Stability and bifurcations of even periodic orbits in the Sitnikov problem

We study different families of even periodic solutions in the classical Sitnikov problem that emanate from the circular case as the eccentricity is increased. The families can be classified by the number N of full revolutions of the primaries and labelled by the number of zeroes p of the vertical coordinate of the massless body in half a period. We give a linear stability criterion of these branches depending on even N, based on the sign for the initial slope of the discriminant function for the associated Hill's equation, in a computable interval of eccentricities. All families for N=2 are linearly stable for small and computable e. The results show a fundamental symmetry-driven difference between the even and odd N cases.