PhysAstro Sims--Interactive Planetary Orbit Exercise

PLANETARY ORBIT EXERCISE

Richard L. Bowman, PhD
937 College Ave., Harrisonburg, VA 22802, USA
richard.bowman@edtechbybowman.net

Kepler's Three Laws of Planetary Motion

Early in the 17th century, Johannes Kepler made a dramatic breakthrough in astronomy by proposing a model for the relative orbits of the planets and the Sun that is still a good model today. He published his model in the form of three laws of planetary motion. With this simulation and exercise, students can explore these three laws of planetary orbits.

Some Definitions

aphelion - point in the orbit of an object in our solar system where it is furthest away from the Sun.
astronomical unit (AU) - the average distance the Earth is from the Sun; 1 AU = 1.496 X 10^11 m..
eccentricity (e) - a quantity indicating how non-circular an ellipse is; its value will be between 0 and 1 where a value nearer to 1 implies a higher degree of non-circularity.
major axis of an ellipse (2a, where a = semi-major axis) - distance from one end of an ellipse to its opposite end at the longest dimension of the ellipse, that is, from perihelion to aphelion; the length of half of this axis (the semi-major axis) is the average distance the object is from the Sun during its orbit, often measured in AU.
minor axis of an ellipse (2b, where b = semi-minor axis) - distance from one end of an ellipse to its opposite end at the narrowest dimension of the ellipse.

percent difference between two vales is given by the following.

% difference =

experimental value - accepted value

accepted value

X 100 %

perihelion - point in the orbit of an object in our solar system where it is closest to the Sun.
period (p) - length of time it takes for an object to orbit the Sun; usually measured in years (yr).

Explorations

Getting Ready

1. Print out a copy of these directions for each member of the lab group, unless your instructor has already made copies for your use. Read the introductory material above before continuing with this exercise!!

2. Go to http://www.edtechbybowman.net/PhysAstroSims/orbits/ , the orbit simulation web page, and execute the default orbit (for the asteroid Geographos). Print out a copy of that page for everyone in the lab group. (All students must hand in their own lab worksheet, but you should collaborate with others in your lab group as you work through this activity.)

3. At the top-right corner of the first page, write your name and the names of the persons in your lab group. Place an asterisk by your name for identification purposes.

Note: Go to http://www.edtechbybowman.net/PhysAstroSims/orbits/KeplerCalc.html to use the "Kepler's Laws Calculator" where appropriate. Any other calculations that are necessary in the steps below must be shown on your worksheet.

Kepler's First Law

4. Write out Kepler's First Law. (Hint: See the "Kepler's Laws Calculator.")

5. Experiment with different values for the semi-major axis on the simulation web page. Try three or four different values. What quality of the orbit does the value of its semi-major axis control?

6. Systematically explore the concept of eccentricity as used to describe orbits. Set the semi-major axis to 1.2 AU and the scaling factors at the default value of "2" and then notice what happens to the orbit as the eccentricity is changed from 0 to 0.10 to 0.50 to 0.90. Describe in words the difference between a highly eccentric orbit and one with a small eccentricity.

7. What is the eccentricity of a circle?

8. On the printout of the orbit of the default object, mark the aphelion and perihelion points in its orbit. (Hint: the major axis for the orbit should be drawn first.)

9. Use a ruler to measure (in cm) the length of the major axis of the default orbit and then mark the center of this major axis with a dot. Mark off the minor axis by drawing a perpendicular line, at this center point on the major axis, going from one edge of the orbit to the opposite edge. Also locate and mark the position of the other focus on the major axis. (Hint: remember that an ellipse is symmetric top to bottom and left to right.)

10. Use the measured length of the major axis in cm, and divide it in half to obtain the length of the semi-major axis. Then equate this value to the known semi-major axis in AU to determine the scaling factor for this plot. Using this conversion factor, measure the length of the minor axis in cm and then determine the length of the semi-minor axis in AU.

Note: Before using the "Kepler's Laws Calculator" for the next steps, re-execute the default orbit in the simulator!

11. Use the "Kepler's Laws Calculator" to find the eccentricity of this orbit by the two equations given there Record your values here.

12. Calculate the percent difference between each of these two values, separately, and the given value for eccentricity used to generate the plot. (The equation for finding this percent difference is given in the introduction to this exercise.)

Kepler's Second Law

13. Write Kepler's Second Law. (Hint: See the "Kepler's Laws Calculator.")

14. Use "Kepler's Laws Calculator." Find the velocities of this object at aphelion and at perihelion and record them here.

15. Calculate the percentage increase of speed at perihelion compared to what it has at aphelion.

16. Now verify that these velocities are inversely related to the distance the object is from the Sun and thus verify Kepler's second law. That is show that the product of an object's orbital velocity and its distance from the Sun is a constant. Do this for aphelion and perihelion.

17. Begin by converting the semi-major axis distance in AU to meters.

18. The distance at aphelion is given by: r_a = a (1+e) . Calculate it.

19. The distance at perihelion is given by: r_p = a (1-e) . Calculate it.

20. Multiply v_a times r_a and v_p times r_p, where v_a and v_p come from "Kepler's Laws Calculator" (#14 above).

21. Find the difference between these two products. According to Kepler, they should be the same. (This is actually verification that Isaac Newton's concept of the conservation of angular momentum is an excellent description of the behavior of an object orbiting the Sun. This was Newton's explanation for the validity of Kepler's second law. )

Kepler's Third Law

22. Write Kepler's Third Law. (Hint: See the "Kepler's Laws Calculator.")

23. Use the "Kepler's Laws Calculator" to calculate the expected period for this object (in years). Record it here.

24. Does this compare closely with the measured value given by your instructor (for Geographos, 1.39 yr)? Calculate the percent difference between these two values.

Conclusion

16. To gain some sense of the brilliance of Kepler's deductions, enter the data for Mars (the planet upon which he based his conclusions) and observe how close to a circle the orbit is. Print out a copy of the plot of this orbit for each member of the lab group.

17. Turn in your worksheet, the printout of the orbit of default object (or that assigned by the instructor), and the printout of the orbit of Mars. Staple them together.

Note: This laboratory is really only an exercise exploring the accepted notions of elliptical orbits. To make it a "real" experimental experience, we should use observed data on the position of an asteroid or planet at various times. This is the actual route that astronomers use in their studies: observation, modeling, prediction, more observations. This activity has been working with the model only.

Bibliography

Any good introductory college astronomy textbook such as the ones below.
John D Fix. Astronomy: Journey to the Cosmic Frontier, 6th ed. McGraw Hill (2011)
Eric Chaisson and Steve McMillan. Astronomy Today, 8th ed. Benjamin Cummings (2013).
More advanced topics are covered in college physics textbooks on classical mechanics such as the one below.
Grant R. Fowles and George L. Cassiday, Analytical Mechanics, 7th ed. Brooks Cole, 2004.

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