Klemperer rosette

AKlemperer rosetteis agravitationalsystem of (optionally) alternating heavier and lighter bodies orbiting in a symmetrical pattern around a commonbarycenter. It was first described byW.B. Klempererin 1962,and is a special case of acentral configuration.
Klemperer described rosette systems as follows:
The simplest rosette would be a series of four alternating heavier and lighter bodies, 90 degrees from one another, in arhombicconfiguration [Heavy, Light, Heavy, Light], where the two larger bodies have the same mass, and likewise the two smaller bodies have the same mass, all orbiting their (empty) geometric center. The more generaltrojan systemhas unequal masses for the two heavier bodies, which Klemperer also calls a "rhombic" system, and is the only version that is not symmetric around the gravitational center.
The number of "mass types" can be increased, so long as the arrangement is symmetrical and cyclic pattern: e.g. [ 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 ], etc.
Klemperer's article specifically analyzes regular polygons with 2–9 corners –dumbbell-shapedthroughnonagon– and non-centrally symmetric "rhombicrosettes" with three orbiting bodies, the outer two stationed at the middle orbiting body'striangular points(L4 and L5), which had already been described and studied byLagrangein 1772.Systems with an even number of 4 or more corners can have alternating heavy and light masses at the corners, though the possible range of mass ratios is constrained by para-stability requirements; systems with odd numbers of corners must have equal masses at every corner.
While Klemperer notes that all the rosettes and the rhombus are vulnerable to destabilization, thehexagonalrosette is the most nearly stable because the "planets" sit in each other's semi-stable triangularLagrangian points, L4 and L5.
The regular polygonal configurations ("rosettes") do not require a central mass (a "sun" at the center is optional, and if present it may bobble above and below the orbital plane), although aLagrange-typerhombus does. If a central body is present, its mass constrains the ranges for the mass-ratio between the orbiting bodies.
The term "Klemperer rosette" (often misspelled "Kemplererrosette") is used to mean a configuration of three or more equal masses, set at the points of anequilateralpolygonand given an equalangular velocityabout theircenter of mass. Klemperer does indeed mention this configuration at the start of his article, but only as an already known set of equilibrium systems before introducing the actual rosettes.
InLarry Niven's novelFleet of Worldsin theRingworldseries, thePuppeteers'eponymous"Fleet of Worlds" is arranged in such a configurationthat Niven calls a "Kemplerer rosette"; this (possibly intentional) misspelling is one viable source of the wider confusion. These fictional planets were maintained in position by large engines, in addition to gravitational force.
Both simple linear perturbation analysis and simulations of rosettesdemonstrate that such systems are unstable: Klemperer explains in his original article, any displacement away from the perfectly symmetrical geometry causes a growing oscillation, eventually leading to the disruption of the system.The system is unstable regardless of whether the center of the rosette is in free space, or is in orbit around a central star.
The short-form reason for the instability is that any perturbation corrupts the geometric symmetry, which increases the perturbation, and further undermines the geometry, and so on. The longer explanation is that any tangential perturbation brings a body closer to one neighbor and further from another; the gravitational imbalance becomes greater towards the closer neighbor and less for the further neighbor, pulling the perturbed object more towards its closer neighbor, amplifying the perturbation rather than damping it. An inward radial perturbation causes the perturbed body to get closer toallother objects, increasing the force on the object and increasing its orbital velocity, which leads indirectly to a tangential perturbation and the argument above.