91Ó°ÊÓ

First, convert the distance from miles to kilometers (since the formula for eccentricity generally uses kilometer units) and take into account that the actual distance is from the center of the Earth, not the surface. Given the radius of the Earth is approximately 6371 km, we find that the closest point (perigee) is 96 miles or approximately 154 km above the Earth's surface, and the farthest point (apogee) is 1865 miles or approximately 3001 km above the surface. Therefore, the actual distances from the center of the Earth would be \(6371 + 154 = 6525 \) km and \(6371 + 3001 = 9372 \) km respectively. The semi-major axis (a) is the distance from the center to the point furthest from the center, which is the apogee. The semi-minor axis (b) is the distance from the center to the point closest to the center, which is the perigee.

The formula for the eccentricity of an ellipse is given by \( e = \sqrt{1 - \frac{b^2}{a^2}} \). Substituting the values of a and b we found, we compute \( e = \sqrt{1 - (\frac{6525}{9372})^2} \)

After carrying out the math, the value of e comes out to be approximately 0.66. This is the eccentricity of the orbit of Explorer 55, and it describes the extent to which the orbit deviates from a perfect circle. An eccentricity of 0 would correspond to a perfect circle, while an eccentricity close to 1 would indicate a highly elongated orbit.