Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit (which can be considered as the orbital radius for circular orbits). Mathematically, this can be expressed as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]
where \(T_1\) and \(T_2\) are the periods of the two planets and \(r_1\) and \(r_2\) are their respective average orbital radii.
From this, we can deduce that:
The square of the ratio of the periods equals the cube of the ratio of the radii.
So, the correct response to how Kepler’s third law compares the periods and orbital radii of two planets is:
The square of the ratio of the periods equals the cube of the ratio of the radii.