How did Newton use his laws of motion to explain Kepler’s elliptical planetary orbits?

Isaac Newton explained Johannes Kepler’s elliptical planetary orbits by applying his laws of motion and his law of universal gravitation to celestial mechanics.12

While Kepler’s laws empirically described planetary motion, they lacked an underlying physical explanation. Newton utilized his second law of motion—which relates force to the time rate of change of momentum—to demonstrate that a planet’s curved path requires a continuous, central force acting toward the Sun.13

By analyzing circular motion and incorporating Kepler’s third law, Newton determined that this centripetal force must vary inversely with the square of the distance ($1/r^2$) from the Sun. He further proved that a body moving under such an inverse-square central force follows an elliptical path, provided its initial velocity is within specific bounds. This synthesis allowed Newton to show that the same gravitational force governing objects on Earth also dictates the motion of planets, comets, and satellites throughout the universe.1345