We recommend that you print out this worksheet and go through the questions while using the following website:

*A. Kepler's First Law*

1. Kepler's First Law states that a
planet moves on an ellipse around the sun. Where is the sun with respect
to that ellipse?

2.a Could a planet move on a *circular*orbit?

2.b If YES, where would the sun be
with respect to that circle?

3. What is the eccentricity of an ellipse?
Give a description (words, not formulae).

4. What happens to the ellipse when
the eccentricity becomes zero?

5. What happens to the ellipse when
the eccentricity becomes one?

6. On planet Blob the *average global*
temperature stays exactly constant throughout the planet's year. What can
you infer about the eccentricity of Blob's orbit?

7. On planet Blip the *average global*
temperature varies dramatically over the planet's year. What can you infer
about the eccentricity of Blip's orbit?

8. For an ellipse of eccentricity e=0.9,
calculate the ratio of periapsis to apoapsis (you may want to look up periapsis
and apoapsis in the "HINTS"). Use the tick-marks to read distances directly
off the screen (to the nearest half-tick).

9. What is the ratio of periapsis to
apoapsis for e=0.5?

10. For e=0.1?

**Now let's talk about our solar system.**

Remember that the orbits of the different planets are not drawn to scale. We have scaled the diagram to the major axis of each orbit.

11. Which planet has the largest eccentricity?

12. What is the ratio of perihelion to aphelion for this planet?

13. Which planet has the second largest eccentricity?

14. What is the ratio of perihelion to aphelion for that planet?

15. If Pluto's perihelion is 30 AU, what is it's aphelion?

16. If Saturn's perihelion is 9.0 AU,
what is it's aphelion?

*B. Kepler's Second Law*

1. For eccentricity e=0.7, count the number of tick-marks on the speedometer between the speed at periapsis and the speed at apoapsis.

2. Same for e=0.4.

3. Again for e=0.1.

4. For eccentricity e=0.7, measure
the time the planet spends

a) to the left
of the vertical line (minor axis)

b) to the right
of the vertical line

5. Again for e=0.2.

a) to the left
of the vertical line (minor axis)

b) to the right
of the vertical line

6. Where does the planet spend most
of it's time, near periapsis or near apoapsis?

*C. Kepler's Third Law*

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

2. By clicking on the UP and DOWN buttons,
run through all the possible combinations of integer exponents available
(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?

3. Using decimal exponents, find the exponent of a (the semi-major axis) that produces the best fit to the data

a) p^{0.6}

b) p^{5.4}

c) p^{78}

4. Using decimal exponents, find the exponent of p (the period) that produces the best fit to the data for

a) a^{0.6}

b) a^{5.4}

c) a^{78}

5. Why do you think Kepler chose to
phrase his third law as he did, in view of the fact that there are many
pairs of exponents that seem to fit the data equally well?

*D. 'Dial-an-Orbit' Applet*

1. Start the planet at X=-80, Y=0.

Find the initial velocity (both X- and Y-components) that will result in a circular orbit (use the tick marks to judge whether the orbit is circular).

Vx __________Vy __________

b) Using the clock, find the period of that orbit.

T: ______________

2. Now start the planet at X=-60, Y=0, and find the velocity that will result in an elliptical orbit of semi-major axis=80 (attention: remember the definition of the semi-major axis; look at the 'BOOK' page if you have forgotten). Use the clock to find the period for that orbit.

T:________________

3. Would you expect the period you
measured in question 2 to be the same as the period you measured in question
1? Why?