The period of revolution of planet A around t | vedclass

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(C) Let $T_A$ and $T_B$ be the time periods of planet $A$ and $B$ around the sun respectively. Given that $T_A = 8 T_B$.According to Kepler's third la

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$4$

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(C) Let $T_A$ and $T_B$ be the time periods of planet $A$ and $B$ around the sun respectively. Given that $T_A = 8 T_B$.According to Kepler's third law of planetary motion,the square of the time period is proportional to the cube of the semi-major axis (distance from the sun),i.e.,$T^2 \propto R^3$.Therefore,$\left(\frac{T_A}{T_B}\right)^2 = \left(\frac{R_A}{R_B}\right)^3$.Substituting the given values:$\left(\frac{8 T_B}{T_B}\right)^2 = \left(\frac{R_A}{R_B}\right)^3$$8^2 = \left(\frac{R_A}{R_B}\right)^3$$64 = \left(\frac{R_A}{R_B}\right)^3$Taking the cube root on both sides:$\frac{R_A}{R_B} = (64)^{1/3} = 4$.Thus,the distance of planet $A$ from the sun is $4$ times the distance of planet $B$ from the sun.

Column-$I$	Column-$II$
$(1)$ Kepler's first law	$(a)$ Law of periods
$(2)$ Kepler's second law	$(b)$ Law of orbits
$(3)$ Kepler's third law	$(c)$ Law of areas

The period of revolution of planet $A$ around the sun is $8$ times that of $B$. The distance of $A$ from the sun is how many times greater than that of $B$ from the sun?

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