“Comments on a Paper of Idrisi, Ullah and Sikkandhar (Effect of Perturbations in Coriolis and Centrifugal Forces on Libration Points in the Restricted Six-Body Problem: JAS (2021) 68:4-25)”

In a recent paper, Idrisi, Ullah and Sikkandhar [1] considered some aspects of the Restricted Six-Body Problem. The problem consists of the motion of an infinitesimal mass under the gravity field of five bodies in a very special configuration: There are four equal points with equal masses m located at the vertices of a regular square that is rotating on its own plane about its center of mass with constant angular velocity ω. Besides, there is a fifth body of mass m0 placed at the center of the square..

On the integration of Cid''s radial intermediary

This paper deals with the integrations of homogeneous quasi-Keplerian Hamiltonians, that is, perturbed Kepler Hamiltonians which perturbation is of the form SUM j=2 N Aj/rj with Aj constant. Although there are many applications of these Hamiltonians in Physics, Astronomy and Astrodynamics, we focus our interest on a particular case in the core of Artificial Satellite Theory, the Cid's radial intermediary. For this problem, we integrate the equations of motion in two different ways, by means of the elliptic P-Weierstrass function and by using the Krylov–Bogoliubov averaging method to integrate a perturbed harmonic oscillator. In this case, the resulting solution is given in terms of the classical Kepler's equation, with no need of introducing more complex generalized Kepler's equation

On the integral solution of elliptic Kepler’s equation

In a recent paper, Philcox, Goodman and Slepian obtain an explicit solution of the elliptic Kepler’s equation (KE) as a quotient of two contour integrals along a Jordan curve C=C(M,e) that contains the unique real solution of KE but not includes other complex zeros of KE in its interior. The aim of this paper is to study the main issues that arise in the practical implementation of this integral solution. Thus, after a study of the complex zeros of KE, several families of Jordan contours C=C(M,e) that are suitable for this integral solution are proposed. Since contours with minimal length turn out to be the more accurate for numerical purposes, several families that minimize their length are constructed. Secondly, the approximation of the contour integrals by the composite trapezoidal rule is considered. Recall that this rule is employed in the fast Fourier transform and, in spite of its lower order, displays a spectral convergence as a function of the number of nodes, which implies a very fast convergence. Finally, the results of some numerical experiments are presented to show that such a combination of appropriate contours with the composite trapezoidal rule leads to a powerful numerical method to solve KE with any desired accuracy for all values of eccentricity

Evolution of the characteristic curves in the restricted three-body problem in terms of the mass parameter

In this work, we study the evolution of the families of simple symmetric periodic orbits in the restricted three-body problem whatever the value of the mass parameter μ. To classify these characteristic curves, we introduce a topological characterization of both orbits and families. Starting from the work of Strömgren for the Copenhagen case, we analyze the evolution of these families, when the mass parameter μ varies in (0, 1/2], focusing on their topological characterization, the existence of asymptotic points and the appearance of certain types of orbits such as horseshoe orbits. Lastly, we consider two samples, the Earth–Moon and Sun–Jupiter systems and classify the different types of orbits for these systems

Stability Conditions for Permanent Rotations of a Heavy Gyrostat with Two Constant Rotors

In this paper, we consider the motion of an asymmetric heavy gyrostat, when its center of mass lies along one of the principal axes of inertia. We determine the possible permanent rotations and, by means of the Energy-Casimir method, we give sufficient stability conditions. We prove that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space, by the action of two spinning rotors, one of them aligned along the principal axis, where the center of mass lies. We also derive necessary stability conditions that, in some cases, are the same as the sufficient ones

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters (Formula presented.) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter (Formula presented.) a set of values (Formula presented.) that lead to the q-convergence in the sense that Newton’s sequence (Formula presented.) generated from (Formula presented.) is well defined, converges to the exact solution (Formula presented.) of KE and further (Formula presented.) holds for all (Formula presented.). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s (Formula presented.)-test with (Formula presented.). Also since in KE the convergence rate of Newton’s method tends to zero as (Formula presented.), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with (Formula presented.) for all (Formula presented.) and a fixed (Formula presented.). Some remarks on the use of this theorem to derive a priori estimates of the error (Formula presented.) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Sufficient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also sufficient, for some configurations of the gyrostat

Superhydrophobic supported Ag-NPs@ZnO-nanorods with photoactivity in the visible range

In this article we present a new type of 1D nanostructures consisting of supported hollow ZnO nanorods (NRs) decorated with Ag nanoparticles (NPs). The 3D reconstruction by high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) electron tomography reveals that the Ag NPs are distributed along the hollow interior of the ZnO NRs. Supported and vertically aligned Ag-NPs@ZnO-NRs grow at low temperature (135 °C) by plasma enhanced chemical vapour deposition on heterostructured substrates fabricated by sputtered deposition of silver on flat surfaces of Si wafers, quartz slides or ITO. The growth mechanisms of these structures and their wetting behavior before and after visible light irradiation are critically discussed. The as prepared surfaces are superhydrophobic with water contact angles higher than 150°. These surfaces turn into superhydrophilic with water contact angles lower than 10° after prolonged irradiation under both visible and UV light. The evolution rate of the wetting angle and its dependence on the light characteristics are related to the nanostructure and the presence of silver embedded within the ZnO NRs. ÂEuropean Union NMP3-CT-2006- 032583Ministerio de Ciencia e Innovación MAT2010-21228, MAT2010-18447, CSD2008-00023Junta de Andalucía P09-TEP-5283, CTS-518

Scaling behavior and mechanism of formation of Si O2 thin films grown by plasma-enhanced chemical vapor deposition

This paper reports a study of the kinetic roughening of Si O2 thin films prepared by plasma-enhanced chemical vapor deposition (PECVD). Tetramethylsilane has been used as a precursor, and the synthesis has been carried out under remote and in-plasma configurations. The analysis of surface topography of the films by atomic force microscopy shows an anomalous scaling behavior that cannot be represented by the Family-Vicsec scaling relation of dynamic scaling theory. The application of different methods for obtaining the roughness exponent α yields different values of this exponent (α=0.7 for the height-height correlation function and α=1.3 for the power spectral density function for long deposition times) in all experimental conditions. Moreover, a strong variation of the α exponent with deposition time has been determined for the samples grown in remote mode. This correlates with the presence of a crossover region of the growth exponent β, which varies from a first value of 1.3 for low deposition times to another of 0.3 for longer deposition times. Such a variation is not found for the samples grown in the plasma, characterized by a β value of 0.28. The results obtained can be explained by the combined effect in the growth process of a low diffusivity of the physisorbed species along with the existence of nonlocal interactions due to shadowing effects. These two assumptions are in agreement with the empirical knowledge existing about the kinetics of the growth of Si O2 thin films by PECVD and establish a link between the scaling properties of the films with the surface chemistry during the film growth.Ministerio de Educación y Ciencia MAT2004-01558 y MAT2007-6576