What is the relationship between a planet's orbital period and the length of its orbit according to Kepler's third law?

• A. The period is proportional to the square of the orbit's semimajor axis
• B. The period is inversely proportional to the length of the orbit
• C. The period is proportional to the length of the orbit
• D. The period is independent of the length of the orbit

Answer: (A)

Kepler's third law states that the square of a planet's orbital period (the time it takes to complete one full revolution around the sun) is directly proportional to the cube of the semi-major axis of its orbit. This can be mathematically described as ( T^2 \propto a^3 ), where ( T ) is the orbital period and ( a ) is the semi-major axis of the ellipse describing the orbit.

The correct relationship can be interpreted as the period being proportional to the square of the semi-major axis of the orbit. This means that if you were to increase the size of the orbit (the semi-major axis), the time it takes to go around the sun increases at a rate that is a function of the square of that increase. As such, this option effectively captures the essence of Kepler's third law and demonstrates the fundamental relationships inherent in celestial mechanics.

The other options do not align with Kepler's findings. For instance, the inverse proportionality or independence suggested in some options misrepresents the established relationship between orbital period and orbital dimensions as defined by Kepler's laws, which specifically emphasize a direct proportionality to the dimensions of the orbit, not just its length.