Stability of some degenerate positions of relative equilibrium in then-body problem

We study, by a fully analytical method, the linear stability of a degenerate configuration of a relative equilibrium where n bodies having the same mass mare at the vertices of a regular polygon with n sides, while a body having a non-zero mass m0 is at the centre of the polygon. It is shown that the characteristic exponents are the roots of n polynomials of the fourth degree. The coefficients of these polynomials depend only on n and the ratio (μ = m0/m. Using this analytical result, we discuss the linear stability of the configuration according to the values of μ as a function of n. It is shown that when 3≤n≤6the instability holds for any value of μ. If n≥7the linear stability holds if μ is larger than some function of n which depends on n and the parity of n. This function can be approximated by 2n3/5 if n is very large and, according to the parity of nby a third degree polynomial of n if n has limited values