How does Newton’s law of gravity relate to Kepler’s laws of planetary motion?

While Johannes Kepler’s empirical laws of planetary motion accurately described how planets move, they lacked a physical explanation for the underlying forces involved. Isaac Newton provided this foundation in the late 17th century by demonstrating that Kepler’s laws are mathematically derivable from his own laws of motion and his law of universal gravitation.123

The relationship between these laws is defined by the following principles:

• Force and Acceleration: Newton’s second law establishes that a force is required to change an object's motion. By analyzing Kepler’s third law, Newton determined that the force pulling planets toward the Sun must vary inversely with the square of the distance ($1/r^2$).13
• Universal Gravitation: Newton concluded that this attractive force is the same gravity that acts on objects on Earth. He formulated the law of universal gravitation, stating that the force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.12
• Orbital Paths: Using these laws, Newton showed that planets follow elliptical paths (Kepler’s first law) as a specific case of motion under a central force. He further demonstrated that objects can also follow parabolic or hyperbolic paths depending on their total energy.145
• Conservation: Kepler’s second law, which states that a planet sweeps out equal areas in equal times, is a direct consequence of the conservation of angular momentum.5

By unifying celestial and terrestrial physics, Newton transformed planetary theory into a branch of physics.23