Spiderweb central configurations

In this paper we study spiderweb central configurations for the $N$-body problem, i.e configurations given by $N=n \times \ell+1$ masses located at the intersection points of $\ell$ concurrent equidistributed half-lines with $n$ circles and a central mass $m_0$, under the hypothesis that the $\ell$ masses on the $i$-th circle are equal to a positive constant $m_i$; we allow the particular case $m_0=0$. We focus on constructive proofs of the existence of spiderweb central configurations, which allow numerical implementation. Additionally, we prove the uniqueness of such central configurations when $\ell \in \{2,\dots,9\}$ and arbitrary $n$ and $m_i$; under the constraint $m_1\geq m_2\geq \ldots \geq m_n$ we also prove uniqueness for $\ell \in \{10,\dots,18\}$ and $n$ not too large. We also give an algorithm providing a rigorous proof of the existence and local unicity of such central configurations when given as input a choice of $n$, $\ell$ and $m_0, . . . ,m_n$. Finally, our numerical simulations highlight some interesting properties of the mass distribution.