Answer:
Given the equation:
\(T^2 =A^3\)
shows the relationship between a planet's orbital period T and the planet's mean distance from the sun, A in Astronomical units
then:
For planet X:
Orbital period is:
\(T_{X} = (A)^{\frac{3}{2}}\) .....[1]
As per the statement:
If planet Y is k times the mean distance from the sun as planet X.
⇒ A planet Y= kA mean distance from the sun as planet X.
then orbital period of Planet Y is:
\(T_{Y} = (kA)^{\frac{3}{2}}=k^{\frac{3}{2}}\cdot (A)^{\frac{3}{2}}\) ....[2]
Divide equation [2] by [1] we have;
\(\frac{T_{Y}}{T_{X}} = \frac{k^{\frac{3}{2}}\cdot (A)^{\frac{3}{2}}}{A^{\frac{3}{2}}}\)
Simplify:
\(\frac{T_{Y}}{T_{X}} =k^{\frac{3}{2}}\)
or
\(T_{Y} =k^\frac{3}{2} T_X\)
Therefore, the orbital period is increased by factor \(k^\frac{3}{2}\)
