Uniqueness of convex central configurations in the planar (1 + 4)-body problem
Abstract
We study strictly convex central configurations of the planar Newtonian five-body problem with one dominant mass and four satellites of arbitrary positive masses. We prove that a single positive bound on the total satellite-to-primary mass ratio guarantees existence and uniqueness for every prescribed directed order of the five bodies on the convex hull, up to translations, rotations and positive scaling. The bound is independent of all satellite mass ratios, including ratios arbitrarily close to the boundary of the mass simplex. The constrained Hessian is nondegenerate on the six-dimensional shape space. We first prove uniqueness and nondegeneracy for the convex coorbital limit by an angular Hessian estimate. We then classify the possible collision limits as the satellite mass ratios vary and construct regularized local continuations for pairs, triples and pairs within triples. An exact factorization of the signed areas preserves strict convexity on independent parameter neighborhoods. These continuations connect every sufficiently small-mass configuration to the unique coorbital configuration and yield the common bound.
Related
- Uniqueness of four-body convex central configurations for all masses
- Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem
- Finiteness of planar central configurations of five bodies with positive masses
- Planar Central Configurations of Four Equal Masses: Classification and Morse Indices
- Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits
BibTeX
@misc{xiao2026uniqueness,
title = {{Uniqueness of convex central configurations in the planar (1 + 4)-body problem}},
author = {Kaitai Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04563},
eprint = {2610.04563},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04563},
}
RIS
TY - UNPB
TI - Uniqueness of convex central configurations in the planar (1 + 4)-body problem
AU - Kaitai Xiao
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04563
UR - https://arxiv.org/abs/2610.04563
AB - We study strictly convex central configurations of the planar Newtonian five-body problem with one dominant mass and four satellites of arbitrary positive masses. We prove that a single positive bound on the total satellite-to-primary mass ratio guarantees existence and uniqueness for every prescribed directed order of the five bodies on the convex hull, up to translations, rotations and positive scaling. The bound is independent of all satellite mass ratios, including ratios arbitrarily close to the boundary of the mass simplex. The constrained Hessian is nondegenerate on the six-dimensional shape space. We first prove uniqueness and nondegeneracy for the convex coorbital limit by an angular Hessian estimate. We then classify the possible collision limits as the satellite mass ratios vary and construct regularized local continuations for pairs, triples and pairs within triples. An exact factorization of the signed areas preserves strict convexity on independent parameter neighborhoods. These continuations connect every sufficiently small-mass configuration to the unique coorbital configuration and yield the common bound.
ER -