Final answer:
Kepler's third law states that the ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distance from the sun. It can be derived from Newton's law of universal gravitation by equating the centripetal force to the gravitational force. Kepler's third law is useful in calculating the relative distances between objects orbiting the sun.
Step-by-step explanation:
Kepler's third law, also known as the law of harmonies, states that the ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distance from the sun. Mathematically, this can be expressed as T₁² / T₂² = ₁³ / ₂³, where T₁ and T₂ are the periods of the planets, and ₁ and ₂ are their average orbital distances from the sun. Kepler's third law can be derived from Newton's law of universal gravitation. By applying Newton's second law of motion to a planet in circular orbit around the sun, we can equate the centripetal force to the gravitational force and solve for the period and orbital radius. With some algebraic manipulation, we can then obtain the mathematical relationship expressed in Kepler's third law. For example, let's consider two planets with periods T₁ = 2 years and T₂ = 4 years, and average orbital distances ₁ = 1 AU and ₂ = 2 AU. Applying Kepler's third law, we have (2 years)² / (4 years)² = (1 AU)³ / (2 AU)³. Simplifying this equation, we get 1/4 = 1/8, which is true. Therefore, Kepler's third law holds for these two planets.