Kepler Problem and Orbits

The Kepler problem is the motion of a body under an attractive inverse-square force, whose bound solutions are the ellipses that describe planetary orbits.

Definition

The Kepler problem is the central-force problem for an attractive force varying as the inverse square of distance, whose orbits are conic sections with the center of force at a focus and whose bound orbits obey Kepler's laws.

Scope

This topic covers the solution of the inverse-square central-force problem: the conic-section orbits (ellipse, parabola, hyperbola) classified by energy, Kepler's three laws of planetary motion, the orbital elements, and the special conserved Laplace-Runge-Lenz vector responsible for the closure and non-precession of bound orbits in a pure inverse-square field.

Core questions

• Why does the inverse-square force produce conic-section orbits classified by energy?
• What do Kepler's three laws state, and how do they follow from the force law?
• What is special about the inverse-square force that keeps bound orbits closed?

Key concepts

• Inverse-square force
• Conic-section orbits
• Kepler's three laws
• Orbital elements (eccentricity, semi-major axis)
• Laplace-Runge-Lenz vector
• Orbital energy and bound/unbound classification

Key theories

Conic-section orbits and Kepler's laws

Bound motion in an inverse-square attraction is an ellipse with the center of force at a focus, sweeping equal areas in equal times, with orbital period squared proportional to semi-major axis cubed.

Laplace-Runge-Lenz vector

The inverse-square force possesses an extra conserved vector that points along the orbit's major axis, explaining why bound Kepler orbits are exactly closed and do not precess.

Clinical relevance

The Kepler solution is the backbone of orbital mechanics for planets, moons, comets, and artificial satellites, underpinning mission design, orbit determination, and transfer maneuvers, while small deviations from pure inverse-square behavior provided early tests of general relativity.

History

Kepler deduced his three empirical laws of planetary motion from Tycho Brahe's observations in the early 1600s, and Newton showed in the 1687 Principia that they follow from an inverse-square law of universal gravitation. The additional conserved vector now associated with Laplace, Runge, and Lenz explained the special degeneracy that keeps Kepler orbits closed.

Key figures

• Johannes Kepler
• Isaac Newton
• Pierre-Simon Laplace

Seminal works

• newton1687
• taylor2005

Frequently asked questions

Why are planetary orbits ellipses rather than other shapes?

Bound motion under an attractive inverse-square force always traces a conic section, and the bound case is specifically an ellipse with the attracting body at one focus, exactly as Kepler observed.

Why do real planetary orbits precess slightly?

A pure inverse-square force gives perfectly closed orbits, but perturbations from other planets and relativistic corrections break that special symmetry, causing the orbit's axis to slowly rotate.