Thomas' Labs



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Deriving Kepler's first law from Newton's law of universal gravitation

The movement of an object with mass orbit_f03b7dc5a0e96ecc5a2f45684c55ab32e0edfbdf.svg orbiting another body with mass orbit_f0026ac69fc3f17756c6dba52187d58b5bf8a948.svg is given by Newton's law of gravitation. If orbit_25bfa51a2ca3fc256cd77b7bd33bde0a3f4bee7a.svg it is possible to consider the position of the larger object constant and use it as the origin of our coordinate system. Then the following equation applies for movement of the smaller object:


In order to solve this differential equation we first consider the angular momentum of or object around its orbit.


In the abscense of external toques, because the only force acting on the object is parallel to its position, the angular momentum is conserved.


We now multiply both sides of our equation from the right by the angular momentum and develop the right side of the equation using vector identities.


We observe that each side of our equation is a derivative of a quantity. We know integrate both sides and take the integrations constant into account.


Our objective is now to solve the equation for orbit_aa83e070fb0c21027a38fc424a7839ed87c71ff3.svg, so we multiply both sides by orbit_08f277a5cd4bf617be804896899e032e50c53f48.svg:


By applying a cyclic permutation of the resulting triple product and using the known property of the scalar product we now express the equation only in terms of the magnitudes of the vectors.


The last steps are to solve for orbit_aa83e070fb0c21027a38fc424a7839ed87c71ff3.svg


Finally we reach our result. Objects orbiting according to Newton's Law of Gravitation follow paths that correspond to the conic sections. Here is Kepler's first Law a special case, where our object has a stable orbit around the larger body.


Deriving a physical interpretation of the excentricity of the orbit