On equilateral central configurations in the 1+4-body problem

Abstract

We investigate central configurations in the planar five-body problem with one dominant mass. The remaining four masses, referred to as coorbital satellites, are infinitesimal and positioned along a circle centered at the big mass. We focus on stacked relative equilibria in which the central body and two fixed satellites form an equilateral triangle, while the two remaining satellites occupy distinct positions on the unit circle. In the limiting case when the small masses tend to zero, the problem naturally divides into three scenarios depending on the location of these remaining bodies relative to the arc formed by the two fixed satellites. We show that the first case, in which both satellites lie inside the arc, cannot occur under the positivity constraint on the masses. The second case, where one satellite lies inside the arc and the other outside, admits solutions that we characterize in detail, while the third case, with both satellites outside the arc, leads to a richer family of admissible configurations.