Generalized Hill's problem: Some cases of complete integrability

In this paper, we are investigating cases of integrability in the planar Hill's problem. The external potential U _extis supposed to be time independent in a given uniformly rotating frame. Cases of integrability of the relative motion of two interacting particles in the vicinity of an equilibrium solution of U _extare found. In all these cases, the form of the second integral is explicitly given, the first being the Jacobian one. Cases in which the interacting potential U between the two particles is of newtonian type are particularized.     

Non—integrability of Hill's lunar problem

We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame.     

On the periodic solutions of a particular Lame equation

We consider the Hill's equation: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqOVdGhabaGaamizaiaadsha% daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWGTbGaai% ikaiaad2gacqGHRaWkcaaIXaGaaiykaaqaaiaaikdaaaGaam4qamaa% CaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaaiykaiabe67a4jabg2% da9iaaicdaaaa!4973!\[\frac{{d^2 \xi }}{{dt^2 }} + \frac{{m(m + 1)}}{2}C^2 (t)\xi = 0\]Where C(t) = Cn (t, {frbuilt|1/2}) is the elliptic function of Jacobi and m a given real number. It is a particular case of theame equation. By the change of variable from t to defined by: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaOWaaiqaaq% aabeqaamaalaaajaaybaGaamizaGGaaiab-z6agbqaaiaadsgacaWG% 0baaaiabg2da9OWaaOaaaKaaGfaacaGGOaqcKbaG-laaigdajaaycq% GHsislkmaaleaajeaybaGaaGymaaqaaiaaikdaaaqcaaMaaeiiaiaa% bohacaqGPbGaaeOBaOWaaWbaaKqaGfqabaGaaeOmaaaajaaycqWFMo% GrcqWFPaqkaKqaGfqaaaqcaawaaiab-z6agjab-HcaOiab-bdaWiab% -LcaPiab-1da9iab-bdaWaaakiaawUhaaaaa!51F5!\[\left\{ \begin{array}{l}\frac{{d\Phi }}{{dt}} = \sqrt {(1 - {\textstyle{1 \over 2}}{\rm{ sin}}^{\rm{2}} \Phi )} \\\Phi (0) = 0 \\\end{array} \right.\]it is transformed to the Ince equation: (1 + · cos(2)) y + b · sin(2) · y + (c + d · cos(2)) y = 0 where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqcaawaaiaadggacq% GH9aqpcqGHsislcaWGIbGaeyypa0JcdaWcgaqaaiaaigdaaeaacaaI% ZaGaaiilaiaabccacaWGJbGaeyypa0Jaamizaiabg2da9aaacaqGGa% WaaSaaaKaaGfaacaWGTbGaaiikaiaad2gacqGHRaWkcaaIXaGaaiyk% aaqaaiaaiodaaaaaaa!4777!\[a = - b = {1 \mathord{\left/{\vphantom {1 {3,{\rm{ }}c = d = }}} \right.\kern-\nulldelimiterspace} {3,{\rm{ }}c = d = }}{\rm{ }}\frac{{m(m + 1)}}{3}\]In the neighbourhood of the poles, we give the expression of the solutions.The periodic solutions of the Equation (1) correspond to the periodic solutions of the Equation (3). Magnus and Winkler give us a theory of their existence. By comparing these results to those of our study in the case of the Hill's equation, we can find the development in Fourier series of periodic solutions in function of the variable and deduce the development of solutions of (1) in function of C(t).     

On the transverse vibrations of a revolving tether

We analyse the transverse vibrations of a tether, modelled as an inextensible cable, and revolving at an average rate equal to the orbital rate. The reference motion is a revolving rigid tether. During this motion the force in the tether (time and location dependent) remains, in a first approximation, aligned with the tether axis. Separation of variables for the vibrations about this motion gives a Legendre equation for the spatial dependency of the deformations and Hill's equations for time dependency of the in- and out-of-plane deformations. The boundary conditions on the Legendre equation generate a series of admissible values of the separation constant that become equidistant. The two Hill's equations generate a series of intervals, contracting to equidistant critical values, where the solutions are unbounded. The admissible values of the separation constant must avoid these intervals. Asymptotic expressions for the separation constant and the critical values are given. The first and second in-plane deformation mode arc unstable for zero end masses. By increasing the ratio of the concentrated over the distributed mass the deformation modes can be stabilised and the values of the separation constant can be made a multiple of the distribution of the critical points. Introducing unequal tip masses does not affect this result.     

Spiral Structures and Chaotic Scattering of Coorbital Satellites

The fractal nature of the transitions between two sets of orbits separated by heteroclinic or homoclinic orbits is well known. We analyze in detail this phenomenon in Hill's problem where one set of orbits corresponds to coorbital satellites exchanging semi-major axis after close encounter (horse-shoe orbits) and the other corresponds to orbits which do not exchange semi-major axis (passing-by orbits). With the help of a normalized approximation of the vicinity of unstable periodic orbits, we show that the fractal structure is intimately tied to a special spiral structure of the Poincaré maps. We show that each basin is composed of a few well behaved areas and of an infinity of intertwined tongues and subtongues winding around them. This behaviour is generic and is likely to be present in large classes of chaotic scattering problems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Generalized hill's problem Lagrangian Hill's case

In this paper, we investigate a generalization of the Hill's problem to the case where no restriction is made about the nature of the field of force perturbing two small bodies in gravitational interaction. We apply the general equations obtained to the dynamics of two bodies located in the vicinity of the triangular lagrangian points of the restricted three-body problem.     

Stochastic Hill's equations for the study of errant rocket burns in orbit

The problem of errant rocket burns in low Earth orbit is of growing interest, especially in the area of safety analysis of nuclear powered spacecraft. The development of stochastic Hill's equations provides a rigorous mathematical tool for the study of such errant rocket maneuvers. These equations are analyzed within the context of a theory of linear dynamical systems driven by a random white noise. It is established that the trajectories of an errant rocket are realizations of a Gauss-Markov process, whose mean vector is given by the solution of a deterministic rocket problem. The time-dependent covariance matrix of the process is derived in an explicit form.     

Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.     

EMCCD电荷倍增驱动电路分析与PCB设计

电子倍增CCD（Electron-Multiplying CCD,EMCCD）的电荷倍增驱动电路是实现这类CCD器件片上增益功能的关键。介绍了TC285SPD电荷倍增驱动器的参考电路的结构,详细分析其电路原理,并对该电路进行了PSpice仿真分析。对电荷倍增驱动电路的3种不同的印刷电路板（Printed Circuit Board,PCB）布局布线方案进行了实际测试,分析了存在的问题,找到了该电路PCB设计的正确方法。     

Estimate of the Transition Value of Librational Invariant Curves

We investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. Hénon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter.     

基于STK的寿命末期GEO卫星轨道演化分析

地球同步轨道上的GEO卫星接近寿命末期时,星上的有效载荷和电子元器件等仍能正常应用,此时如果只做东西方向的位置保持,可以大幅度延长卫星的在轨工作寿命。简要介绍了引起GEO卫星轨道变化的主要摄动力,利用STK软件分析寿命末期GEO卫星的运动规律,并给出这类GEO卫星轨道演化的仿真分析流程,用以指导寿命末期GEO卫星的开发应用工作。     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Optimal deflection of NEOs en route of collision with the Earth

Recently, a method for the n-body analysis of the velocity change required to deflect a hazardous near-Earth object (NEO) was presented by Carusi et al. [Carusi, A., Valsecchi, G.B., D'Abramo, G., Boattini A., 2002. Icarus 159, 417-422]. We extent this method in order to optimize the velocity change vector instead of its along-track magnitude. From an application of both methods to a fictitious NEO we find Carusi's parallel approach to be reasonable for phases of unperturbed two-body motion. But, for orbit phases inhering third-body perturbations, i.e., for planetary close approaches or prior to a collision, the results obtained from the new method show the radial component of deflection impulse to play a major role. We show that a fivefold greater efficiency can be achieved by a deflection impulse being non-parallel to orbital velocity. The new method is applied to two possible 99942 Apophis impact trajectories in order to provide constraints for future Apophis deflection mission analysis.     

Spectropolarimetry of a complete infrared-selected sample of Seyfert 2 galaxies

We report the results of a spectropolarimetric survey of a complete far-infrared-selected sample of Seyfert 2 galaxies. We have found polarized broad H α emission in one new source, NGC 5995. In the sample as a whole, there is a clear tendency for galaxies in which we have detected broad H α in polarized light to have warm mid–far-infrared colours     in agreement with our previous results. However, a comparison of the optical, radio and hard X-ray properties of these systems leads us to conclude that this is a secondary consequence of the true mechanism governing our ability to see scattered light from the broad-line region. We find a strong trend for galaxies showing such emission to lie above a critical value of the relative luminosity of the active core to the host galaxy (as measured from the [O  iii ] 5007-Å equivalent width) which varies as a function of the obscuring column density as measured from hard X-ray observations. The warmth of the infrared colours is then largely due to a combination of the luminosity of the active core, the obscuring column and the relative importance of the host galaxy in powering the far-infrared emission, and not solely orientation as we inferred in our previous paper. Our data may also provide an explanation as to why the most highly polarized galaxies, which appear to have tori that are largely edge-on, are also the most luminous and have the most easily detectable scattered broad H α .     

Fast rotation of neutron stars and equation of state of dense matter

Fast rotation of compact stars (at sub-millisecond period) and, in particular, their stability, are sensitive to the equation of state (EOS) of dense matter. Recent observations of XTE J1739-285 suggest that it contains a neutron star rotating at 1122 Hz. At such rotational frequency the effects of rotation on star’s structure are significant. We study the interplay of fast rotation, EOS, and gravitational mass of a sub-millisecond pulsar. We discuss the EOS dependence of spin-up to a sub-millisecond period, via mass accretion from a disk in a low-mass X-ray binary.     

Relativistic simulations of the phase-transition-induced collapse of neutron stars

An increase in the central density of a neutron star may trigger a phase transition from hadronic matter to deconfined quark matter in the core, causing it to collapse to a more compact hybrid star configuration. We present a study of this, building on previous work by Lin et al.. We follow them in considering a supersonic phase transition and using a simplified equation of state, but our calculations are general relativistic (using 2D simulations in the conformally flat approximation) as compared with their 3D Newtonian treatment. We also improved the treatment of the initial phase transformation, avoiding the introduction of artificial convection. As before, we find that the emitted gravitational wave spectrum is dominated by the fundamental quasi-radial and quadrupolar pulsation modes but the strain amplitudes are much smaller than suggested previously, which is disappointing for the detection prospects. However, we see significantly smaller damping and observe a non-linear mode resonance which substantially enhances the emission in some cases. We explain the damping mechanisms operating, giving a different view from the previous work. Finally, we discuss the detectability of the gravitational waves, showing that the signal-to-noise ratio for current or second generation interferometers could be high enough to detect such events in our Galaxy, although third generation detectors would be needed to observe them out to the Virgo cluster, which would be necessary for having a reasonable event rate.