Kepler′s Ellipse Observed from Newton′s Evolute (1687), Horrebow′s Circle (1717), Hamilton′s Pedal Curve (1847), and Two Contrapedal Curves (28.10.2018)

摘要

Johannes Kepler discovered the very elegant elliptical path of planets with the Sun in one focus of that ellipse in 1605. Kepler inspired generations of researchers to study properties hidden in those elliptical paths. The visible elliptical paths belong to the Aristotelian World. On the other side there are invisible mathematical objects in the Plato's Realm that might describe the mechanism behind those elliptical paths. One such curve belonging to the Plato's Realm discovered Isaac Newton in 1687 - the locus of radii of curvature of that ellipse (the evolute of the ellipse). Are there more curves in the Plato's Realm that could reveal to us additional information about Kepler's ellipse? W.R. Hamilton in 1847 discovered the hodograph of the Kepler's ellipse using the pedal curve with pedal points in both foci (the auxiliary circle of that ellipse). This hodograph depicts the moment of the tangent momentum of orbiting planets. Inspired by the hodograph model we propose newly to use two contrapedal curves of the Kepler's ellipse with contrapedal points in both the Kepler's occupied and Ptolemy's empty foci. Observers travelling along those contrapedal curves might bring new valuable experimental data about the orbital angular velocity of planets and a new version of the Kepler's area law. Based on these contrapedal curves we have defined the moment of the normal momentum. The first derivation of the moment of the normal momentum reveals the torque of the ellipse. This torque of ellipse should contribute to the precession of the Kepler's ellipse. In the Library of forgotten works of Old Masters we have re-discovered the Horrebow's circle (1717) and the Colwell's anomaly H (1993) that might serve as an intermediate step in the solving of the Kepler's Equation (KE). Have we found the Arriadne's Thread leading out of the Labyrinth or are we still lost in the Labyrinth?