Klemperer rosette

A Klemperer rosette is a gravitational system of a heavier and lighter bodies orbiting in a regular repeating pattern around a common barycenter. It was first described by W. B. Klemperer in 1962. [cite journal| journal=Astronomical Journal| volume=67| issue= 3| month=April| year= 1962| pages=162–7| url=http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1962AJ.....67..162K|title= Some Properties of Rosette Configurations of Gravitating Bodies in Homographic Equilibrium|doi= 10.1086/108686|author= Klemperer, W. B.]

Klemperer described the system as follows:

"Such symmetry is also possessed by a peculiar family of geometrical configurations which may be described as 'rosettes'. In these an even number of 'planets' of two (or more) kinds, one (or some) heavier than the other, but all of each set of equal mass, are placed at the corners of two (or more) interdigitating regular polygons so that the lighter and heavier ones alternate (or follow each other in a cyclic manner)"

The simplest rosette would be series of four alternating heavier and lighter bodies, 90 degrees from one another, in a rhombic configuration [Heavy, Light, Heavy, Light] , where the two heavier masses weigh the same, and likewise the two lighter masses weigh the same. The number of "mass types" can be increased, so long as the arrangement pattern is cylic: 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 also mentioned octagonal and rhombic rosettes.

Misuse and misspelling

The term "Klemperer rosette" (often mis-spelled "Kemplerer" rosette") is often used to mean a stable configuration of three or more equal masses, set at the points of an equilateral polygon and given an equal angular velocity about their center of mass.

Klemperer does indeed mention this configuration at the start of his article, but only as an already known set of stable systems before introducing the actual rosettes.

In Larry Niven's novel "Ringworld", the Puppeteers' "Fleet of Worlds" is arranged in such a configuration (5 planets spaced at the points of a pentagon) which Niven calls a "Kemplerer rosette"; this mis-spelling (and misuse) is probably the main source of confusion. Simulations of this system [cite web| url=http://burtleburtle.net/bob/physics/kempler.html| title=Klemperer Rosettes| first=Bob| last= Jenkins| accessdate=2007-01-12] (or a simple linear perturbation analysis) demonstrate that such systems are definitely not stable, as any motion away from the perfect geometric configuration causes an oscillation eventually leading to the disruption of the system. (Klemperer's original article also states this fact.) This is the case whether or not the center of the Rosette is in free space, or itself in orbit around a star. The short-form reason is because any perturbation destroys the symmetry. The longer explanation is that any tangential perturbation causes a body to get closer to one neighbor and farther from another; the gravitational force becomes greater towards the closer neighbor and less for the farther neighbor, thus further pulling the perturbed object towards its closer neighbor, enhancing the perturbation rather than damping it. An inward radial perturbation causes theperturbed body to get closer to all other objects, increasing the force on the object and increasing its orbital velocity---which leads indirectly to a tangential perturbation and the argument above. (A similar argument is used to demonstrate the orbital instability of the Ringworld itself.) Thus, there is no doubt that the Puppeteers's Rosette of farming worlds, as described in Niven's Ringworld, would require artificial stabilizing, as does the Ringworld itself.

References

External links

* [http://burtleburtle.net/bob/physics/kempler.html Rosette simulations using Java applets]
*http://www.technovelgy.com/ct/content.asp?Bnum=605