Kepler's Laws Of Planetary Motion

NCERT Class 11, Chapter 8 (Gravitation) states Kepler's third law: the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit, written $T^2\propto r^3$. This empirical law, later explained by Newton's gravitation, lets us compare orbits using only distance ratios. Taking Earth as the reference with $T=1$ year and $r=1$ astronomical unit, the relation becomes ${\left(\frac{T_a}{T_E}\right)}^2={\left(\frac{r_a}{r_E}\right)}^3$. Substituting $\frac{r_a}{r_E}=4$ gives $T_a^2=4^3=64$, so $T_a=\sqrt{64}=8$ years. The option $4$ years assumes the period scales linearly with distance, ignoring the law altogether. The option $16$ years uses $4^2$, confusing the cube and the square. The option $64$ years forgets to take the square root of the period-squared value. As a plausibility check, a more distant orbit must have a longer period, and Kepler's law correctly predicts that the asteroid takes several times longer than Earth to circle the Sun, consistent with the eight-year result.