Editor’s Note: In many places throughout this lesson, the orbits of the planets about the Sun is said to be circular. This is done only for purposes of discussion and ease in reading. In actuality, the orbits of the Earth and Mars about the Sun are elliptical paths, and the Sun is positioned at one focus of those ellipses. Though the assumption of circular orbits is not entirely correct, computations and classroom discussions will be less cumbersome as a result; moreover, the paths are close enough to circular that it is not unreasonable to make these assumptions. (At the end of this lesson, there is a discussion about the actual paths of both planets and how the situation changes as a result.)
Display an image of the solar system—perhaps the one below, which shows the Sun, Earth, and Mars—and ask the class, "Who can explain how the Sun, the Earth, and the other planets in our solar system move?" Allow a student volunteer to explain that the Sun is the center of the solar system and the other planets rotate about the Sun in elliptical orbits.
Explain that this heliocentric model (with the Sun at the center) of the solar system is a somewhat recent discovery. For many years, it was believed that the Earth was the center of the universe, and the Sun and the other planets orbited the Earth. This geocentric model (with the Earth at the center) of the universe was still widely accepted as recently as the 16th century. In fact, in his 1596 book Mysterium Cosmographicum, Johannes Kepler proposed a model of the planets in which the Earth was the center, and each planet orbited the Earth in a planetary sphere described by one of the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). His model is shown below:
You may also want to highlight the following historical fact, which is not widely known. Though Nicolai Copernicus (1473‑1543) is often credited with first proposing that the Sun is the center of the universe, the idea is actually much older:
Aristarchus of Samos (310‑230 BC) … proposed that the Sun was the center of the universe … and even tried to estimate the distance from the Earth to the Sun, using trigonometry and the known Earth‑Moon distance.
| ||– "Aristarchus and the Motion of the Earth,"|
Webster’s Interactive Encyclopedia, 1999
It seems that both the Greeks and the Romans suspected that the Sun was the center of the solar system, but this idea appears to have been lost during the Dark Ages. It was not until the Renaissance that the heliocentric model was resurrected, when Copernicus proposed it.
After this introduction, say to students, "We know that the path of the Earth around the Sun is roughly a circle. But if people used to think that the Earth was stationary with the Sun traveling around the Earth, what would they have thought the path of the Sun around the Earth would be?" Remind them of what we now know—that the Earth travels around the Sun in a roughly circular orbit. Then, give them a moment to think about this question, give them another moment to discuss the question with a classmate, and then have students share their thoughts with the entire class. [The Earth rotates about the Sun in a circular path, which means that the distance between the Earth and Sun is constant. Therefore, when people assumed that the Earth was the center of the universe, it appeared that the Sun made a circular orbit around the Earth, because the distance between them never changed.]
The question about the path of the Sun around the Earth is intended to get students thinking about relative paths. Therefore, you needn’t spend much time on the discussion. Further, the belief that the Sun rotates around the Earth seems reasonable. But problems occur when we assume that the other planets in the solar system also rotate about the Earth. The focus of this lesson is to answer the question, "If Earth were the center of the solar system (instead of the Sun), and if Mars rotated about the Earth, what would it have appeared that the path of Mars was?"
Allow students several minutes to discuss this question. Have them work in pairs, and walk around the room as students talk, offering suggestions as necessary. One suggestion is that they draw a picture of the Sun, Earth, and Mars, given what they know about the solar system; and then, consider the relative positions of Earth and Mars as both make their orbits about the Sun. To fully consider the situation, students may ask for data about the distance of each planet from the Sun, as well as the length of time it takes each planet to orbit the Sun. The actual data is as follows:
|Distance from Earth to Sun||149,600,000 kilometers|
|Distance from Mars to Sun||227,900,000 kilometers|
|Ratio of Distances||1.524|
|Period of Mars Orbit About the Sun||687 days|
|Ratio of Orbital Periods||1.881|
For the purposes of constructing a model, you may want to suggest that students use the approximations listed below. These approximations will be used for much of this lesson, because they allow for elegant results when a model is created.
|Distance from Earth to Sun||150 million kilometers|
|Distance from Mars to Sun||225 million kilometers|
|Ratio of Distances||3:2|
|Period of Mars Orbit About the Sun||2 years|
|Ratio of Orbital Periods||2|
Allow students ample time to construct a model and think about this situation. As it is a difficult question, not every student will discover a correct solution. In fact, some students may even have trouble beginning the exploration. If that happens, allow a pair of struggling students to work with a pair of students who have made some progress.
After all groups have made a conjecture as to the path of Mars relative to Earth, ask several students to draw their conjectures on the chalkboard. (These drawings could also be done on transparency sheets, but the chalkboard or a whiteboard actually works better in this situation. Students can compare the drawings more easily when they are displayed side‑by‑side.) Allow students to explain their conjectures, and allow others to comment on the drawings. Have the class discuss which model seems most likely, and encourage students to modify their conjectures as they gather information and listen to their classmates’ suggestions.
To then test student conjectures, parametric equations can be used to describe the position of Mars relative to the Earth. To set up these equations, you will need to use the data provided earlier in the lesson regarding the distance from the Sun and the orbital period of each planet. The ratio of the distances to the Sun is 3:2, and the ratio of the time it takes to orbit the Sun is 1:2.
In addition, you will also want to invent a new unit, equal to 75 million kilometers. Adopting such units will prevent the use of ugly numbers in later computation. It can be fun to name this unit after a student in the class, especially a student who offered interesting insight during the previous discussions. For instance, the class might decide that 75 million kilometers = 1 emma. That way, the distance from Sun to Earth can be described as 2 emmas, and the distance from Sun to Mars can be described as 3 emmas.
Putting all of these pieces together, the location of each planet can be described by the parametric equations shown below:
Allow students ample time to develop these equations. Do not show them the correct equations until they have attempted to generate them on their own. Further, once you share the correct equations, rather than explain where the numbers come from, allow a student to offer an explanation. [The coefficients in front of sin and cos represent the ratio of the distances from the Sun, 3:2. The coefficient in front of πt within the parentheses represent the ratio of the speeds for the two planets, 1:2. Specifically, the ratio of speed indicates that Earth completes an orbit of the Sun in about half the time that it takes Mars.]
Students might be more comfortable (or familiar with) expressing the parametric equations as follows, where the x‑ and y‑coordinates are written separately:
|x = 2 cos(2πt)|
|y = 2 sin(2πt)|| |
|x = 3 cos(πt)|
|y = 3 sin(πt)|
The relative position of the two planets, then, can be found by finding the difference between their coordinates. This results in the following parametric equations:
| ||x = 3 cos(πt) – 2 cos(2πt)|
|y = 3 sin(πt) – 2 sin(2πt)|
Using a graphing calculator or computer software, allow students to graph these equations. Students may be surprised to see the result—a limaçon with period 2π/3. However, if these equations are simplified through the use of double‑angle formulae and written in a slightly different representation, students should see why they give a limaçon when graphed:
|x ||=||3 cos(πt) – 2 cos(2πt)|
| ||=||3 cos(πt) – 2 [2 cos2(πt) – 1]|
| ||=||3 cos(πt) – 4 cos2(πt) + 2|
| ||=||2 + (3 – 4 cos(πt))(cos(πt))|
|y ||=||3 sin(πt) – 2 sin(2πt)|
| ||=||3 cos(πt) – 2 [2 sin(πt) cos(πt)]|
| ||=||3 sin(πt) – 4 sin(πt) cos(πt)|
| ||=||0 + (3 – 4 cos(πt))(sin(πt))|
Notice that the first term in each equation gives the coordinates of the point where the loop of the limaçon crosses the x‑axis, (2,0).
Once the equations have been generated, conduct a class discussion using the Questions for Students below.
To conclude the lesson, allow students to use the Mars Orbit ‑ Model and Mars Orbit ‑ Actual online activities.
The difference between these activities is that Model uses the approximate numbers for the distance from the Sun and the orbital period, whereas Actual uses the exact values. When the applets are run, Model gives a limaçon that continually repeats, whereas Actual gives a limaçon that shifts with each orbit of Mars.
- Saari, Donald G. "Mathematics is Everywhere." Keynote address, Math Olympiad Awards Ceremonies, Washington, DC, June 27, 2005.
- Saari, Donald G. Collisions, Rings, and Other Newtonian N‑Body Problems. Washington, DC: American Mathematical Society, 2005.
This lesson prepared by Samuel E. Zordak, based on an idea presented in the keynote speech of Dr. Donald Saari at the 2005 Math Olympiad Awards Ceremonies, 27 June 2005.
Questions for Students
1. In the parametric equations describing the orbit of Mars relative to Earth, what does the variable t represent?
2. Relative to the Earth, when would it appear that Mars is moving the fastest? …the slowest? Where do these points occur on the graph?
[Mars will appear to move fastest when it is farthest from the Earth, and it will appear to move slowest when it is closest to Earth. On the graph, the point of greatest speed occurs at (-5,0), and the point of least speed occurs at (0,1).]
3. If a 15th century astronomer had seen this representation of Mars relative to Earth, do you think he would still believe that Earth is the center of the solar system? Explain your answer.
[Astronomers in the 15th century (and, in fact, since the 2nd century) already knew that this was the path that Mars seemed to take around the Earth — they had mapped it out plenty of times, yet they still believed that the Earth was the center of the universe. They rationalized this apparent contradiction by putting circles on circles. Their basic model was that Mars travels on one circle, the center of which travels on another circle centered at the Earth. That way Mars doesn't travel in a circle exactly, but it still orbits the earth. Astronomers spent about 15 centuries trying to refine this model to get a better match with observations.
When Galileo proposed his theory about the Sun as the center of the solar system, he showed that Mars would take the path that the others had found. However, that was not enough to convince his peers that the Sun was the center. Yet his contemporaries were neither unobservant nor stupid. They simply had deeply ingrained beliefs and unyielding scientific paradigms.]
- What were some of the ways that the students illustrated that they were actively engaged in the learning process?
- Was your lesson developmentally appropriate? Were the concepts in the lesson beyond the reach of your students?
- What adjustments did you make while teaching the lesson? Why did you have to adjust, and were your adjustments effective?
- How did the students demonstrate understanding of the materials presented?