Central configurations of the collinear three-body problem and singular surfaces in the mass space

This Letter is to provide a new approach to study the phenomena of degeneracy of the number of the collinear central configurations under geometric equivalence. A direct and simple explicit parametric expression of the singular surface H3 is constructed in the mass space (m1,m2,m3) element of (R+)3. The construction of H3 is from an inverse respective, that is, by specifying positions for the bodies and then determining the masses that are possible to yield a central configuration. It reveals the relationship between the phenomena of degeneracy and the inverse problem of central configurations. We prove that the number of central configurations is decreased to 3!/2-1=2, m1, m2, and m3 are mutually distinct if m element of H3. Moreover, we know not only the number of central configurations but also what the nonequivalent central configurations are. -- Highlights: → Provide a new method to study the degeneracy of number of CC. → Results advanced the understanding of number of central configurations. → Singular mass surface H3 is given by a direct and simple parametric expression. → The proof only requires some basic knowledge of linear algebra. → The method can be applied to some other collinear n-body problem.