Kepler's Third Law

Kepler's third law states that the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit, written as $T^2\propto a^3$. Using the Earth's orbit as the reference where $a=1$ AU gives $T=1$ year, the constant of proportionality becomes one in these units, so $T^2=a^3$. For $a=4$ AU, $T^2=4^3=64$, giving $T=\sqrt{64}=8$ years. The option of 16 years wrongly squares the distance instead of taking the three-halves power. The option of 64 years mistakes the period for $a^3$ itself, forgetting the square root. The option of 4 years assumes a linear relation between period and distance. This is exactly the NCERT and historical Kepler relationship governing planetary motion, which Newton later derived from his universal law of gravitation combined with the centripetal requirement of circular orbits. The proportionality constant $\frac{4{\pi }^2}{GM}$ depends only on the mass of the central star, so the identical numerical relationship holds for every planet orbiting that star, which is precisely why expressing distances in astronomical units and periods in years makes the constant equal to one. This universality is what allows astronomers to deduce orbital periods purely from measured distances. A plausibility check confirms that a more distant planet must take longer to orbit, and the $T=a^{3/2}$ scaling gives a sensible eight-year period for a fourfold-larger orbit.