The mass dependence of the period of the periodic solutions of the Sitnikov problem

We study the problem in which $N$ bodies, called primaries, of equal masses $m$ are describing circular keplerian solutions in the $xy$ plane and a body $\mu$, of zero mass, moves on a line perpendicular to the plane of motion of the primaries and passing through their center of mass. We show that such a problem is equivalent to the Classical Circular Sitnikov Problem, in which $N=2$ and $m=\frac{1}{2}$. We also show that the main parameter in searching for periodic solutions is $M=mN$, the total mass of all the primaries. We add an analytic study of the period, $T(h)$, as a function of the negative energy $h$. We generalize some results of [2] and we show the dependence of $T(h)$ on the mass parameter $M$. Finally, we confirm, the expected result that the case of the Newtonian potential for a homogeneous circular ring of mass $M$ is just the limit case of the problem we have studied, in which we let $N$ go to infinity, while keeping the product $mN$ finite.