Solution of canonical equations which result from elimination of short-periodic terms in second-order planetary theory

We solve the first order non-linear differential equation and we calculate the two quadratures to which are reduced the canonical differential equations resulting from the elimination of the short period terms in a second order planetary theory carried out through Hori's method and slow Delaunay canonical variables when powers of eccentricities and the sines of semi-inclinations which are >3 are neglected and the eccentricity of the disturbing planet is identically equal to zero. The procedure can be extended to the case when the eccentricity of the disturbing planet is not identically equal to zero. In this latter general case, we calculatedthe two quadratures expressing angular slow Delaunay canonical variable ₁ of the disturbed planet and angular slow Delaunay canonical variable ₂ of the disturbing planet in terms of timet.     

Complete Zonal Problem of the Artificial Satellite: Generic Compact Analytic First Order in Closed Form

This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J _n (n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n ^th zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which contained results for particular zonal harmonics, let’s say typically from J ₂ to J ₈), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator associated with this transformation is given into the same closed form, and has also been validated.     

The Extended Delaunay Method Applied to First Order Resonance

This paper is concerned with the extended Delaunay method as well as the method of integration of the equations, applied to first order resonance. The equations of the transformation of the extended Delaunay method are analyzed in the (p + 1)/p type resonance in order to build formal, analytical solutions for the resonant problem with more than one degree of freedom. With this it is possible to gain a better insight into the method, opening the possibility for more generalized applications. A first order resonance in the first approximation is carried out, giving a better comprehension of the method, including showing how to eliminate the ‘Poincaré singularity’ in the higher orders. This revised version was published online in July 2006 with corrections to the Cover Date.     

Normal form,Lie-Poisson structure and reduction for the Henon-Heiles system

The reduced Henon-Heiles system is investigated as a Hamiltonian dynamical system obtained by applying the normalization of the HamiltonianH=1/2(p ₁ ² +p ₂ ² +q ₁ ² +q ₂ ² )+1/3q ₁ ³ –q ₁ q ₂ ² to fourth-degree terms. The related equations of motion are bi-Hamiltonian and possess the Lie-Poisson structure. Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions, the equations of motion take various symplectic forms. The various reductions give different coordinate representations of the solutions. These coordinate representations are used to seek the simplest representation of the solutions.     

Analytical method for perturbed frozen orbit around an asteroid in highly inhomogeneous gravitational fields: a first approach

This article provides a method for finding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the heterogeneous body is firstly stated, in polar-nodal coordinates, which takes into account the coefficients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coefficients. The project has been developed in collaboration with the European Space Agency (ESA).     

The three-dimensional motion of Trojan asteroids

The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL ₄,L ₅, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed.     

The uniqueness of normal forms via Lie transforms and its applications to Hamiltonian systems

A necessary and sufficient condition is given for the unique normal forms about critical elements-equilibrium points and periodic orbits. With some applications to the Birkhoff's normal form along with its generalized form by K.R. Meyer, the restricted problem of three bodies near L₄, the Birkohoff's normalization procedure, and the singular perturbation, of Hamiltonian systems are discussed.     

Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ¹ action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J ₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J ₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J ₂ effect as geometric phases.     

Gyldén Model: The Slowly Changing Equivalent Gravitational Parameter

In the Gyldén problem, the case of slowly changing equivalent gravitational parameter (e.g.p.) is studied. Assuming the following law for the variation of the e.g.p.: μ (t) = μ₀+ εμ (εt), ε< μ₀, we obtained a) a O(ε⁴ T⁴)-approximation of the solution, on a shortened time scale (0,T), with T of order o(ε^-1),for the general case (i.e. the support function μ is O(1)-valued and admits an expansion in power series of its argument), and b) a O(ε²)-approximation of the solution, on a natural timescale of order O(1/ε), for the case of a bounded variation rate (i.e. the support function μ and its derivative are both O(1)-valued). For the study of the first problem we introduced a type of Lissajous variables and used the Lie-Hori normalization scheme; for the second problem we used the Delaunay variables and applied the von Zeipel method for approximate integration. The physical interpretation of the results is in both cases the same: within the corresponding limits of approximation, the variation of e.g.p. (i) has no effect on the size of the osculating ellipse, (ii) it sets the pericenter in slow rotation and (iii) it introduces a secular variation in the longitudes. This revised version was published online in July 2006 with corrections to the Cover Date.     

Short Term Evolution of Artificial Satellites

When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.This revised version was published online in October 2005 with corrections to the Cover Date.     

New results on large-scale structures

A new version of the magnetic-tape catalog of ABELL clusters of galaxies is used to obtain redshift estimators and to generate two samples of clusters. A procedure for searching for superclusters of galaxies is applied and the results are given in tabular and graphic form. For a lmited homogeneous sample (distance 60–275 Mpc, galactic latitude B > 35°), 12 multiplets, having member clusters with known redshifts, are found. It is shown that the spatial covariance function for rich clusters has the form ξ = (r₀/r)γ with r₀ = 22.4 ± 1.8 Mpc and γ = 1.90 ± 0.25 for 3 Mpc ≲ r ≲ 80 Mpc.     

Formal solution of the asteroid motion near the 2:1 commensurability

In a previous paper (Ammar in Proc. Math. Phys. 77:99, 2002) the statement of the problem was formulated and the basic equations of motion were formed in terms of variables suitable for the applications in the problem of asteroid motion close to 2:1 commensurability. The short period terms has been eliminated up to the first order in masses O(μ), using a perturbation approach based on the Lie series, the problem is reduced to that of secular resonance one. In the present work the extended Delaunay method has been applied to develop the Hamiltonian and the generator as a power series in rather than the power of ε, where ε is a small parameter of order of the relative mass of the perturber. Hamilton–Jacobi method were used as a method of integration of the equations of the dynamical system in order to build a formal solution for the resonant problem of the type 2:1 with one degree of freedom.     

Fourier analysis of the light curves of eclipsing variables,XII

The aim of the present paper will be to make use of the expressions, established in Paper XI, for the fractional loss of light _l ⁰ of arbitrarily limb-darkened stars in the form of Hankel transforms of zero order, in order to evaluate the explicit forms of the _l ⁰'s for different types of eclipses (Section 2), as well as of the momentsA _2mof the respective light curves (Section 3)-in a closed form; or in terms of expansions that converge under all circumstances envisaged. Particular attention will be directed to a connection between these expansions and other functions already available in tabular form; or to alternative forms amenable to automatic computation.     

Closed form Vaidya-Tikekar type charged fluid spheres with pressure

Recently, Bijalwan (Astrophys. Space Sci. doi:, 2011) discussed all important solutions of charged fluid spheres with pressure and Gupta et al. (Astrophys. Space Sci. doi:, 2010) found first closed form solutions of charged Vaidya-Tikekar (V-T) type super-dense star. We extend here the approach evolved by Bijalwan (Astrophys. Space Sci. doi:, 2011) to find all possible closed form solutions of V-T type super-dense stars. The existing solutions of Vaidya-Tikekar type charged fluid spheres considering particular form of electric field intensity are being used to model massive stars. Infact at present maximum masses of the star models are found to be 8.223931M _Θ and 8.460857M _Θ subject to ultra-relativistic and non-relativistic conditions respectively. But these stars with such are large masses are not well behaved due to decreasing velocity of sound in the interior of star. We present new results concerning the existence of static, electrically charged perfect fluid spheres that have a regular interior. It is observed that electric intensity used in this article can be used to model superdense stars with ultrahigh surface density of the order 2×10¹⁴ gm/cm³ which may have maximum mass 7.26368240M _Θ for ultra-relativistic condition and velocity of sound found to be decreasing towards pressure free interface. We solve the Einstein-Maxwell equations considering a general barotropic equation of state with pressure. For brevity we don’t present a detailed analysis of the derived solutions in this paper.     

Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound _* ^r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe ^–(*₁/) ^a , with positive constantsA,a and ₁ ^* exponential estimate of Nekhoroshev's type.     

Notes on Hori Method for Canonical Systems

In this paper a slightly different approach is proposed for the process of determining the functions S _m and H _m ^* of the algorithm of the canonical version of Hori method. This process will be referred to as integration theory of the mth order equation of the method. It will be shown that the ordinary differential equation with an auxiliary parameter t ^* as independent variable, introduced through Hori auxiliary system, can be replaced by a partial differential equation in the time t. In this way, the mth order equation of the algorithm assumes a form very similar to the one of other perturbation methods. In virtue of this new approach of the integration theory for Hori method, Lagrange's variational equations introduced by Sessin are revised. As an example, the Duffing equation is solved through this new approach.     

Reconstruction of a Monthly Homogeneous Sunspot Area Series Since 1832

In this work, a procedure to elaborate a homogeneous sunspot area series using the Royal Greenwich Observatory/USAF/NOAA data (from 1874 to the present) and the De la Rue and co-workers data (from 1832 to 1868) is presented. These two data series correspond to time intervals that do not overlap and a direct comparison between them could not be carried out. We used the International Sunspot Number (R_i) and the Group Sunspot Number (R_G) as a link between the two original series. Thus, two homogeneous sunspot area series have been built using a simple mathematic procedure based on linear relations.     

Nonuniform Target Ionization and Fitting Thick Target Electron Injection Spectra to RHESSI Data

Past analyses of flare hard X-ray (HXR) spectra have largely ignored the effect of nonuniform ionization along the electron paths in the thick-target model, though it is very significant for well-resolved spectra. The inverse problem (photon spectrum to electron injection spectrum F ₀(E ₀)) is disturbingly non-unique. However, we show that it is relatively simple to allow for the effect in forward fitting of parametric models of F ₀(E ₀)) and provide an expression to evaluate it for the usual single power-law form of F ₀(E ₀)).The expression involves the column depth N _* of the transition region in the flare loop as one of the parameters so data fitting can enable derivation of N _* (and its evaporative evolution) as part of the fitting procedure. The fit to RHESSI data on four flares for a single power law F ₀(E ₀)) is much improved when ionization structure is included compared to when the usual fully ionized approximation is used. This removes the need, in these events at least, to invoke broken power laws, or other forms, of the acceleration spectrum F ₀(E ₀)) to explain the observed photon spectrum     

On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Solution of non-coherent spectral line transfer problem for finite atmosphere by theF n -method

By use of the orthogonality and normalization integrals developed by McCormick and Siewert (1970) a set of singular integral equations suitable forF _n -method is derived for non-coherent spectral line formation problem in finite media.F _n -equations for exit distributions are used to develop some algebraic equations with suitable recursion relations.