Central configurations on the plane with N heavy and k light bodies

The paper proves a continuation theorem for planar central configurations with N heavy masses and k light masses scaling as µj = Θ µ̃j. Its Theorem 11 asserts existence and asymptotics qi = q̂i + O(Θ), pj = x* + Θ^{1/3} p̂j + O(Θ^{2/3}), built via a reduced system RS and an implicit-function argument in the scaled parameter δ = Θ^{1/3}, using the induced potential W̃ and a block-triangular Jacobian (see the definition of W̃ and the statement/proof sketch including (66)) . The candidate solution reaches the same conclusion by a Lyapunov–Schmidt-style project–rescale formulation, enforcing a weighted projection to remove the cluster center drift and a phase condition to kill SO(2), then applying the IFT. The only notable difference is that the model records x(Θ) − x* = O(Θ^{2/3}) (weaker) whereas the paper sharpens this to O(Θ); otherwise the constructions, nondegeneracy hypotheses, and asymptotics agree. The proofs are substantively equivalent though organized differently.