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| Keywords: Newtons proof of Keplers second law.gif Isaac Newton's proof of Kepler's second law of planetary motion At left a large yellow disk represents a star and at right the smaller blue disk represents a planet The blue arrow represents the planet's instantaneous orbital velocity Assuming no force acting upon it it will move in a straight line at a fixed rate After some fixed amount of time it will have reached a new position Then at this position an instantaneous centripetal force red arrow is acted upon the planet towards the star altering its path as shown with the addition of the blue and red arrows violet arrow The planet is shown traversing this new path as well as a shadow faint gray disk of the planet describing its trajectory during the same amount of time if the attractive force hadn't been acted upon it that is if it had remained in its original trajectory We have then three different positions for the planet and an alternate position given the lack of the centripetal force These four positions describe three triangles which share the star as one of their vertices We then select the first and second triangles We can see they both share the same length for a base red double arrow Tracing parallel lines we can see they also share the same height in orange Since the area of a triangle is given by <math>A \tfrac base \times height 2 </math> we conclude that both triangles have the same area A similar process is performed with the second and the third triangle showing that the area of the triangle defined by the altered trajectory is the same as the area of the unaltered trajectory The time interval can be made arbitrarily small until the instantaneous force can be considered acting continuously Therefore the line connecting the planet to the sun will always sweep an equal area as described by Kepler's second law own Lucas V Barbosa 2012-07-23 Animation for astrophysics Animations of physics Animations of Kepler motions Mathematical proof Isaac Newton | ||||
