the equation (t^{2}=a^{3}) shows the relationship between a planets orbital period, (t), and the planets…

Explanation:

Step1: Define variables for planets X and Y

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

Step2: Use the given relationship between periods

We know that $T_Y = 2T_X$. Substitute $T_Y$ into the equation $T_Y^2 = A_Y^3$: $(2T_X)^2=A_Y^3$.

Step3: Expand the left - hand side

$(2T_X)^2 = 4T_X^2$. Since $T_X^2 = A_X^3$, we can rewrite the equation as $4A_X^3 = A_Y^3$.

Step4: Solve for the ratio of distances

We want to find $\frac{A_Y}{A_X}$. From $4A_X^3 = A_Y^3$, we can get $A_Y^3/A_X^3 = 4$. Then $\left(\frac{A_Y}{A_X}\right)^3=4$, and $\frac{A_Y}{A_X}=4^{\frac{1}{3}}=(2^2)^{\frac{1}{3}} = 2^{\frac{2}{3}}$.

Answer:

$2^{\frac{2}{3}}$