Some results are provided in support of Moeckel’s dominant mass conjecture [R. Moeckel, J. Dyn. Differ. Equations 6, No. 1, 37-51 (1994; Zbl 0793.70008)] that any linearly stable relative equilibrium contains a mass much larger than the others. One result is that for n≥24,306, any equal mass relative equilibrium is spectrally unstable. By adding masses of smaller order in the appropriate places,
... [Show full abstract] we also show that it is possible to thicken the ring around the (1+n)-gon relative equilibrium and to maintain its linear stability.