Spiderweb central configurations

In this paper we study spiderweb central configurations for the $N$-body problem, i.e configurations given by $N=n\times \mathrm{\ell }+1$ masses located at the intersection points of $\mathrm{\ell }$ concurrent equidistributed half-lines with $n$ circles and a central mass $m_0$, under the hypothesis that the $\mathrm{\ell }$ 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 $\ell \in {2,\dots,9}$andarbitrary$n$and$m_i$;undertheconstraint$m_1\geq m_2\geq \ldots \geq m_n$wealsoproveuniquenessfor$\ell \in {10,\dots,18}$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 $n$, $\mathrm{\ell }$ and $m_0, . . . ,m_n$. Finally, our numerical simulations highlight some interesting properties of the mass distribution.