Here we discuss the evolution of binaries around MBH in nuclear stellar clusters. We focus on their secular evolution due to the perturbation by the MBH, while simplistically accounting for their collisional evolution. Binaries with highly inclined orbits in respect to their orbit around the MBH are strongly affected by secular processes, which periodically change their eccentricities and inclinations (e.g. Kozai-Lidov cycles). During periapsis approach, dissipative processes such as tidal friction may become highly efficient, and may lead to shrinkage of a binary orbit and even to its merger. Binaries in this environment can therefore significantly change their orbital evolution due to the MBH third-body perturbative effects. Such orbital evolution may impinge on their later stellar evolution. Here we follow the secular dynamics of such binaries and its coupling to tidal evolution, as well as the stellar evolution of such binaries on longer time-scales. We find that stellar binaries in the central parts of NSCs are highly likely to evolve into eccentric and/or short period binaries, and become strongly interacting binaries either on the main sequence (at which point they may even merge), or through their later binary stellar evolution. The central parts of NSCs therefore catalyze the formation and evolution of strongly interacting binaries, and lead to the enhanced formation of blue stragglers, X-ray binaries, gravitational wave sources and possible supernova progenitors. Induced mergers/collisions may also lead to the formation of G2-like cloud-like objects such as the one recently observed in the Galactic center.

It has been suggested that the 170 day period in the light curve of the low mass X-ray binary 4U 1820-30 arises from the presence of a third body with a large inclination to the binary orbit. We show that this long period motion arises if the system is librating around the stable fixed point in a Kozai resonance. We demonstrate that mass transfer drives the system toward this fixed point, and calculate, both analytically and via numerical integrations, that the period of libration is of order 170 days when the mutual inclination is near the Kozai critical value. The non-zero eccentricity of the binary, combined with tidal dissipation, implies that the rate of change of the binary period would be slower than, or even of opposite sign to, that implied by standard mass transfer models. If the 170 day period results from libration, then, contrary to appearances, the orbital period of the inner binary is increasing with time; in that case, (e/0.009) 2 Q/k 2 2.5 × 10 9 , where k 2 ≈ 0.01 is the tidal Love number and e = 0.009 is the fiducial eccentricity of the inner binary. It appears unlikely that the observed negative period derivative results from the smaller than expected (but positive) value ofṖ combined with the previously suggested acceleration of the system in the gravitational field of the host globular cluster NGC 6624. The discrepancy between the observed and expected period derivative requires further investigation.

To date, 17 circumbinary planets have been discovered. In this paper, we focus our attention on the stability of the Kepler circumbinary planetary systems with only one planet, i.e. . In addition to their intrinsic interest, the study of such systems is an opportunity to test our understanding of planetary system formation and evolution around binaries. The investigation is done by means of numerical simulations. We perform numerical integrations of the full equations of motion of each system with the aim of checking the stability of the planetary orbit. The investigation of the stability of the above systems consists of three numerical experiments. In the first one we perform a long term (1Gyr) numerical integration of the nominal solution of the six Kepler systems under investigation. In the second experiment, we look for the critical semimajor axis of the six planetary orbits, and finally, in the third experiment, we construct two dimensional stability maps on the eccentricity-pericentre distance plane. Additionally, using numerical integrations of the nominal solutions we checked if this solutions were close to the exact resonance.