11. Kern E. Kenyon, Kepler's First Law: A Short Derivation

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Volume 6: Pages 564-566, 1993

Kepler's First Law: A Short Derivation

Kern E. Kenyon 1

14632 North Lane, Del Mar, California 92014 U.S.A.

Kepler's first law is derived analytically by combining two nonlinear equations in four unknowns: (1) conservation of angular momentum and (2) the force balance for the revolving mass in the orbital plane between the outward centrifugal force and the inward gravitational force component normal to the orbit. This derivation, which is shorter and hopefully clearer than existing ones, does not involve conservation of energy or integration of differential equations. The derivation also results in a simpler form of the orbit equation: R sin3psgr; = const, where R is the radius of curvature of the orbit, ψ is the angle between the radius and velocity vectors of the orbiting mass, and there is only one constant.

Keywords: Kepler's first law, planetary motion, angular momentum, centrifugal force

Received: May 6, 1992; Published Online: December 15, 2008