Question

the equation ( t^{2}=a^{3} ) shows the relationship between a planets orbital period, ( t ), and the planets mean distance from the sun, ( a ), in astronomical units, au. if planet ( y ) is twice the mean distance from the sun as planet ( x ), by what factor is the orbital period increased? ( 2^{\frac{1}{3}} ) ( 2^{\frac{1}{2}} ) ( 2^{\frac{2}{3}} ) ( 2^{\frac{3}{2}} )

Explanation:

Step1: Set up equations for both planets

Let the mean - distance of planet \(X\) from the sun be \(A_X\) and its orbital period be \(T_X\), so \(T_X^{2}=A_X^{3}\). Let the mean - distance of planet \(Y\) from the sun be \(A_Y = 2A_X\) and its orbital period be \(T_Y\), so \(T_Y^{2}=A_Y^{3}=(2A_X)^{3}\).

Step2: Express \(T_Y\) in terms of \(T_X\)

From \(T_X^{2}=A_X^{3}\), we have \(A_X=T_X^{\frac{2}{3}}\). Substitute \(A_X=T_X^{\frac{2}{3}}\) into \(T_Y^{2}=(2A_X)^{3}\):

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Answer:

\(2^{\frac{3}{2}}\)