Word problem

Question

With an eccentricity of 0.25, Pluto's orbit is the most eccentric in the solar system. The length of the minor axis of its orbit is approximately 10,000,000,000 km. Find the distance between Pluto and the sun at perihelion (where the planet is closest to the sun) and aphelion (where the planet is furthest away). 

 

This problem comes with a picture of an ellipse around the sun. Please help! Thanks

Answer 1

Solve using the properties of an ellipse.

 

aphelion:  r_max = a(1 + e)

 

perihelion:  r_min = a(1 - e)

 

b = 10¹⁰ km

 

b = a√(1 - e²) ⇒ a = b / √(1 - e²)

 

a = 10¹⁰ / √[1 - (0.25)²] km ≅ 1.033 x 10¹⁰ km

 

r_min = (1.033 x 10¹⁰)(1 - 0.25) km ≅ 7.75 x 10⁹ km

 

r_max = (1.033 x 10¹⁰)(1 + 0.25) km ≅ 1.29 x 10¹⁰ km