Two asteroids,A and B,orbit the Sun.Asteroid A follows a circular orbit with a radius of 6 AU.
Asteroid B follows an elliptical orbit with a perihelion distance of 4 AU and an aphelion distance of 8 AU.
How do their orbital periods compare to each other?
A) Asteroid B has about half the period of asteroid A.
B) Asteroid B has about 1.5 times the period of asteroid A.
C) Asteroid B has about twice the period of asteroid A.
D) The asteroids have identical periods.

Question

Quizplus · Accepted Answer

According to Kepler's third law of planetary motion, the square of the orbital period of a planet (or asteroid, in this case) is directly proportional to the cube of the semi-major axis of its orbit. For asteroid A, the radius of its circular orbit is 6 AU, which is also its semi-major axis. For asteroid B, the semi-major axis of its elliptical orbit is the average of its perihelion and aphelion distances, which is (4 AU + 8 AU) / 2 = 6 AU. Since both asteroids have the same semi-major axis length, their orbital periods are identical, regardless of the shape of their orbits.

Two Asteroids,A and B,orbit the Sun