Which statement correctly states Keplers Third Law of Planetary Motion?

Answer 1

Kepler's laws cannot be proven. They are empirical laws not derived from axioms.

The answer above is nonsense.

Yes Kepler could not prove his laws and had only derived them empirically. But after Newton formulated his Universal law of gravitation, Kepler's laws could and have been proven.

The working is quite cumbersome and cannot be repeated in this forum. But if you Google "Keplers Laws Proof" or similar you will find numerous explanations. But be prepared that complete proofs require quite some math skills. To do it rigorously you need Calculus and Vector Algebra.

In essence, the first two laws are equivalent to the law of conservation of Angular Momentum (which is the cross product of the position and velocity vectors). The third law calculates the orbital period as a function of semi-major axis of the ellipse and the central mass, basically defining elliptic motion.

It is amazing that Kepler was able to formulate these laws empirically (mainly by studying the motion of Mars) without having the physics and mathematics he would have needed, at his disposal.

Answer 2

Kepler's third law of planetary orbits states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.