Due:  Wednesday, February 19

Note:  Both of the problems below are modifications of laboratory exercises in Discovering Calculus with Mathematica by Knoll, Shaw, Johnson, and Evans, Wiley (1995).
I   If the Sun is located at the origin, then the orbits of Neptune and Pluto are approximated by the polar equations:

respectively, where distances are measured in kilometers.

(A)  At which point in its orbit is Neptune nearest the Sun?  What is the distance to the Sun at that point?

(B)   On the same coordinate axes plot the graphs of the orbits of Neptune and Pluto.

(C)   It would appear from the graphs that there are two points of potential disaster.  Find them.

Note:   Neptune and Pluto will not collide even though the graphs intersect.   The orbits of the inner eight planets lie in a plane called the Ecliptic Plane.  The orbit of Pluto is inclined at an angle of approximately 17 degrees to it.

(D)   Is there any planet other than Pluto that does not lie on the ecliptic plane?  If so, what is its angle of inclination?
 

II    If the Sun is located at the origin, then the orbit of Mars is approximated by the ellipse

where distance is measured in kilometers.

(A)   Plot the graph of the orbit of Mars and calculate the area swept out in one revolution by a "radius" joining the Sun to Mars.

(B)   The period of Mars is 1.88 years.  It follows from Kepler's second law of planetary motion that in one year, a radius joining the Sun to Mars will sweep out an area of A/1.88, where A is the area found in part (A).  Calculate this area.

(C)   On January 1, 1960, Mars was located at the polar point corresponding to q = 1.89665.   Where was Mars on January 1, 1961?   Hint:  Find a such that the area swept out from q = 1.89665 to  q = a is the area that you found in part (B).

(D)   Again, plot the graph of the orbit of Mars and locate the two points on the graph that indicate the positions of Mars on January 1, 1960 and on January 1, 1961.

(E)    What was the total distance traveled by Mars from January 1, 1960 to January 1, 1961.

(F)    Describe briefly Kepler's three laws of planetary motion and give a brief historical account of their development.