Final answer:
Kepler's third law states that the period squared is proportional to the cube of the semi-major axis of an orbit. By rearranging the equation, we can derive the period squared as a function of the radius. The proportionality constant depends on the specific system being considered.
Step-by-step explanation:
Kepler's third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. In other words, the period squared can be expressed as a function of the radius. Let's consider the general equation for Kepler's third law: T^2 = k * a^3, where T is the period, a is the semi-major axis, and k is the proportionality constant.
If we rearrange the equation to solve for T^2, we have T^2 = k * a^3. Since the semi-major axis of a circular orbit is equal to the radius, we can substitute r for a in the equation. Therefore, the period squared as a function of radius is simply T^2 = k * r^3.
It's important to note that the value of the proportionality constant, k, depends on the specific system being considered.