Continua of Central Configurations in the Newtonian n-body Problem with Negative Masses

A central configuration is an arrangement of masses in the Newtonian n-body problem in which the acceleration vector of any single mass points toward the center of mass and is proportional to its displacement from the center of mass with the same proportionality constant. Finding central configurations is useful in practical applications, such as placing a satellite in orbit. If the Earth, Sun, and satellite form a central configuration, then the satellite does not require any thrusters to remain in orbit. It is an open question if, given any collection of n > 4 positive masses, there exists only a finite number of central configurations (up to rotation, scaling, and translation). With the introduction of a negative mass, Gareth Roberts was able to construct a continuum of central configurations for the 5-body case. It seems unlikely that this is the only continuum that exists. We investigate other highly symmetric configurations to see if the same phenomenon may occur, or if Roberts' configuration has some unique property which allows a continuum to exist. In our search, we use techniques from algebraic geometry, such as Groebner bases.