The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness

Abstract

We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed.

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries

“…Thus, the present work extends the results of Perdios and Markellos (1988), Perdios (2007) and Kalantonis et al (2008) by revealing a new type of dynamical behavior, since for primaries of zero or positive oblateness (A ≥ 0) additional equilibrium points on the Z-axis and associated separate Z-axis straight line oscillations generating 3D periodic orbits do not exist. A characteristic feature of this new dynamical behavior is that the manifolds of families generated consists of three-dimensional halo type periodic orbits located either entirely above or entirely below the orbital plane of the primaries.…”

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

“…But only some of the authors, given above, have taken into account the effect of the radiation pressure i.e., the mechanical force of direct sunlight on artificial satellites which produces major changes in their orbital elements. Several authors have examined the secular perturbations arising from this cause are: Shapiro (1962), Kozai (1962), Wyatt (1962), Elipe and Lara (1997), Ishwar and Elipe (2001), Poddar (2002), Perdios (2003), Aggarwal et al (2006), AbdulRaheem et al (2008 and Kalantonis et al (2008). But our procedure is different from them in the sense that we have used the mobile coordinates to draw the periodic orbits.…”

Revealing the Newton–Raphson basins of convergence in the circular pseudo-Newtonian Sitnikov problem

“…Several types of perturbations, such as primaries with either prolate [12] or oblate shape [13], as well as the radiation pressure [14], have been added for making the system of three bodies more realistic. Another well-studied aspect of the Sitnikov three-body problem is the study of the families of periodic orbits and the corresponding bifurcations [7,15,16]. In addition, the stability of motion in the same system has been investigated in [17], where it was found that in the case where the mass of the test particle is not negligible the energetically allowed regions of motion grow with increasing value of the third body, while at the same time the amplitude of the oscillation, along the vertical z-axis, gradually increases.…”