Kepler's 3rd law is mathematically represented as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]
This implies that the square of the period of one planet (\(T_1\)) divided by the square of the period of another planet (\(T_2\)) is equal to the cube of their average distances from the sun (\(r_1\) and \(r_2\)).
To condense this to a simpler format related to ratios:
\[ T^2 \propto r^3 \]
This means that \(T^2\) (the square of the period) is proportional to \(r^3\) (the cube of the distance).
However, none of the options provided match this relationship exactly when representing the proportionality and specific usage of constants.
The correct choice that aligns with the notion that the square of the period relates to the cube of the distance is:
A. \(T^2 = k \cdot r^3\) (where \(k\) is a constant, but that option is not listed)
Considering your options: A. \(T^2 = r^3\) closely resembles the relationship but is not complete as it doesn't express the proportionality or the necessary constant.
So, if interpreting the information purely based on the options and their closest relation to Kepler's Law, A. \(T^2 = r^3\) would be the most fitting answer, though it misses the constant aspect.
3 answers