Inhomogeneous Dark Radiation Dynamics on a de Sitter Brane

Assuming spherical symmetry we analyse the dynamics of an inhomogeneous dark radiation vaccum on a Randall and Sundrum 3-braneworld. Under certain natural conditions we show that the effective Einstein equations on the brane form a closed system. On a de Sitter brane and for negative dark energy density we determine exact dynamical and inhomogeneous solutions which depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial configuration. We also identify the conditions leading to the formation of a singularity or of regular bounces inside the dark radiation vaccum. This revised version was published online in July 2006 with corrections to the Cover Date.     

Cosmological dynamics of a bulk scalar field in the DGP setup

We reconsider the issue of cosmological dynamics in a DGP setup with a bulk scalar field. The ghost-free, normal branch of this DGP-inspired braneworld scenario has the potential to realize a self-consistent phantom-like behavior. The roles played by the bulk canonical scalar field on this phantom-like dynamics are explored. Within a dynamical system approach, the effective phantom nature of the scenario is investigated with details. This analysis shows that there is a stable, late-time de Sitter phase.     

Gravitational energy, momentum and angular momentum of Schwarzschild Anti-de Sitter space-times in the teleparallel equivalent of general relativity

We give a class of spherically symmetric-Anti de Sitter (Ads), exact solution in the teleparallel equivalent of general relativity (TEGR). The solution depends on an arbitrary function F(R)\mathcal{F}(R) and reproduce the metric of Schwarzschild Ads space-time. In the context of the Hamiltonian formulation of the TEGR we compute the gravitational energy of this class. The calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. We show that the form of the energy depends on the arbitrary function. We make a constrain on this arbitrary function to give the correct form of energy.     

The Evolution of the Stable and Unstable Manifold of an Equilibrium Point

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter,ν. The eigenvalues of the linearized system are complex for ν < 0 and purely imaginary for ν > 0. Thus for ν < 0 the equilibrium has a two‐dimensional stable manifold and a two‐dimensional unstable manifold, but for ν > 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation. One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as ν tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for ν < 0 and the whole structure of these manifolds shrinks to the equilibrium as ν → 0 and disappears. These considerations have interesting implications about Strömgren's conjecture in celestial mechanics and the blue sky catastrophe of Devaney.     

Attitude stability of artificial satellites subject to gravity gradient torque

The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability requirements scenario.     

Dynamical evolution of an unstable gravastar with zero mass

Using the conventional gravastar model, that is, an object constituted by two components where one of them is a massive infinitely thin shell and the other one is a de Sitter interior spacetime, we physically interpret a solution characterized by a zero Schwarzschild mass. No stable gravastar is formed and it collapses without forming an event horizon, originating what we call a massive non-gravitational object. The most surprise here is that the collapse occurs with an exterior de Sitter vacuum spacetime. This creates an object which does not interact gravitationally with an outside test particle and it may evolve to a point-like topological defect.     

Energy and stability in the Full Two Body Problem

The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.      

Application of Hamiltonian structure-preserving control to formation flying on a J 2-perturbed mean circular orbit

The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J ₂ perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J ₂-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories.     

Vaidya black hole in non-stationary de Sitter space: Hawking’s temperature

In this paper we present a class of non-stationary solutions of Einstein’s field equations describing embedded Vaidya-de Sitter black holes with a cosmological variable function Λ(u). The Vaidya-de Sitter black hole is interpreted as the radiating Vaidya black hole is embedded into the non-stationary de Sitter space with variable Λ(u). The energy-momentum tensor of the Vaidya-de Sitter black hole is expressed as the sum of the energy-momentum tensors of the Vaidya null fluid and that of the non-stationary de Sitter field, and satisfies the energy conservation law. We study the energy conditions (like weak, strong and dominant conditions) for the energy-momentum tensor. We find the violation of the strong energy condition due to the negative pressure and leading to a repulsive gravitational force of the matter field associated with Λ(u) in the space-time. We also find that the time-like vector field for an observer in the Vaidya-de Sitter space is expanding, accelerating, shearing and non-rotating. It is also found that the space-time geometry of non-stationary Vaidya-de Sitter solution with variable Λ(u) is Petrov type D in the classification of space-times. We also find the Vaidya-de Sitter black hole radiating with a thermal temperature proportional to the surface gravity and entropy also proportional to the area of the cosmological black hole horizon.     

Quantum statistical entropy of Schwarzchild-de Sitter spacetime

Using the quantum statistical method, we calculate quantum statistical entropy between the black hole horizon and the cosmological horizon in Schwarzchild spacetime and derive the expression of quantum statistical entropy in de Sitter spacetime. Under the Unruh-Verlinde temperature of Schwarzchild-de Sitter spacetime in the entropic force views, we obtain the expression of quantum statistical entropy in de Sitter spacetime. It is shown that in de Sitter spacetime quantum statistical entropy is the sum of thermodynamic entropy corresponding black hole horizon and the one corresponding cosmological horizon. And the correction term of de Sitter spacetime entropy is obtained. Therefore, it is confirmed that the black hole entropy is the entropy of quantum field outside the black hole horizon. The entropy of de Sitter spacetime is the entropy of quantum field between the black hole horizon and the cosmological horizon.     

Geostationary secular dynamics revisited: application to high area-to-mass ratio objects

The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree two, lunisolar perturbations with a realistic model for the Sun and Moon orbits, and solar radiation pressure. The long-term dynamics of the GEO region has been studied both numerically and analytically, in view of the relevance of such studies to the issue of space debris or to the disposal of GEO satellites. Past studies focused on the orbital evolution of objects around a nominal solution, hereafter called the forced equilibrium solution, which shows a particularly strong dependence on the area-to-mass ratio. Here, we (i) give theoretical estimates for the long-term behavior of such orbits, and (ii) we examine the nature of the forced equilibrium itself. In the lowest approximation, the forced equilibrium implies motion with a constant non-zero average ‘forced eccentricity’, as well as a constant non-zero average inclination, otherwise known in satellite dynamics as the inclination of the invariant ‘Laplace plane’. Using a higher order normal form, we demonstrate that this equilibrium actually represents not a point in phase space, but a trajectory taking place on a lower-dimensional torus. We give analytical expressions for this special trajectory, and we compare our results to those found by numerical orbit propagation. We finally discuss the use of proper elements, i.e., approximate integrals of motion for the GEO orbits.     

A model of the late universe with viscous Zel’ldovich fluid and decaying vacuum

Many have speculated about the presence of a stiff fluid in very early stage of the universe. Such a stiff fluid was first introduced by Zel’dovich. Recently the late acceleration of the universe was studied by taking bulk viscous stiff fluid as the dominant cosmic component, but the age predicted by such a model is less than the observed value. We consider a flat universe with viscous stiff fluid and decaying vacuum energy as the cosmic components and found that the model predicts a reasonable background evolution of the universe with de Sitter epoch as end phase of expansion. More over, the model also predicts a reasonable value for the age of the present universe. We also performed a dynamical system analysis of the model and found that the end de Sitter phase predicted by the model is stable.     

On the Centre Manifold of Collinear Points in the Planar Three-Body Problem

We study the existence of invariant tori in a neighbourhood of the collinear equilibrium points of the planar three-body problem. To this end some properties of the normal form of the Hamiltonian reduced to the 4D central manifold are proved. Using this normal form, we show that the nondegeneracy conditions of KAM theorem are satisfied for all positive masses, including the 2:1 resonance case. The evaluation of the conditions is done numerically.     

Normal forms for the epicyclic approximations of the perturbed Kepler problem

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.     

Thermodynamics of the five-dimensional Schwarzschild-de Sitter black hole

Considering the fact that there is a correlation between the black hole horizon and cosmological horizon, we discuss the thermodynamic properties of de Sitter spacetime. The equivalent temperature and energy of de Sitter spacetime are obtained. It is shown that the upper limit energy of de Sitter spacetime is equal to the energy of a pure de Sitter spacetime. The thermodynamic entropy of de Sitter spacetime is the sum of the black hole horizon thermodynamic entropy and the one of cosmological horizon.     

Low correlation between redshift and intrinsic brightness for quasars and galaxies in a de sitter universe

Since the discovery of quasars, it has been recognized that these objects must have either an extraordinary intrinsic brightness or a nonlinear redshift. The most widely accepted current belief incorporates a linear (Hubble) redshift- distance relation and time evolution within a Big Bang model. We reconsider the possibility of a nonlinear (de Sitter) redshift-distance relation and find quasar intrinsic brightness to be not at all extraordinary. Given a de Sitter law, intrinsic brightness is found to be independent of redshift over five orders of magnitude.     

Higher dimensional dust collapse with a cosmological constant

The general solution of the Einstein equation for higher dimensional (HD) spherically symmetric collapse of inhomogeneous dust in presence of a cosmological term, i.e., exact interior solutions of the Einstein field equations is presented for the HD Tolman–Bondi metrics embedded in a de Sitter background. The solution is then matched to exterior HD Schwarzschild–de Sitter. A brief discussion on the causal structure singularities and horizons is provided. It turns out that the collapse proceed in the same way as in the Minkowski background, i.e., the strong curvature naked singularities form and that the higher dimensions seem to favor black holes rather than naked singularities.      

Rotation of a rigid satellite with a fluid component: a new light onto Titan’s obliquity

We revisit the rotation dynamics of a rigid satellite with either a liquid core or a global subsurface ocean. In both problems, the flow of the fluid component is assumed inviscid. The study of a hollow satellite with a liquid core is based on the Poincaré–Hough model which provides exact equations of motion. We introduce an approximation when the ellipticity of the cavity is low. This simplification allows to model both types of satellite in the same manner. The analysis of their rotation is done in a non-canonical Hamiltonian formalism closely related to Poincaré’s “forme nouvelle des équations de la mécanique”. In the case of a satellite with a global ocean, we obtain a seven-degree-of-freedom system. Six of them account for the motion of the two rigid components, and the last one is associated with the fluid layer. We apply our model to Titan for which the origin of the obliquity is still a debated question. We show that the observed value is compatible with Titan slightly departing from the hydrostatic equilibrium and being in a Cassini equilibrium state.     

Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues

This note gives a concise algorithm for computing a normal form for a real linear Hamiltonian differential equatin which has purely imaginary eigenvalues. This algorithm is then applied to the differential equation which comes from the quadratic terms of the Hamiltonian of the restricted three body problem at a Lagrange equilateral triangle equilibrium point.