Question

how does kepler’s third law compare the periods and orbital radii of two planets within a solar system? (1 point) the square of the ratio of the periods equals the cube of the ratio of the radii. the cube of the ratio of the periods equals the square of the ratio of the radii. the ratio of the periods equals the square of the ratio of the radii. the square of the ratio of the periods equals the ratio of the radii.

Explanation:

Kepler's third law (harmonic law) states that for two planets orbiting the same star (in a solar system), the relationship between their orbital periods (\(T\)) and orbital radii (\(r\)) is given by \(\frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}\), which can be rephrased as the square of the ratio of the periods equals the cube of the ratio of the radii. Let's analyze each option:

• Option 1: Matches the statement of Kepler's third law.
• Option 2: Incorrectly swaps the exponents (cube of period ratio and square of radius ratio) which goes against the law.
• Option 3: Does not follow the cubic relationship for radii and square for periods as per the law.
• Option 4: Incorrectly relates the square of period ratio to the simple ratio of radii, ignoring the cubic dependence of radius.

Answer:

The square of the ratio of the periods equals the cube of the ratio of the radii.