No abstract

In this work, we investigate the eclipse timing of the polar binary HU Aquarii that has been observed for almost two decades. Recently, Qian et al. attributed large (O-C) deviations between the eclipse ephemeris and observations to a compact system of two massive jovian companions. We improve the Keplerian, kinematic model of the Light Travel Time (LTT) effect and re-analyse the whole currently available data set. We add almost 60 new, yet unpublished, mostly precision light curves obtained using the time high-resolution photo-polarimeter OPTIMA, as well as photometric observations performed at the MONET/N, PIRATE and TCS telescopes. We determine new mid--egress times with a mean uncertainty at the level of 1 second or better. We claim that because the observations that currently exist in the literature are non-homogeneous with respect to spectral windows (ultraviolet, X-ray, visual, polarimetric mode) and the reported mid--egress measurements errors, they may introduce systematics that affect orbital fits. Indeed, we find that the published data, when taken literally, cannot be explained by any unique solution. Many qualitatively different and best-fit 2-planet configurations, including self-consistent, Newtonian N-body solutions may be able to explain the data. However, using high resolution, precision OPTIMA light curves, we find that the (O-C) deviations are best explained by the presence of a single circumbinary companion orbiting at a distance of ~4.5 AU with a small eccentricity and having ~7 Jupiter-masses. This object could be the next circumbinary planet detected from the ground, similar to the announced companions around close binaries HW Vir, NN Ser, UZ For, DP Leo or SZ Her, and planets of this type around Kepler-16, Kepler-34 and Kepler-35.Comment: 20 pages, 18 figures, accepted to Monthly Notices of the Royal Astronomical Society (MNRAS

Abstract. In this paper we apply a new technique alternative to the numerically computed Lyapunov Characteristic Number (LCN) for studying the dynamical behaviour of planetary systems in the framework of the gravitational N -body problem. The method invented by P. Cincotta and C. Simó is called the Mean Exponential Growth of Nearby Orbits (MEGNO). It provides an efficient way for investigation of the fine structure of the phase space and its regular and chaotic components in any conservative Hamiltonian system. In this work we use it to study the dynamical behaviour of the multidimensional planetary systems. We investigate the recently discovered υ And planetary system, which consists of a star of 1.3 M and three Jupiter-size planets. The two outermost planets have eccentric orbits. This system appears to be one of the best candidates for dynamical studies. The mutual gravitational interaction between the two outermost planets is strong. Moreover the system can survive on a stellar evolutionary time scale as it is claimed by some authors (e.g., Rivera & Lissauer 2000b). As the main methodological result of this work, we confirm important properties of the MEGNO criterion such as its fast convergence, and short motion times (of the order of 10 4 times the longest orbital period) required to distinguish between regular and chaotic behaviors. Using the MEGNO technique we found that the presence of the innermost planet may cause the whole system to become chaotic with the Lyapunov time scale of the order of 10 3 -10 4 yr only. Chaos does not induce in this case visible irregular changes of the orbital elements, and therefore its presence can be overlooked by studying variations of the elements. We confirm explicitly the strong and sensitive dependence of the dynamical behaviour on the companion masses.

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable.

It is shown that in the Rabi model, for an integer value of the spectral parameter $x$, in addition to the finite number of the classical Judd states there exist infinitely many possible eigenstates. These eigenstates exist if the parameters of the problem are zeros of a certain transcendental function; in other words, there are infinitely many possible choices of parameters for which integer $x$ belongs to the spectrum. Morover, it is shown that the classical Judd eigenstates appear as degenerate cases of the confluent Heun function.Comment: 7 pages, 4 figure