Sure, let's walk through this step-by-step to find the correct relationship according to Kepler's third law.
Kepler's third law states that the square of the period of revolution of a planet ([tex]$T^2$[/tex]) is directly proportional to the cube of the semi-major axis of its orbit, which we can denote as the mean distance ([tex]$R$[/tex]) from the Sun. This relationship can be written mathematically as:
[tex]\[ T^2 \propto R^3 \][/tex]
This means that if you take the period of revolution [tex]$T$[/tex] and square it, it will be proportional to the cube of the mean distance [tex]$R$[/tex]. In other words, if you double the mean distance [tex]$R$[/tex], then the period [tex]$T$[/tex] will increase by a factor such that the square of the new period corresponds to the cube of the new distance.
Given the options:
A. [tex]\( R \)[/tex]
B. [tex]\( R^2 \)[/tex]
C. [tex]\( R^3 \)[/tex]
D. [tex]\( R^4 \)[/tex]
E. [tex]\( R^5 \)[/tex]
The correct relationship, as stated by Kepler's third law, is:
[tex]\[ T^2 \propto R^3 \][/tex]
Therefore, the correct answer is:
C. [tex]\( R^3 \)[/tex]
