Why does an inverse-square gravitational force help explain planetary orbits?

In the context of Newton’s laws of motion, an inverse-square gravitational force is essential for explaining planetary orbits because it mathematically aligns with Kepler’s third law of planetary motion.

According to Newton’s second law, a planet requires a centripetal force to curve its path toward the Sun, preventing it from moving in a straight line due to its inertia. By analyzing the motion of objects in circular orbits, Newton demonstrated that if the periods of revolution vary as $r^{3/2}$—as Kepler’s third law dictates—the required acceleration and force must vary inversely with the square of the distance ($1/r^2$).12

Newton showed that this gravitational force acts as if all the mass of a spherical body is concentrated at its center. This inverse-square relationship allows for stable elliptical orbits, as well as parabolic or hyperbolic paths depending on the object's initial velocity and energy. Thus, the inverse-square law provides the physical mechanism that governs the motion of planets, satellites, and other celestial bodies.13