An artificial satellite in a circular orbit around the Sun has a period of 8 years. Determine the ratio of the satellite's orbital radius to that of the earth's orbital radius. Assume that the earth's orbit around the Sun is circular.
A)1
B)2
C)4
D)8
E)23

Question

Quizplus · Accepted Answer

We know that the period of an orbit is related to its radius by the following formula:
T^2 = k r^3
where T is the period, r is the radius, and k is a constant that depends on the mass of the central body (in this case, the Sun) and the gravitational constant.

We can use this formula for both the satellite and the Earth and set up a ratio:
(T_satellite / T_Earth)^2 = (r_satellite / r_Earth)^3

We are given that T_satellite = 8 years, and we know that T_Earth = 1 year. So:

(8/1)^2 = (r_satellite / r_Earth)^3
64 = (r_satellite / r_Earth)^3

Taking the cube root of both sides:

4 = r_satellite / r_Earth

Therefore, the ratio of the satellite's orbital radius to that of the Earth's is 4, so the answer is (C).

An Artificial Satellite in a Circular Orbit Around the Sun