Question

1. what is the relationship between a planets mean distance from the sun and its period of revolution?

Explanation:

The relationship between a planet's mean distance (\(a\), in astronomical units) from the Sun and its period of revolution (\(T\), in years) is described by Kepler's Third Law of Planetary Motion. The law states that the square of the orbital period of a planet is proportional to the cube of the semi - major axis (mean distance) of its orbit. Mathematically, it is expressed as \(T^{2}=a^{3}\) (for our solar system, when \(T\) is in years and \(a\) is in astronomical units). This means that as a planet's mean distance from the Sun increases, its period of revolution (the time it takes to complete one orbit around the Sun) also increases, and the relationship follows the cubic - square proportionality given by Kepler's Third Law.

Answer:

The relationship is given by Kepler's Third Law of Planetary Motion, which states that the square of a planet's period of revolution (\(T\)) around the Sun is proportional to the cube of its mean distance (\(a\)) from the Sun, mathematically expressed as \(T^{2}\propto a^{3}\) (or \(T^{2}=ka^{3}\), where \(k\) is a constant; for the solar system with \(T\) in years and \(a\) in astronomical units, \(k = 1\) and \(T^{2}=a^{3}\)). In general terms, as the mean distance of a planet from the Sun increases, its period of revolution also increases, following the cubic - square proportionality.