° traces out a thousand miles of the circumference! So the entire circumference, traced out by 360°, would have to be 360*4*1000 = 1,440,000 miles. So the radius (the distance from the Earth to the moon, if you're following) has to satisfy 2*Pi*r = 1,440,000 miles, so r = 1,440,000/(2*Pi) miles, or r = 229,000 miles.) Next time, use this information to guide the students in determining the distance to the Moon!

Five to ten minutes for review:

Take a few minutes to remind everyone where we were. Make sure all the groups are up to speed and ready to finish answering the question at hand: How far is it to the Moon?

Fifteen minutes to calculate the distance:

Break up into groups and have them finish the task at hand. In fifteen minutes, each group should know the distance to the Moon!

The rest of the hour for discussion and seguey into the next task, distance to the Sun:

Well, how far is it to the Moon? What did the kids come up with? (Since the Moon's path is elliptical, the actual distance varies between 225,000 and 252,000 miles. If you trusted my numbers and calculated the Earth's circumference to be 24,000 miles, then an hour's movement of 0.25° yields a distance of 229,000 miles. Using the accepted circumference of 24,812 miles, you get a distance of about 237,000 miles, which is very near the average distance.)

Now let's see if our experiences with the moon are helpful in figuring out the distance to the Sun. In determining the distance to the Moon, we measured angles to figure out the shape of the triangle made by two points on the Earth's surface and a point on the Moon. To finally determine the distance, we had to know how to scale this triangle, which we determined by knowing how big one of its sides is, the side on the Earth, and thinking of the triangle as a wedge cut out of a big circle. When astronomers first measured the distance to the Sun, they couldn't use the exact same method because the angle they would have had to measure would have been way too small to measure, something like 0.0005°! One idea they used to fix this was to look for a triangle that didn't require such a tiny angle. Over 2000 years ago, the Greek astronomer Aristarchus used the Moon to make such a triangle:

[To be honest, it should be pointed out that this method has some practical problems. In particular, it was historically really hard to tell when precisely half of the moon was illuminated, and hence hard to know when the Earth-moon-Sun angle was really 90°. Because of this problem, Aristarchus' initial measurement of the distance to the Sun was not very accurate, and more elaborate measurement schemes were required. However, with today's instrumentation this wouldn't be such a problem (eg, with a telescope such as the one made with Scopes for Schools, you could easily tell exactly when a point half-way across the face of the moon marks the division between light and dark).]

About twenty minutes to discuss as a class the problem of measuring the distance to the Sun.

What question were we working on last time? How do we measure the distance to the Sun? Can anyone remind us what general direction we were headed in?

Why were we looking at this triangle rather than one made by two points on the surface of the Earth? Two points on the Earth are too close together - the small angle at the Sun would be too small to measure precisely. If we can figure out the shape of this triangle and how to scale it (the length of one side), then we'll be able to determine the distance to the Sun. Can anyone get us started?

A Couple of points they might notice:

1. The length of the small side is just the distance to the Moon, which we now know;
2. The way it's drawn, the angle at the Moon is 90° How would we know when this angle really is 90°? It's when exactly half of the Moon is lit by the Sun.
3. If we can measure the angle at the Earth, then the angle at the Sun is just 90° minus the angle at the Earth.

We now have a side length, so we know how to scale this triangle. We just need to finish figuring out it's shape. Can anyone see a way to measure the angle at the Earth? With enough prodding, they will notice that the point on Earth where the Moon is directly overhead at Half-Moon, and the point where the Sun is directly overhead at the same time seem to be relevant. In fact, the angle at the Earth is exactly the difference in longitude between the point where the moon is directly overhead and the point where the Sun is directly overhead at precisely the time of half-moon.

The rest of the hour to determine how far away the Sun is!

It turns out the angle at the Earth would measure about 89.85° (That doesn't leave much for the angle at the Sun - only 0.15°!) From here, the distance to the Sun can be measured in a way similar to how we measured the distance to the moon. See how much the students can figure out for themselves. For the teacher's own reference, here's how I think through this one: This time, it's a giant circle centered at the Sun, and passing through the Earth. An angle of 0.15° corresponds to about 237,000 miles, so the whole circumference is about 360/0.15 times 237,000 miles, or 568,800,000 miles (Whoa!) So the radius, the distance from the Sun to the Earth, is about 568,800,000/(2*Pi) miles, or about 91,000,000 miles.

Hopefully the students now have a clearer conception of the fact that science is a human endeavor, something carried out by real people. In particular, the astonishing scale of the solar system may seem a bit more believable now that the students have had a chance to see that measuring distances to objects in the solar system requires no magic, but is really a fairly down-to-Earth task.

Some of the more observant students may also have noticed a subtle but important trend in the exercises: We started out with a distance we could measure directly (Milwaukee to Green Bay) and used this distance to indirectly measure the distance around the Earth. We then used the distance around the Earth to indirectly measure the distance to the moon, and the distance to the moon was used in turn to measure the distance to the Sun. We could have kept going - the distance to the Sun can be used to measure the distance to objects even further away, like stars and outer planets! This general trend is an example of what scientist sometimes call "boot-strapping," using careful reasoning to build up the things we can measure. It is central to many scientific sagas.

To help focus the question of whether this activity has met its goals in the classroom, Scopes for Schools has created this guide for assessment.

Assessment is a vital tool for Scopes for Schools as we seek to improve and evaluate our activities and our program as a whole. After you have had a chance to assess this activity, we would greatly appreciate a chance to learn from both your specific experience with the activity and your general experience as an educator.

We invite any feedback you care to give, and would especially like to hear about:

• How well the activity met it's goals, as demonstrated by the assessment;
• What we can do in terms of altering the activity itself or the assessment techniques to better meet our goals;
• Any suggestions you have ("shoulds" or "should nots") for future teachers who use this activity.

This questionaire fleshes out these questions in a way that we have found particularly helpful.