Newton's Law of Universal Gravitation

Isaac Newton's law of universal gravitation, published in his Principia Mathematica (1687), provides the physical explanation for all three of Kepler's laws.

The Law

Every object with mass (M) attracts every other object with mass (m) with a force (F) that is proportional to the product of their masses and inversely proportional to the square of the distance (r) between them. G is the gravitational constant (approximately 6.674 × 10⁻¹¹ N·m²/kg²).

How It Explains Kepler's Laws

• First Law: Newton showed that an inverse-square force law produces conic-section orbits. For bound objects (negative total energy), the orbit is an ellipse.
• Second Law: Any central force (one directed along the line between two bodies) conserves angular momentum, which leads directly to equal areas in equal times.
• Third Law: For two bodies interacting via inverse-square gravity, T² = (4π²/GM)a³ follows from the math. The constant of proportionality depends on the mass of the central body, which is why we can weigh planets with it.

For the step-by-step derivation, see Deriving Kepler's Laws from Newton.

Beyond Kepler

Newton's gravity also explains phenomena Kepler's laws alone cannot handle, such as orbital perturbations from other planets and the N-body problem.

See also: Kepler and Newton (historical context).