Rosetta Klemperera
Rosetta Klemperera – the gravitational system of light and heavy bodies revolving in regularly repeating orbits around a common center of mass. It was first described by Wolfgang Klemperer in 1962. Klemperer described the system as follows: "This symmetry is also inherent in a peculiar family of geometric configurations that can be described as 'rosettes’. They contain an even number of „planets" of two (or more) types, one (or several) sets of which are heavier than others, and all the planets belonging to the same set (having the same mass) are located in the corners of two (or more) alternating regular polygons so that light and heavy alternate (or follow each other in a cyclical manner). " The simplest rosette will consist of a row of four alternating heavy and light bodies, located at an angular distance of 90 degrees from each other, in a rhombic configuration [тяжёлое, лёгкое, тяжёлое, лёгкое], moreover, two heavy bodies have the same mass, as well as two light bodies. The number of body types by mass can be increased as long as the arrangement remains cyclical: for example, [1,2,3 … 1,2,3 ], [ 1,2,3,4,5 … 1,2,3,4,5 ], [ 1,2,3,3,2,1 … 1,2,3,3,2,1 ]… Klemperer mentioned octagonal and rhombic rosettes. Inappropriate use The term „Klemperer rosette" (often misspelled: „Kemperer rosette") is often used to describe a configuration of three or more equal masses located at the vertices of an equilateral polygon that have the same angular velocity relative to their center of mass. Klemperer mentions such a configuration at the beginning of his article, but only as a representative of the already known set of systems in equilibrium, before describing the rosette itself. In Larry Niven’s novel „The Ringworld", the „fleet of worlds" by Pearson’s puppeteers are arranged in such a configuration (5 planets at the vertices of a pentagon), which Niven calls „Kemplerer’s rosette." This (possibly intentional) malformed spelling (and misuse) could be the source of such a misunderstanding. Another possible source of spelling distortion is the similarity between the names of Kemplerer and Johannes Kepler, who described the laws of planetary motion in the 17th century. Instability Simulation of this system (or simple linear analysis of perturbations) shows that such a system is certainly unstable: any deviation from the ideal geometric configuration causes oscillations that ultimately lead to the destruction of the system (in the original article, Klemperer also notes this fact). The result does not depend on whether there is empty space in the center of the rosette, or whether it revolves around a star. The explanation for the instability is that any tangential perturbation leads to the fact that one of the bodies approaches one of its neighbors and moves away from the other, as a result of which the force of attraction to the nearest neighbor becomes greater, and in relation to the distant one – less, as a result which causes the disturbed object to move towards the nearest neighbor, which increases the disturbance, and does not compensate for it. Radial perturbation directed inward leads to the fact that the perturbed body becomes closer to all other objects, as a result of which the force of their interaction and orbital velocity increase, which indirectly leads to tangential perturbation (the result of which is described above). Thus, the Puppeteers rosette described by Larry Niven would require artificial stabilization.