Final answer:
To solve Kepler's third law, calculate the square of the period (T²) which is proportional to the cube of the semi-major axis (a³) of the planet's orbit. The orbital period can be determined by rearranging the law as T = √(a³). This relationship can be visualized graphically, plotting the orbital periods of planets against their semi-major axis distances to compare with the theoretical Keplerian curve.
Step-by-step explanation:
To solve Kepler's third law of periods, we begin with the law's central statement that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis (a) of the orbit. Mathematically, we can express Kepler's third law as:
T² ≈ a³
In the case of planets orbiting the sun, we can refine this relationship further, representing the semi-major axis in astronomical units (AU) and the period in years. If we want to calculate the period for a planet, we rearrange this equation to solve for T:
T = √(a³)
For a hypothetical planet with a given semi-major axis, say a = 5 AU, we would compute the period as T = √(5³) = √(125) = 11.18 years approximately. To visualize this relationship for our solar system, we would plot orbital periods against their corresponding semi-major axis distances on a graph, usually on a logarithmic scale, and mark the position of each planet to observe how they align with the Keplerian curve predicted by this law.
Kepler's third law is significant as it applies universally, indicating that all objects with mass, regardless of their size, orbit at the same speed given a constant orbital radius. The period T represents the time it takes for one complete orbit, and the average orbital speed (υ) is given by the circumference of the orbit divided by T:
υ = (2πr) / T
Through this formulation, we can unearth the beautiful simplicity behind celestial mechanics, dictated by natural laws.