According to Kepler's third law, T is proportional to which power of the semi-major axis a when the central body's gravity parameter μ is constant?

According to Kepler's third law, T is proportional to which power of the semi-major axis a when the central body's gravity parameter μ is constant?

• A. T is proportional to a
• B. T is proportional to a^2
• C. T is proportional to a^(3/2)
• D. T is proportional to 1/a

Answer: (C)

The relation you’re testing is that the orbital period grows with the size of the orbit in a specific way: for a central body with constant gravity parameter μ, the period squared is proportional to the semi-major axis cubed. In formula form, T^2 = (4π^2/μ) a^3, which means T = 2π sqrt(a^3/μ). Since μ is fixed, T scales as a^(3/2). So larger orbits take longer, and the increase follows the 3/2 power, not linearly or more steeply. This is why doubling the semi-major axis makes the period increase by 2^(3/2) ≈ 2.83, not by a factor of 2 or 4.

The relation you’re testing is that the orbital period grows with the size of the orbit in a specific way: for a central body with constant gravity parameter μ, the period squared is proportional to the semi-major axis cubed. In formula form, T^2 = (4π^2/μ) a^3, which means T = 2π sqrt(a^3/μ). Since μ is fixed, T scales as a^(3/2). So larger orbits take longer, and the increase follows the 3/2 power, not linearly or more steeply. This is why doubling the semi-major axis makes the period increase by 2^(3/2) ≈ 2.83, not by a factor of 2 or 4.