On target for Venus – set oriented computation of energy efficient low thrust trajectories

Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented and compute approximations to trajectories for a mission to Venus.     

具抛物不动点的二维保测度映射及其三维扩张的KS熵

我们已经研究了分别具椭圆和双曲不动点的二维保测度映射及其受摄三维扩张的ＫＳ熵。本文研究一类具抛物不动点的二维保测度映射：及其受摄扩张：的ＫＳ熵随参数Ａ、Ｂ、Ｃ、Ｄ、Ｅ的变化．数值探索结果表明：适当定义区域内的二维映射Ｔ２的ＫＳ熵与Ａ无关，与我们的理论分析结果相一致。受摄扩张映射Ｔ３的ＫＳ熵随摄动参数Ｂ、Ｃ、Ｄ的增大而增大，却随Ｅ的增大而减小．我们还发现，随着摄动的逐渐增强，映射Ｔ３的不变环面将逐渐破裂，使更多的轨道逃逸，从而可能使映射Ｔ３的ＫＳ熵减小。另外，不变环面存在的判别式在大范围内仍在一定程度上有效。     

Gravity from Spacetime Thermodynamics

The Einstein-Hilbert action (and thus the dynamics of gravity) can be obtained by: (i) combining the principle of equivalence, special relativity and quantum theory in the Rindler frame and (ii) postulating that the horizon area must be proportional to the entropy. This approach uses the local Rindler frame as a natural extension of the local inertial frame, and leads to the interpretation that the gravitational action represents the free energy of the spacetime geometry. As an aside, one obtains an insight into the peculiar structure of Einstein-Hilbert action and a natural explanation to the questions: (i) Why does the covariant action for gravity contain second derivatives of the metric tensor? (ii) Why is the gravitational coupling constant positive? Some geometrical features of gravitational action are clarified.     

Stickiness in three-dimensional volume preserving mappings

We investigate the orbital diffusion and the stickiness effects in the phase space of a 3-dimensional volume preserving mapping. We first briefly review the main results about the stickiness effects in 2-dimensional mappings. Then we extend this study to the 3-dimensional case, studying for the first time the behavior of orbits wandering in the 3-dimensional phase space and analyzing the role played by the hyperbolic invariant sets during the diffusion process. Our numerical results show that an orbit initially close to a set of invariant tori stays for very long times around the hyperbolic invariant sets near the tori. Orbits starting from the vicinity of invariant tori or from hyperbolic invariant sets have the same diffusion rule. These results indicate that the hyperbolic invariant sets play an essential role in the stickiness effects. The volume of phase space surrounded by an invariant torus and its variation with respect to the perturbation parameter influences the stickiness effects as well as the development of the hyperbolic invariant sets. Our calculations show that this volume decreases exponentially with the perturbation parameter and that it shrinks down with the period very fast.     

Area-preserving Poincaré mappings of the unit disk

The best way to investigate the long-time behaviour of dynamical systems is to introduce an appropriate Poincaré mapping P and study its iterates.Two cases of physical interest arise: Conservative and dissipative systems. While the latter has been considered by a great many authors, much less is known for the first one (according to Liouville's theorem, here the mapping leaves a certain measure in phase space invariant). In this paper, we concentrate our attention on compact phase spaces (or, rather, surfaces of section). This assumption is mathematically useful and physically reasonable.We consider the simplest possible (2-dimensional) systems whehre the phase space is the compact unit disk D in ². A family of simple area-preserving mappings from D onto itselves will be given and discussed in detail.It is shown that general characteristics of the dynamics are quite similar to those of e.g. the Hénon-Heiles system, while other features, as the structure of invariant curves, are different.     

Enigmatic aspects of entropy inside the black hole: what do falling comoving observers see?

In a recent paper it was suggested that inclusion of mutual gravitational interactions can give a possible scenario for reversing gravitation collapse and averting a singular phase. We extend this idea to the still unsolved problem of matter collapsing beyond black hole event horizons. For a comoving observer there is no change in entropy as he goes through the horizon. Matter collapses to a minimum radius, and then can re-expand with the same entropy. It is shown that phase space inside a collapsing black hole is also invariant.     

Trajectory design strategies applied to temporary comet capture including Poincaré maps and invariant manifolds

Temporary satellite capture (TSC) of Jupiter-family comets has been a focus of investigation within the astronomy community for decades. More recently, TSC has been approached from the perspective of dynamical systems theory, within the context of the circular restricted three-body problem (CR3BP). Thus, this problem serves as a testbed for exploring techniques that support trajectory design in similar dynamical regimes. In particular, an association between the invariant manifolds of libration point orbits and the paths of comets that experience TSC has been explored. In this investigation, TSC is further examined from the perspective of transit, that is, transition through the gateways associated with the collinear libration points, in the three-body problem. Periapsis Poincaré maps, previously employed for trajectory design in several investigations, are used to deliver insight into the nature of transit trajectories for energy levels near those associated with several Jupiter-family comets. The evolution of transit trajectories with increasing energy is explored, and the existence of solutions with similar characteristics to the paths of comets P/1996 R2, 82P/Gehrels 3, and 147P/Kushida–Muramatsu is demonstrated within the context of the planar CR3BP using planar periapsis maps. During TSC, the path of comet 111P/Helin–Roman–Crockett is highly inclined with respect to Jupiter; the motion of this comet is examined relative to invariant manifolds in the spatial CR3BP. A method to display the information contained in higher-dimensional Poincaré maps is also demonstrated, and is employed to locate a trajectory possessing the same qualitative characteristics as the path of 111P/Helin–Roman–Crockett.     

The route to chaos during a pulsation event

A time series analysis of a pulsation event in solar radio emission provides an evolution from a regular doubly periodic phase to an irregular behaviour. Applying some techniques developed in the theory of nonlinear dynamic systems to this irregular stage suggests that there exists a low-dimensional attractor. Estimates of the maximum Lyapunov exponent give some evidence to deterministic chaos. The sudden transition from a regular to a chaotic structure is identified as a part of the Ruelle-Takens-Newhouse route to chaos which is typical in nonlinear systems. It is checked whether this pulsation event may be interpreted in terms of known pulsation models. Consequences for models, which are suitable to describe such an evolution, are discussed.     

Planck’s law and the light quantum hypothesis

The phase space of a light quantum in a given volume is subdivided into “cells” of magnitudeh ³. The number of possible distributions of the light quanta of a macroscopically defined radiation over these cells gives the entropy and with it all thermodynamic properties of the radiation.     

Conformally invariant model of the early universe

A model of the early universe in the Einstein theory of gravitation, supplemented by a conformalty invariant version of the Weinberg—Salam model, is considered. The conformai symmetry principle leads to the need to eliminate the Higgs potential from the expression for gravitational action, using the Lagrangian density of the model of Weinberg—Salam electroweak interactions as the material source, and to incorporate the conformally invariant Penrose—Chernikov—Tagirov term. In the limit of flat space, we arrive at the a version of the Weinberg—Salam model without Higgs particle-like excitations. In the conformalty invariant model under consideration, Higgs fields are absorbed by the spatial metric, so one can assume that the masses of elementary particles originate at the time when the evolution of the universe begins. Translated from Astrofizika, Vol. 41, No. 3, pp. 459–471, July–September, 1998.     

Phase information and the evolution of cosmological density perturbations

The Fourier transform of cosmological density perturbations can be represented in terms of amplitudes and phases for each Fourier mode. We investigate the phase evolution of these modes using a mixture of analytical and numerical techniques. Using a toy model of one-dimensional perturbations evolving under the Zel'dovich approximation as an initial motivation, we develop a statistic that quantifies the information content of the distribution of phases. Using numerical simulations beginning with more realistic Gaussian random-phase initial conditions, we show that the information content of the phases grows from zero in the initial conditions, first slowly and then rapidly when structures become non-linear. This growth of phase information can be expressed in terms of an effective entropy. Gaussian initial conditions are a maximum entropy realization of the initial power spectrum; gravitational evolution decreases the phase entropy. We show that our definition of phase entropy results in a statistic that explicitly quantifies the information stored in the phases of density perturbations (rather than their amplitudes), and that this statistic displays interesting scaling behaviour for self-similar initial conditions.     

Dynamic Structure of the GLONASS and GPS Orbital Space: Problem of Disposal of Retired Objects

The paper presents the results of a study of the dynamic structure of the orbital space of the navigation systems GLONASS and GPS. It is shown that the dynamic structure of the GLONASS region is determined by the action of one stable Lidov–Kozai secular resonance. The motion of almost all the retired objects of the GLONASS system is stable throughout the 100-year study period. In the GPS region, there is an orbital resonance and a large number of secular resonances. Their combined influence leads to a rapid increase in the eccentricity of the orbits of the retired objects of the system. Features of the dynamic structure of the orbital space are used to find the graveyard (parking) orbits of the retired objects of navigation systems.     

Phase-space volume of regions of trapped motion: multiple ring components and arcs

The phase-space volume of regions of regular or trapped motion, for bounded or scattering systems with two degrees of freedom respectively, displays universal properties. In particular, sudden reductions in the phase-space volume or gaps are observed at specific values of the parameter which tunes the dynamics; these locations are approximated by the stability resonances. The latter are defined by a resonant condition on the stability exponents of a central linearly stable periodic orbit. We show that, for more than two degrees of freedom, these resonances can be excited opening up gaps, which effectively separate and reduce the regions of trapped motion in phase space. Using the scattering approach to narrow rings and a billiard system as example, we demonstrate that this mechanism yields rings with two or more components. Arcs are also obtained, specifically when an additional (mean-motion) resonance condition is met. We obtain a complete representation of the phase-space volume occupied by the regions of trapped motion.     

Thermodynamical analysis of acoustic Schwarzschild black hole

In this paper, we analyze the thermodynamic properties of acoustic Schwarzschild black holes with its parameter. The study is conducted in the extended phase space displaying the phase transition. This phase transition is examined through Gibbs free energy, specific heat, and heat capacity. Later, we discuss the thermal stability of acoustic black holes through Hawking temperature by identifying their stable and unstable regions. We calculate the corrected entropy to examine the thermal fluctuations. Through the corrected entropy we observe that there is no fluctuation in the case of small black holes. We also discuss the energy emission process from acoustic black holes. Moreover, we employ the generalized uncertainty principle to obtain a modified Lagrangian equation. We analyze the tunneling and Hawking temperature of the acoustic Schwarzschild black hole after solving the field equations.     

On the use of the autonomous Birkhoff equations in Lie series perturbation theory

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.     

Kolmogorov entropy as a measure of disorder in some non-integrable Hamiltonian systems

This paper is devoted to study the stochastic behaviour of some Hamiltonian systems with closed velocity curves. We investigate Hamiltonians already studied by Ali and Somorjai (1). These authors, by discussing Poincaré's surfaces of section for several energy values, gave a qualitative evaluation of the stochasticity of the systems.Here we present a quantitative study of this stochastic behaviour. For each energy we compute the Lyapunov characteristic exponents of fifty orbits chosen at random, in order to calculate the Kolmogorov entropy by Pesin's formula. Our results are in agreement with those of Ali and Somorjai: the disorder does not increase monotonically with increasing energy. However, we find that the largest entropy does not necessarily correspond to the maximum of the stochastic volume. The Kolmogorov entropy thus appears to be a good measure of the degree of disorder of dynamical systems.     

A Hopf variables view on the libration points dynamics

This study analyzes a recently discovered class of exterior transfers to the Moon. These transfers terminate in retrograde ballistic capture orbits, i.e., orbits with negative Keplerian energy and angular momentum with respect to the Moon. Yet, their Jacobi constant is relatively low, for which no forbidden regions exist, and the trajectories do not appear to mimic the dynamics of the invariant manifolds of the Lagrange points. This paper shows that these orbits shadow instead lunar collision orbits. We investigate the dynamics of singular, lunar collision orbits in the Earth–Moon planar circular restricted three-body problem, and reveal their rich phase space structure in the medium-energy regime, where invariant manifolds of the Lagrange point orbits break up. We show that lunar retrograde ballistic capture trajectories lie inside the tube structure of collision orbits. We also develop a method to compute medium-energy transfers by patching together orbits inside the collision tube and those whose apogees are located in the appropriate quadrant in the Sun–Earth system. The method yields the novel family of transfers as well as those ending in direct capture orbits, under particular energetic and geometrical conditions.     

A Complete Catalogue of High-Speed Solar Wind Streams during Solar Cycle 23

High-speed solar wind streams (HSSWSs) are ejected from the Sun and travel into the interplanetary space. Because of their high speed, they carry out energetic particles such as protons and heavy ions, which leads to an increase in the mean interplanetary magnetic field (IMF). Since the Earth is in the path of those streams, Earth’s magnetosphere interacts with the disturbed magnetic field, leading to a significant radiation-induced degradation of technological systems. These interactions provide an enhanced energy transfer from the solar wind/IMF system into the Earth’s magnetosphere and initiate geomagnetic disturbances that may have a possible impact on human health. Solar cycle 23 was a particularly unusual cycle with many energetic phenomena during its descending phase and also had an extended minimum. We have identified and catalogued the HSSWSs of this cycle and determined their characteristics, such as their maximum velocity, beginning and ending time, duration, and possible sources. We identified 710 HSSWSs and compared them with the corresponding characteristics of the streams of previous solar cycles. For first time, we used the CME data to study the stream sources, which led to useful results for the monitoring and forecasting of space weather effects.     

A Friedmann universe in the quantization scheme of reduced phase space

It is shown how, using action-angle variables in the scheme of quantization of reduced phase space, to construct a wave function for a Friedmann universe filled with harmonic excitations of photons and massive fermions, as well as to find physically observable quantities. Such an approach leads to an equation of the Schrödinger type, admits of a simple interpretation of the wave Junction, and enables one to trace the connection with classical evolution, which is fully reproduced in the proposed model. Within the framework of the reduced theory, massive fermions are described by action of the Nambu-Jona-Lasinio type with spontaneous breaking of chiral symmetry. The proposed scheme leads to consequences that are consistent with the Mach principle and Dirac’s hypothesis of large numbers.     

The fundamental cosmic distance scale: state of the art and the Gaia perspective

In recent papers we had developed a unified picture of black hole entropy and curvature which was shown to lead to Hawking radiation. It was shown that for any black hole mass, holography implies a phase space of just one quantum associated with the interior of the black hole. Here we study extremal rotating and charged black holes and obtain unique values for ratios of angular momentum to entropy, charge to entropy, etc. It turns out that these ratios can be expressed in terms of fundamental constants in nature, having analogies with other physical systems, like in condensed matter physics.