Reconstruction of A Non-Stationary Potential of Gravitating System on Given Evolving Orbits

We study the problem of the reconstruction of a non-stationary space symmetrical regular planar potential of the gravitating system on a family of evolving types of orbits being used in the dynamics of stationary stellar systems. An application of such an inverse problem to the dynamical evolution of stellar systems with variable masses is given. The general form of the evolving orbit which we use when writing out the differential equations for non-stationary potential may also be interpreted as an osculating orbit of the perturbed Keplerian motion. In this case we are making an additional transformation of the basic equation of the problem and demonstrating an appropriate example of the construction of a non-stationary potential of a gravitating system. In connection with the stellar dynamical character of our inverse problem, we also give a generalized form of its basic equation in a rotating coordinate system.     

Equations of radiation gas dynamics in tetrad form for astrophysical applications

This paper deals with the representation of relativistic equations of gas dynamics with due regard to the general relativity theory effects in the form accepted and widely applied in the special relativity theory. With this purpose, a strict formal definition of a non-inertial co-moving reference frame without rotation is carried out on the basis of a tetrad formalism by use of the Fermi—Walker rules of transport of 4-frame. The equations of physical kinetics, relativistic collapse, Einstein's equations, equations of relatiivistic radiation gas dynamics for ideal and dissipative gases, Taub's equations for a shock wave, which allow for radiation and electron-positron pairs, are obtained in this reference frame. On the basis of the local Lorentz transformation and the Ricci rotation coefficients, these equations are written in the laboratory reference frame, in order to illustrate the fact that the general relativity effects can be simply taken into account in the equations having a form accepted in the special relativity theory.     

High order normal form construction near the elliptic orbit of the Sitnikov problem

We consider the Sitnikov problem; from the equations of motion we derive the approximate Hamiltonian flow. Then, we introduce suitable action–angle variables in order to construct a high order normal form of the Hamiltonian. We introduce Birkhoff Cartesian coordinates near the elliptic orbit and we analyze the behavior of the remainder of the normal form. Finally, we derive a kind of local stability estimate in the vicinity of the periodic orbit for exponentially long times using the normal form up to 40th order in Cartesian coordinates.     

Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincaré's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.     

Dynamics of multibody chains in circular orbit: non-integrability of equations of motion

This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups.     

Thin-Shell Magnetohydrodynamic Equations for the Solar Tachocline

We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity.     

Global Solution of Dynamical Systems a Report to Z. Kopal, on what Followed Since

Two basic problems of dynamics, one of which was tackled in the extensive work of Z. Kopal (see e.g. Kopal, 1978, Dynamics of Close Binary Systems, D. Reidel Publication, Dordrecht, Holland.), are presented with their approximate general solutions. The ‘penetration’ into the space of solution of these non-integrable autonomous and conservative systems is achieved by application of ‘The Last Geometric Theorem of Poincaré’ (Birkhoff, 1913, Am. Math. Soc. (rev. edn. 1966)) and the calculation of sub-sets of ‘solutions précieuses’ that are covering densely the spaces of all solutions (non-periodic and periodic) of these problems. The treated problems are: 1. The two-dimensional Duffing problem, 2. The restricted problem around the Roche limit. The approximate general solutions are developed by applying known techniques by means of which all solutions re-entering after one, two, three, etc, revolutions are, first, located and then calculated with precision. The properties of these general solutions, such as the morphology of their constituent periodic solutions and their stability for both problems are discussed. Calculations of Poincaré sections verify the presence of chaos, but this does not bear on the computability of the general solutions of the problems treated. The procedure applied seems efficient and sufficient for developing approximate general solutions of conservative and autonomous dynamical systems that fulfil the PoincaréBirkhoff theorems. The same procedure does not apply to the sub-set of unbounded solutions of these problems.     

A note on the application of a generalized canonical approach to non-linear Hamiltonian systems

In the author's treatment of the ideal resonance problem (1988), a non-canonical transformation was employed to bring the original Hamiltonian to a form amenable to the use of standard action-angle variables. Though the strictly Hamiltonian form of equations of motion was thus compromised, their general form was maintained, allowing transformation of the system to arbitrary order and forestalling the introduction of elliptic functions until a final explicit integration required in this approach. The general theory of such transformations is presented, and some points regarding their application are discussed, leading to the conclusion that the approach is practically limited to systems with a single degree of freedom only.     

The motion of artificial satellites in the set of Eulerian redundant parameters

In this paper, the connections between orbit dynamics and rigid body dynamics are established throughout the Eulerian redundant parameters, the perturbation equations for any conic motion of artificial satellites are derived in terms of these parameters. A general recursive and stable computational algorithm is also established for the initial-value problem of the Eulerian parameters for satellites prediction in the Earth's gravitational field with axial symmetry. Applications of the algorithm are considered for the two cases of short and long term predictions. For the short-term prediction, we consider the problem of the final state prediction of some typical ballistic missiles in the geopotential model with zonal harmonic terms up to J ₃₆, while for the long-term prediction, we consider the perturbed J ₂ motion of Explorer 28 over 100 revolutions.     

Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque

The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state.     

Solution of an infinite number of inequations depending on a continuous parameter and application to the solution of equations in dynamical astronomy

The investigation of a large class of problems in physics and astrophysics requires the dtermination of the ranges of some parameters z, E, ... for which inequations of the form F(r; z, E, ... )0 are satisfied for all r in some interval )a, b(. The solution of this problem is given under the form of three general theorems and, resulting from them, a very simple numerical procedure. This can also be used to solve equations of the form df/dt=0 where f is some function of variables and derivatives of these variables (functions of t) with respect to t, for instance Vlasov-type equations in dynamics of flat stellar disks.     

Regularization of the circular restricted three-body problem using ‘similar’ coordinate systems

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.     

Dynamics of the 3/1 planetary mean-motion resonance: an application to the HD60532 b-c planetary system

In this paper, we use a semi-analytical approach to analyze the global structure of the phase space of the planar planetary 3/1 mean-motion resonance. The case where the outer planet is more massive than its inner companion is considered. We show that the resonant dynamics can be described using two fundamental parameters, the total angular momentum and the spacing parameter. The topology of the Hamiltonian function describing the resonant behaviour is investigated on a large domain of the phase space without time-expensive numerical integrations of the equations of motion, and without any restriction on the magnitude of the planetary eccentricities. The families of the Apsidal Corotation Resonances (ACR) parameterized by the planetary mass ratio are obtained and their stability is analyzed. The main dynamical features in the domains around the ACR are also investigated in detail by means of spectral analysis techniques, which allow us to detect the regions of different regimes of motion of resonant systems. The construction of dynamical maps for various values of the total angular momentum shows the evolution of domains of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems.     

Post-Newtonian satellite orbits

The first post-Newtonian approximation of general relativity is used to account for the motion of solar system bodies and near-Earth objects which are slow moving and produce weak gravitational fields. The \(n\)-body relativistic equations of motion are given by the Einstein-Infeld-Hoffmann equations. For \(n=2\), we investigate the associated dynamics of two-body systems in the first post-Newtonian approximation. By direct integration of the associated planar equations of motion, we deduce a new expression that characterises the orbit of test particles in the first post-Newtonian regime generalising the well-known Binet equation for Newtonian mechanics. The expression so obtained does not appear to have been given in the literature and is consistent with classical orbiting theory in the Newtonian limit. Further, the accuracy of the post-Newtonian Binet equation is numerically verified by comparing secular variations of known expression with the full general relativistic orbit equation.     

The Generalized Uncertainty Principle and the Friedmann equations

The Generalized Uncertainty Principle (or GUP) affects the dynamics in Plank scale. So the known equations of physics are expected to get modified at that very high energy regime. Very recently authors in Ali et al. (Phys. Lett. B 678:497, 2009) proposed a new Generalized Uncertainty Principle (or GUP) with a linear term in Plank length. In this article, the proposed GUP is expressed in a more general form and the effect is studied for the modification of the Friedmann equations of the FRW universe. In the midway the known entropy-area relation get some new correction terms, the leading order term being proportional to \(\sqrt{\mathrm{Area}}\).     

Small virial oscillations of axisymmetric gravitating systems. I

A closed system of equations describing the gross dynamics of axisymmetric, collisionless gravitating systems is proposed. The equations are converted to dimensionless form. This system of equations is reduced to a system of linear equations with periodic coefficients. Translated from Astrofizika, Vol. 43, No. 2, pp. 293-302, April–June, 2000.     

Quadratic analytical solution of general systems through formal hamiltonization

A technique for the quadratic analytical solution of general nonlinearly perturbed periodic systems is presented. It relies on a device recognized as early as Birkhoff (1927), through which any system of ordinary differential equations can be cast in Hamiltonian form through the introduction of a set of auxiliary conjugate variables. The particular implementation applies the author's quadratic Hamiltonian approach, utilizing Lie transforms (so admitting of easy inversion), and featuring the ability to determine the frequencies of the system to twice the order of the solution at the last step. The method is exemplified through an analysis of the van der Pol equation to find the solution to second order, and frequencies to fourth, of the limit cycle of the system. Finally, the relationship of the approach to other perturbation techniques, particularly the vector/matrix Lie transform method, is discussed.     

Universal unfolding of symmetric resonances

We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff–Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.     

A New Formulation for General Relativistic Force-Free Electrodynamics and Its Applications

We formulate the general relativistic force-free electrodynamics in a new 3 1 language. In this formulation,when we have properly defined electric and magnetic fields,the covariant Maxwell equations could be cast in the traditional form with new vacuum con-stitutive constraint equations. The fundamental equation governing a stationary,axisymmet-ric force-free black hole magnetosphere is derived using this formulation which recasts the Grad-Shafranov equation in a simpler way. Compared to the classic 3 1 system of Thorne and MacDonald,the new system of 3 1 equations is more suitable for numerical use for it keeps the hyperbolic structure of the electrodynamics and avoids the singularity at the event horizon. This formulation could be readily extended to non-relativistic limit and find applica-tions in flat spacetime. We investigate its application to disk wind,black hole magnetosphere and solar physics in both flat and curved spacetime.