Periodic orbits of the planar anisotropic Kepler problem

Agraïments: The second author of this work was partially supported by Fundación Séneca de la Región de Murcia grant number 19219/PI/14.In this paper we prove that at every energy level the anisotropic problem with small anisotropy has two periodic orbits which bifurcate from elliptic orbits of the Kepler problem with high eccentricity. Moreover we provide approximate analytic expressions for these periodic orbits. The tool for proving this result is the averaging theory

The dynamics of the relativistic Kepler problem

Altres ajuts: Fundación Séneca (Spain), grant 20783/PI/18We deal with the Hamiltonian system (HS) provided by the correction given by the special relativity to the motion of the two-body problem, or by the first order correction to this problem coming from the general relativity. This Hamiltonian system is completely integrable with the angular momentum C and the Hamiltonian H. We have two objectives. First, we describe the global dynamics of the Hamiltonian system (HS) in the following sense. Let Ih and Ic are the subset of the phase space where H = h and C = c. Since C and H are first integrals, the sets Ic, Ih and Ihc = Ih∩Ic are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when h and c vary. Second, recently Tudoran in [21] provided a criterion which detects when a non-degenerate equilibrium point of a completely integrable system is Lyapunov stable. Every equilibrium point q of the completely integrable Hamiltonian system (HS) is degenerate and has zero angular momentum, so the mentioned criterion cannot be applied to it. But we will show that this criterion is also satisfied when it is applied to the Hamiltonian system (HS) restricted to zero angular momentum

Periodic Orbits of Quantised Restricted Three-Body Problem

In this paper, perturbed third-body motion is considered under quantum corrections to analyse the existence of periodic orbits. These orbits are studied through two approaches to identify the first (second) periodic-orbit types. The essential conditions are given in order to prove that the circular (elliptical) periodic orbits of the rotating Kepler problem (RKP) can continue to the perturbed motion of the third body under quantum corrections where a massive primary body has excessive gravitational force over the smaller primary body. The primaries moved around each other in circular (elliptical) orbits, and the mass ratio was assumed to be sufficiently small. We prove the existence of the two types of orbits by using the terminologies of Poincaré for quantised perturbed motion

X-ray observations of highly obscured τ_(9.7 μm) > 1 sources: an efficient method for selecting Compton-thick AGN?

Observations with the IRS spectrograph onboard Spitzer have found many sources with very deep Si features at 9.7 μm, that have optical depths of τ > 1. Since it is believed that a few of these systems in the local Universe are associated with Compton-thick active galactic nuclei (hereafter AGN), we set out to investigate whether the presence of a strong Si absorption feature is a good indicator of a heavily obscured AGN. We compile X-ray spectroscopic observations available in the literature on the optically-thick (τ_(9.7 μm) > 1) sources from the 12 μm IRAS Seyfert sample. We find that the majority of the high-τ optically confirmed Seyferts (six out of nine) in the 12 μm sample are probably Compton-thick. Thus, we provide direct evidence of a connection between mid-IR optically-thick galaxies and Compton-thick AGN, with the success rate being close to 70% in the local Universe. This is at least comparable to, if not better than, other rates obtained with photometric information in the mid to far-IR, or even mid-IR to X-rays. However, this technique cannot provide complete Compton-thick AGN samples, i.e., there are many Compton-thick AGN that do not display significant Si absorption, with the most notable example being NGC1068. After assessing the validity of the high 9.7 μm optical-depth technique in the local Universe, we attempt to construct a sample of candidate Compton-thick AGN at higher redshifts. We compile a sample of seven high-τ Spitzer sources in the Great Observatories Origins Deep Survey (GOODS) and five in the Spitzer First-Look Survey. All these have been selected to have no PAH features (EW_(6.2 μm) 10^(42) erg s^(−1)) of the detected GOODS sources corroborates that these are AGN. For FLS, ancillary optical spectroscopy reveals hidden nuclei in two more sources. SED fitting can support the presence of an AGN in the vast majority of sources. Owing to the limited photon statistics, we cannot derive useful constraints from X-ray spectroscopy on whether these sources are Compton-thick. However, the low L_(X)/L_(6 μm) luminosity ratios, suggest that at least four out of the six detected sources in GOODS may be associated with Compton-thick AGN

On the Periodic Orbits of the Perturbed Two- and Three-Body Problems

In this work, a perturbed system of the restricted three-body problem is derived when the perturbation forces are conservative alongside the corresponding mean motion of two primaries bodies. Thus, we have proved that the first and second types of periodic orbits of the rotating Kepler problem can persist for all perturbed two-body and circular restricted three-body problems when the perturbation forces are conservative or the perturbed motion has its own extended Jacobian integral