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How did Newton’s laws explain Kepler’s laws of planetary motion?

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A diagram with labeled vector illustrations. Panel A shows vector addition with two green arrows A and B forming a parallelogram, and their sum C as a red diagonal. Panel B shows vector subtraction, with A and negative B forming resultant D. Panel C demonstrates vector components using angles and projections. Panel D uses the right-hand rule with thumbs to show the direction of the cross product E = A x B and its opposite, -E = B x A.
mechanics Figure 1: (A) The vector sum C = A + B = B + A. (B) The vector difference A + (−B) = A − B = D. (C, left) A cos θ is the component of A along B and (right) B cos θ is the component of B along A. (D, left) The right-hand rule used to find the direction of E = A × B and (right) the right-hand rule used to find the direction of −E = B × A.

Isaac Newton explained Kepler’s laws of planetary motion by establishing the foundations of classical mechanics and the law of universal gravitation. While Kepler’s empirical laws described how planets move, they lacked an underlying physical explanation; Newton provided this by defining force and momentum through his three laws of motion.1

Newton demonstrated that a central gravitational force varying as the inverse square of the distance (1/r2) accounts for these motions:1

  • Kepler’s First Law: Newton showed that an object subject to an inverse-square central force follows a path defined by a conic section. Depending on the body's total energy, this path can be an ellipse, a parabola, or a hyperbola.231
  • Kepler’s Second Law: This law, which states that a planet sweeps out equal areas in equal times, is a direct consequence of the conservation of angular momentum in a central force field.23
  • Kepler’s Third Law: By applying his second law of motion to circular orbits, Newton showed that if the period of revolution varies as r3/2, the required centripetal force must vary as 1/r2, consistent with his law of gravitation.1