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.     

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.     

A minimal dynamical model for tidal synchronization and orbit circularization

We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations, in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in p:q (p and q being co-prime integers) synchronous rotation.     

Bruns' Theorem: The Proof and Some Generalizations

We give here a proof of Bruns’ Theorem which is both complete and as general as possible: Generalized Bruns’ Theorem.In the Newtonian (n+1)-body problem in ℝ ^p with n≥2 and 1≤p≤n+1, every first integral which is algebraic with respect to positions, linear momenta and time, is an algebraic function of the classical first integrals: the energy, the p(p−1)/2 components of angular momentum and the 2p integrals that come from the uniform linear motion of the center of mass. Bruns’ Theorem only dealt with the Newtonian three-body problem in ℝ³; we have generalized the proof to n+1 bodies in ℝ^p with p≤n+1. The whole proof is much more rigorous than the previous versions (Bruns, Painlevé, Forsyth, Whittaker and Hagiara). Poincaré had picked out a mistake in the proof; we have understood and developed Poincaré’s instructions in order to correct this point (see Subsection 3.1). We have added a new paragraph on time dependence which fills in an up to now unnoticed mistake (see Section 6). We also wrote a complete proof of a relation which was wrongly considered as obvious (see Section 3.3). Lastly, the generalization, obvious in some parts, sometimes needed significant modifications, especially for the case p=1 (see Section 4). This revised version was published online in July 2006 with corrections to the Cover Date.     

A theory of integration for the extended Delaunay method

In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincaré. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincaré. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincaré is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincaré.     

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).     

Determination of Libration Characteristics of Asteroids near Mean Motion Commensurabilities 1/1, 4/3, 3/2, 2/1, 7/3, 5/2, and 3/1

The possibility of using a generalized perfect resonance for the study of libration motions of asteroids near the (p+ q)/p-type commensurabilities of the mean motions of asteroids and Jupiter is considered. Based on the equations of the planar circular restricted three-body problem, the libration-motion equations are derived and their solutions for the intermediate Hamiltonian, as well as a solution taking into account perturbations of the order O(m ^3/2), are determined.     

The vector approach to the problem of physical libration of the Moon. II. The nonlinear problem

By the new vector method in a nonlinear setting, a physical libration of the Moon is studied. Using the decomposition method on small parameters we derive the closed system of nine differential equations with terms of the first and second order of smallness. The conclusion is drawn that in the nonlinear case a connection between the librations in a longitude and latitude, though feeble, nevertheless exists; therefore, the physical libration already is impossible to subdivide into independent from each other forms of oscillations, as usually can be done. In the linear approach, ten characteristic frequencies and two special invariants of the problem are found. It is proved that, taking into account nonlinear terms, the invariants are periodic functions of time. Therefore, the stationary solution with zero frequency, formally supposing in the linear theory a resonance, in the nonlinear approach gains only small (proportional to e) periodic oscillations. Near to zero frequency of a resonance there is no, and solution of the nonlinear equations of physical libration is stable. The given nonlinear solution slightly modifies the previously unknown conical precession of the Moon’s spin axis. The character of nonlinear solutions near the basic forcing frequency Ω₁, where in the linear approach there are beats, is carefully studied. The average method on fast variables is obtained by the linear system of differential equations with almost periodic coefficients, which describe the evolution of these coefficients in a nonlinear problem. From this follows that the nonlinear components only slightly modify the specified beats; the interior period T ≈ 16.53 days appears 411 times less than the exterior one T ≈ 18.61 Julian years. In particular, with such a period the angle between ecliptic plane and Moon orbit plane also varies. Resonances, on which other researches earlier insisted, are not discovered. As a whole, the nonlinear analysis essentially improves and supplements a linear picture of the physical libration.     

Bianchi type-I cosmological models for binary mixture of perfect fluid and dark energy

We consider a self consistent system of Bianchi Type-I cosmology and Binary Mixture of perfect fluid and dark energy. The perfect fluid is taken to be obeying equations of state p _PF =γρ _PF with γ∈[0,1]. The dark energy is considered to be obeying a quintessence-like equation of state where the dark energy obeys equation of state p _DE =ωρ _DE where ω∈[−1,0]. Exact solutions to the corresponding Einstein field equations are obtained. Some special cases are discussed and studied. Further more power law models and exponential models are investigated.     

Dissipative cylindrical fast magnetoacoustic waves in planetary magnetospheres

Nonlinear cylindrical fast magnetoacoustic waves are investigated in a dissipative magnetoplasma comprising of electrons, positrons, and ions. In this regard, cylindrical Kadomtsev-Petviashvili-Burgers (CKPB) equation is derived using the small amplitude perturbation expansion method. Furthermore, cylindrical Burgers-Kadomtsev-Petviashvili (Cyl Burgers-KP) for a fast magnetoacoustic wave is derived, for the first time, for spatial scales larger than the electron/positron skin depths, c/ω _p(e,p). Using the tangent hyperbolic method, the solutions of both planar KPB and Burgers-KP equations are obtained and then subsequently used as an initial profile to solve their respective counterparts in the cylindrical geometry. The effect of positron concentration, kinematic viscosity, and plasma β are explored both for the KPB and the Burgers-KP shock waves and the differences between the two are highlighted. The temporal evolution of the cylindrical fast magnetoacoustic wave is also numerically investigated. The present study may be beneficial to study the propagation characteristics of nonlinear electromagnetic shock waves in planetary magnetospheres.     

An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem

The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically obtained solutions.     

Bodily tides near spin–orbit resonances

Spin–orbit coupling can be described in two approaches. The first method, known as the “MacDonald torque”, is often combined with a convenient assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because derivation of the expression for the MacDonald torque tacitly fixes the rheology of the mantle by making Q scale as the inverse tidal frequency. Spin–orbit coupling can be treated also in an approach called “the Darwin torque”. While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been fully exploited in the literature, where Q is often assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence it is necessary to enrich the theory of spin–orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles of solid-state mechanics, i.e., from the expression for the mantle’s compliance in the time domain. We also explain that the tidal torque includes not only the customary, secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin–Kaula expansion for the tidal torque smoothly passes zero, when the secondary traverses the lmpq resonance (e.g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). Thus, we prepare a foundation for modeling entrapment of a despinning primary into a resonance with its secondary. The roles of the primary and secondary may be played, e.g., by Mercury and the Sun, correspondingly, or by an icy moon and a Jovian planet. We also offer a possible explanation for the “improper” frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR.     

Stable Manifolds and Homoclinic Points Near Resonances in the Restricted Three-Body Problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses 1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for μ small enough in the unaveraged restricted three-body problem.     

Observational Signatures of Impulsively Heated Coronal Loops: Power-Law Distribution of Energies

It has been established that small scale heating events, known as nanoflares, are important for solar coronal heating if the power-law distribution of their energies has a slope α steeper than −2 (α<−2). Forward modeling of impulsively heated coronal loops with a set of prescribed power-law indices α is performed. The power-law distribution is incorporated into the governing equations of motion through an impulsive heating term. The results are converted into synthetic Hinode/EIS observations in the 40″ imaging mode, using a selection of spectral lines formed at various temperatures. It is shown that the intensities of the emission lines and their standard deviations are sensitive to changes in α. A method based on a combination of observations and forward modeling is proposed for determining whether the heating in a particular case is due to small or large scale events. The method is extended and applied to a loop structure that consists of multiple strands.     

Cavitation by Nonlinear Interaction Between Inertial Alfvén Waves and Magnetosonic Waves in Low Beta Plasmas

This paper presents the model equations governing the nonlinear interaction between dispersive Alfvén wave (DAW) and magnetosonic wave in the low-β plasmas (β≪m _e/m _i; known as inertial Alfvén waves (IAWs); here \upbeta = 8pn₀T /B₀²\upbeta = 8\pi n_{0}T /B_{0}^{2} is thermal to magnetic pressure, n ₀ is unperturbed plasma number density, T(=T _e≈T _i) represents the plasma temperature, and m _e(m _i) is the mass of electron (ion)). This nonlinear dynamical system may be considered as the modified Zakharov system of equations (MZSE). These model equations are solved numerically by using a pseudo-spectral method to study the nonlinear evolution of density cavities driven by IAW. We observed the nonlinear evolution of IAW magnetic field structures having chaotic behavior accompanied by density cavities associated with the magnetosonic wave. The relevance of these investigations to low-β plasmas in solar corona and auroral ionospheric plasmas has been pointed out. For the auroral ionosphere, we observed the density fluctuations of ∼ 0.07n ₀, consistent with the FAST observation reported by Chaston et al. (Phys. Scr. T84, 64, 2000). The heating of the solar corona observed by Yohkoh and SOHO may be produced by the coupling of IAW and magnetosonic wave via filamentation process as discussed here.     

Seismology of the solar convection zone

An attempt is made to infer the structure of the solar convection zone from observedp-mode frequencies of solar oscillations. The differential asymptotic inversion technique is used to find the sound speed in the solar envelope. It is found that envelope models which use the Canuto-Mazzitelli (CM) formulation for calculating the convective flux give significantly better agreement with observations than models constructed using the mixing length formalism. This inference can be drawn from both the scaled frequency differences and the sound speed difference. The sound speed in the CM envelope model is within 0.2% of that in the Sun except in the region withr > 0.99R _⊙. The envelope models are extended below the convection zone, to find some evidence for the gravitational settling of helium beneath the base of the convection zone. It turns out that for models with a steep composition gradient below the convection zone, the convection zone depth has to be increased by about 6 Mm in order to get agreement with helioseismic observations.     

The Main Problem in Satellite Theory Revisited

Using the elimination of the parallax followed by the Delaunay normalization, we present a procedure for calculating a normal form of the main problem (J ₂ perturbation only) in satellite theory. This procedure is outlined in such a way that an object-oriented automatic symbolic manipulator based on a hierarchy of algebras can perform this computation. The Hamiltonian after the Delaunay normalization is presented to order six explicitly in closed form, that is, in which there is no expansion in the eccentricity. The corresponding generating function and transformation of coordinates, too lengthy to present here to the same order; the generator is given through order four. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interpretation of the solar-like pulsational behaviour of η Bootis

HR 5235, better known as η Bootis, is a bright and well-known star for which very accurate observations have recently enabled Kjeldsen et al. (2003) and Carrier, Bouchy, and Eggenberger (2003) not only to confirm the presence of solar-like oscillations, but also to identify the excitation in the oscillation spectrum of several p-mode frequencies with harmonic degrees l = 0 – 2. Here we show how such observational success, through the calculation and the investigation of theoretical structure models and the comparison of the observed oscillation spectra with the predicted p-mode frequencies of oscillations, permits one to draw conclusions about the actual evolutionary state of this star and on the physical properties of its internal structure. The computation of the structure models is based on the use of updated global parameters and includes overshooting from the convective core. In particular, we consider the effect on the stellar structure, and hence on the theoretical frequencies, of employing different equations of state and different formalisms to describe the convective energy transport.     

Strained coordinate methods in rotating stars

It was shown in a previous paper (Smith, 1976) that the method of strained coordinates may be usefully employed in the determination of the structure of rotating polytropes. In the present work this idea is extended to Main-Sequence stars with conservative centrifugal fields. The structure variables, pressure, density and temperature are considered pure functions of an auxiliary coordinates (the strained coordinate) and the governing equations written in a form that closely resembles the structure equations for spherical stars but with correction factors that are functions ofs. A systematic, order-by-order derivation of these factors is outlined and applied in detail to a Cowling-model star in uniform rotation. The technique can be extended beyond first order and external boundary conditions are applied, as they should be, at the true surface of the star. Roche approximations are not needed.     

Dynamics of Kantowski-Sachs universe with magnetized anisotropic Dark Energy

Kantowski-Sachs cosmological model in the presence of magnetized anisotropic dark energy is investigated. The energy-momentum tensor consists of anisotropic fluid with anisotropic EoS p=ωρ and a uniform magnetic field of energy density ρ _B . We obtain exact solutions to the field equations using the condition that expansion is proportional to the shear scalar. The physical behavior of the model is discussed with and without magnetic field. We conclude that universe model as well as anisotropic fluid does not approach isotropy through the evolution of the universe.