**Exercise for Kepler's Laws**

Before you begin, everyone in the group needs to make a cover sheet. Be sure to include your name, your section, today's date, this assignment, and the other members of your group. Leave enough space to write a few things on the cover page.

For the rest of this worksheet, you will need to have assigned tasks and take turns "mousing." This can be done at the end of the sections on each law. Chose now who will start as mouse driver. The other tasks in the group will be the recorder (the one to fill in the work sheet), and one or more archivers to take detailed notes.

Go to the Solar System Collaboratory Home Page at the hyperlink listed above. Hit one of the Enter Website links. This will open several new windows. Hit the link to enter the Kepler's Laws module. You should now see four links - one to each of Kepler's three laws, and one to a dial-an-orbit application. We will be spending much of the time in the dial-an-orbit application.

**Kepler's First Law**

Mouse driver ____________________________

Follow the First Law link. As soon as you enter this page, move the eccentricity slider to 0.5. (The initial orbit shown on the page does NOT have an eccentricity of zero!) Now click on "Give me a Hint" to read about the first law.

1. Write out Kepler's First Law.

2. Use the information on the Hints page and on the Show Me the Math page to label the following information on the ellipse shown below.

a. Major Axis b. Semi-Major Axis c. Minor Axis

d. Semi-Minor Axis e. Focus #1 with the "sun" f. Focus #2 with the "X"

g. Perihelion (Periapsis) h. Aphelion (Apoapsis)

3. In a real orbit, what is at the focus #2? __________________________

Now go to the dial-an-orbit application. In this application, the sun is always at the origin (i.e. x=0,y=0). Enter the following values for location and velocity of the orbiting planet and then press the SELECT button. For accuracy, enter the values in the initial position and initial speed windows using the keyboard. Note that each small tick mark represents 10 units of length. Answer the questions about the resulting orbit.

**Orbit #1:**

y = 0

v

v

4. Did you start the orbit at periapsis or at apoapsis? __________________________

5. Where does orbit #1 cross the x axis again? __________________________

(This will be the distance from the sun at
that point.)

6. What is the Major axis of this orbit? __________________________

7. What is the Semi-major axis of this orbit? __________________________

8. Where is the second, empty, focus? __________________________

9. What is the distance between the two foci (focuses)? __________________________

Use the pause button and the time increments shown below the picture to measure the period of the orbit.

10. What is the period of this orbit? __________________________

The eccentricity of an ellipse is defined as the ratio of the distance between the foci to the major axis.

eccentricity = (distance between the foci)÷(major axis)

11. What is the eccentricity of orbit #1? __________________________

**Orbit #2:**

y = 0

v

v

13. Where is the second, empty, focus? __________________________

14. What is the distance between the two foci (focuses)? __________________________

15. What is the period of this orbit? __________________________

16. What is the eccentricity of orbit #2? __________________________

**Orbit #3:**

y = 0

v

v

18. Where is the second, empty, focus? __________________________

19. What is the distance between the two foci (focuses)? __________________________

20. What is the period of this orbit? __________________________

21. What is the eccentricity of orbit #3? __________________________

22. This ellipse has a special shape. What is this shape? __________________________

Go back to the First Law page. Browse through the planets and find:

23. What planet has the highest eccentricity? __________________________

24. What is that eccentricity? __________________________

25. What is the eccentricity of Earth's orbit? __________________________

**Kepler's Second Law**

New Mouse Driver ____________________________

Follow the Second Law link. Look at the Hints page and the Show me the Math page.

26. Write out Kepler's Second Law.

Go back to the Dial an Orbit application.

**Orbit #1:**

Now you will try to recreate orbit #1, but from a different starting location. Remember that when the satellite gets to the location given above, it will have the same velocity.

Start the planet at:

x = -160Your orbit #1 should have crossed the axis here. The -160 is the distance from the sun at that location in the orbit. Be sure to use the negative sign in the calculations below. Use trial and error to find the velocity, v

y = 0

v_{x }= 0

27. v_{y} = ____________________

28. Which speed is faster, periapsis or apoapsis? ___________________________

Look at the starting conditions in the First Law section for the periapsis distance and velocity.

29. For orbit #1, what is the product of the distance at periapsis times the velocity at periapsis? ___________________________

30. For orbit #1, what is the product of the distance at apoapsis times the velocity at apoapsis? ________________________________

31. There may be a small bit of rounding error, but are these numbers the same? ___________________

**Orbit #2:**

Start the planet at (x=-130,y=0, and v_{x}=0).
Find the velocity, v_{y}, that will give you the same orbit (semi-major
axis, period, and direction) as orbit #2 in the First Law exercises.

32. v_{y} = ________________________

33. For orbit #2, what is the product of the speed at periapsis times the distance at periapsis? ________________________________

34. For orbit #2, what is the product of the speed at apoapsis times the distance at apoapsis? ________________________________

35. There may be a small bit of rounding error, but are these numbers the same? ________________________________

**Orbit #3:**

36. Knowing the special situation for orbit #3, would you expect the speed to vary anywhere? ________________________________

**Kepler's Third Law**

New Mouse Driver ____________________________

Follow the Third Law link. Look at the Hints page.

37. Write out Kepler's Third Law.

38. Using INTEGER exponent buttons, run through all
the available integer combinations (1/1, 1/2,..., 1/9; 2/1, 2/2,..., 2/9;
3/1, 3/2,... 3/9;...). Which combinations give you a good fit to the data?
(There will be several that give you a good fit.)

39. Although there are many pairs of exponents that
seem to fit the data equally well, why do you think Kepler formulated his
third law as he did?

40. The period of Halley's comet is 76 years. From the graph, what is the semi-major axis of Halley's comet? __________________________

41. Look at the answers to the earlier exercises. How do the semi-major axes of orbits #1, #2, and #3 compare? __________________________

42. From Kepler's Third Law, how would you expect the periods of orbits #1, #2, and #3 to compare? __________________________________

In the Dial-an-orbit application, start the planet
at (x=30, y=0, and v_{x}=0). Find the velocity, v_{y},
that will give you the same semi-major axis as the orbits in the First
Law exercises.

43. v_{y} = ____________________

44. How would you expect the period of this orbit
to compare to the others?

In the Dial-an-orbit application, start the planet
at (x=90, y=0, and v_{x}=0). Find the velocity, v_{y},
that will give you a semi-major axis of 150.

45. v_{y} = ____________________

46. How would you expect the period of this orbit
to compare to the others?

In the Dial-an-orbit application, start the planet
at (x=60, y=0, and v_{x}=0). Find the velocity, v_{y},
that will give you a semi-major axis of 80.

47. v_{y} = ____________________

48. How would you expect the period of this orbit
to compare to the others?

**Before turning in this exercise, everyone needs
to write out all of Kepler's Laws on their cover page.**