Problem Diagram


Aphelion and Perihelion Distance between the Earth and the Sun

During a field trip to a national astrology center, a park ranger posed a challenging question to the 12^th graders visiting the center. The earth was closest to the sun on that day, and he wanted to know what is the shortest (i.e., perihelion distance) and longest distance (i.e., aphelion distance) between the earth and the sun. It is known that the elliptical orbit of earth around the sun in polar coordinate is:

     

where e is the eccentricity and a is the distance of the major axis.

Derive the equations for the perihelion distance and aphelion distance between the earth and the sun. What is the closest and farthest distance between the earth and the sun?

• The eccentricity is approximately 0.02 and the distance of the major axis is 3 x 10⁸ km.