The herein presented analytical framework fully describes the motion of coplanar systems consisting of a stellar binary and a planet orbiting both stars on orbital as well as secular timescales. Perturbations of the Runge-Lenz vector are used to deri
ve short period evolution of the system, while octupole secular theory is applied to describe its long term behaviour. A post Newtonian correction on the stellar orbit is included. The planetary orbit is initially circular and the theory developed here assumes that the planetary eccentricity remains relatively small (e_2<0.2). Our model is tested against results from numerical integrations of the full equations of motion and is then applied to investigate the dynamical history of some of the circumbinary planetary systems discovered by NASAs Kepler satellite. Our results suggest that the formation history of the systems Kepler-34 and Kepler-413 has most likely been different from the one of Kepler-16, Kepler-35, Kepler-38 and Kepler-64, since the observed planetary eccentricities for those systems are not compatible with the assumption of initially circular orbits.
H${acute{e}}$non [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point particle bouncin
g elastically on two disks. A one parameter family of orbits, named h-orbits, was obtained by starting the particle at rest from a given height. We obtain an analytical expression for the escape distribution of the h-orbits, which is also compared with results from numerical simulations. Finally, some discussion is made about possible applications of the h-orbits in connection with Hills problem.
In previous papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. We considered systems with well separated components and different initial setups (e.g.
coplanar and non-coplanar orbits). However, the systems we examined had comparable masses. In the present paper, the validity of some of the formulae derived previously is tested by numerically integrating the full equations of motion for systems with smaller mass ratios (from ${10^{-3} hspace{0.2cm} mbox{to} hspace{0.2cm} 10^{3}}$, i.e. systems with Jupiter-sized bodies). There is also discussion about HD217107 and its planetary companions.
In a previous paper, we developed a technique for estimating the inner eccentricity in coplanar hierarchical triple systems on initially circular orbits, with comparable masses and with well separated components, based on an expansion of the rate of
change of the Runge-Lenz vector. Now, the same technique is extended to non-coplanar orbits. However, it can only be applied to systems with ${I_{0}<39.23^{circ}}$ or ${I_{0}>140.77^{circ}}$, where ${I}$ is the inclination of the two orbits, because of complications arising from the so-called Kozai effect. The theoretical model is tested against results from numerical integrations of the full equations of motion.
We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well separated components and comparable m
asses. The derivation of short period terms is based on an expansion of the rate of change of the Runge-Lenz vector. Then, the short period terms are combined with secular terms, obtained by means of canonical perturbation theory. The validity of the theoretical equations is tested by numerical integrations of the full equations of motion.
