How does Kepler’s third law compare the periods and orbital radii of two planets within a solar system?(1 point)
The square of the ratio of the periods equals the ratio of the radii.
The square of the ratio of the periods equals the ratio of the radii.
The ratio of the periods equals the square of the ratio of the radii.
The ratio of the periods equals the square of the ratio of the radii.
The square of the ratio of the periods equals the cube of the ratio of the radii.
The square of the ratio of the periods equals the cube of the ratio of the radii.
The cube of the ratio of the periods equals the square of the ratio of the radii.
The square of the ratio of the periods equals the cube of the ratio of the radii.
The square of the ratio of the periods equals the cube of the ratio of the radii.
The correct answer is: The square of the ratio of the periods equals the cube of the ratio of the radii.
Kepler's third law, also known as the law of harmonies, mathematically relates the periods (T) and orbital radii (R) of two planets within a solar system. It states that the ratio of the square of the periods is equal to the cube of the ratio of the radii.
Mathematically, this can be written as:
(T₁/T₂)² = (R₁/R₂)³
In this equation, T₁ and T₂ are the periods of the two planets, and R₁ and R₂ are their respective orbital radii.
The key idea behind this law is that the period of a planet (or any object) is directly proportional to the radius of its orbit. Furthermore, the relationship between the two is not linear, but rather a power law relationship. Specifically, the period scales with the orbital radius raised to the three-halves power (T ∝ R^(3/2)).
To understand this relationship, you need to know that the gravitational force between two objects decreases with distance. As a result, planets farther from the sun experience weaker gravitational forces and need more time to complete their orbits.
Knowing Kepler's third law allows astronomers to calculate the periods and orbital radii of planets within a solar system if they know only one of these values for a particular planet. By comparing the periods and orbital radii of different planets, astronomers can gain insights into the overall structure and dynamics of a solar system.