Isochrone Mechanics

Abstract

After years of meticulous analysis of Tycho Brahe’s observations of Mars, Johannes Kepler discovered, at the beginning of the seventeenth century, three laws that now bear his name. The first two, published in the Astronomia Nova (1609), state that (1) planets follow elliptical orbits with the Sun at one of their focii, and (2) the time elapsed between two points on an orbit is proportional to the swept area in that time. Kepler’s third law, was published later in his Harmonices Mundi (1619). It states that the cube of the semi-major axis of these ellipses is proportional to the square of the planet’s orbital period. This chapter is dedicated to a generalization of Kepler’s laws to all isochrone orbits. In particular, we will show that any periodic orbit in any isochrone potential satisfies Kepler’s third law (Sect. 9.1) for the radial period T. We will interpret this law in various geometrical contexts and will also point out a similar and unified law for the apsidal angle $$\Theta $$ . In Sect. 9.2, we devise a geometrical method for solving the equations of motion in an isochrone potential, and show that all isochrone orbits can be parametrized by a Keplerian ellipse. Finally, we shall use all these results to exhibit and classify isochrone orbits in Sect. 9.3.