Some properties of rosette configurations of gravitating bodies in homographic equilibrium
Abstract
A particular case of the multibody problem which admits of simple solutions has been reexamined. It concerns hypothetical plane arrays of an even number of mutually gravitating bodies revolving about their common barycenter in dynamic equilibrium while maintaining the shape of their configuration (though not necessarily its size). The common property of this family of arrays is the existence of mirror symmetry about the radius vector from the barycenter to each of the massive bodies. Dynamic equilibrium is possible for limited ranges of rhombic configurations and of arrays in the shape of hexagonal and octagonal (and higher polygonal) regular rosettes. Such arrangements comprise alternating heavier bodies, all alike, and an equal number of lighter bodies, also all alike, in regularly alternating fashion. Relationships between permissible radius ratios and mass ratios exist such that, while the mass ratio may assume any positive value, the com patible radii can differ only within narrow limits. A sun body may or may not be located in the barycenter. The orbits are all circles, or, in general, conic sections of equal eccentricity with the barycenter at one focus. The influences of the mass and radius ratios upon the angular velocity of the orbital motion and upon per missible oscillations in and out of the mean orbital plane are delineated.
 Publication:

The Astronomical Journal
 Pub Date:
 April 1962
 DOI:
 10.1086/108686
 Bibcode:
 1962AJ.....67..162K