Geometric Derivation of the Delaunay Variables and Geometric Phases

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ¹ action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J ₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J ₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J ₂ effect as geometric phases.     

Periodic orbits of the planar collision restricted 3-body problem

Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also continue to this restricted problem the so called “comet orbits”. An erratum to this article can be found at     

联合GRACE卫星轨道及距离变率数据反演地球重力场方法研究

基于动力学法,研究联合GRACE卫星精密轨道及距离变率数据反演地球重力场的方法,该方法可对重力位系数及卫星初始状态误差同时进行有效校正。通过对各观测值模拟不同的随机误差,研究了不同精度观测值联合反演所能达到的精度,以及用相同精度的观测值进行联合反演时不同采样率对反演结果的影响,模拟计算结果表明：联合反演模式下,当距离变率精度为1 μm/s,卫星位置精度为2~3 cm,速度精度为0.1~0.5 mm/s时,加速度计精度为(1.0×10 ^-10~1.0×10^-9 m/s ²比较适合;将距离变率精度由1 um/s提高到0.1 um/s时,反演精度可获得相应提高;在观测值精度一定的情况下,联合反演算法宜采用5 s采样率。     

恒星内部电荷屏蔽对电子俘获反应的影响

本文分析了在恒星环境下，电荷强屏蔽对原子核电子俘获反应的影响，分析中忽略了电荷屏蔽对核跃迁矩阵元的影响．作为例子，分别讨论了电荷屏蔽对核素33S、30P和57Fe的电子俘获率的影响．结果表明：在较低温度和较高密度下，由于电荷屏蔽，其电子俘获率相对于无屏蔽情形有明显降低，这可能对恒星晚期演化和超新星爆发理论带来明显的影响．     

Regular and chaotic orbits in a self-consistent triaxial stellar system with slow figure rotation

We created a self-consistent triaxial stellar system through the cold disipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. The resulting system rotates slowly even though its total angular momentum is zero, i.e., it offers an example of figure rotation. The potential of the system was subsequently approximated with interpolating formulae yielding a smooth potential stationary in the rotating frame. The Lyapunov exponents could then be computed for a randomly selected sample of 3,472 of the bodies that make up the system, allowing the recognition of regular and partially and fully chaotic orbits. The regular orbits were Fourier analyzed and classified using their locations on the frequency map. A comparison with a similar non-rotating model showed that the fraction of chaotic orbits is slightly but significantly enhanced in the rotating model; alternatively, there are no significant differences between the corresponding fractions neither of partially and fully chaotic orbits nor of long axis tubes, short axis tubes, boxes and boxlets among the regular orbits. This is a reasonable result because the rotation causes a breaking of the symmetry that may increase chaotic effects, but the rotation velocity is probably too small to produce any other significant differences. The increase in the fraction of chaotic orbits in the rotating system seems to be due mainly to the effect of the Coriolis force, rather than the centrifugal force, in good agreement with the results of other investigations.     

Impact of the Mass Parameter on Particle Dynamics in a Ring Configuration of N Bodies

Parameters play a very important and determinative role in the dynamics of a dynamical system as well as in the formation of its particular characteristics. In this paper we investigate the way in which a large scale variation of the mass parameter, influences the behavior of a mass-less particle which moves in the vicinity of a ring arrangement of N-bodies. More precisely, we study the impact of this parameter on periodic motions and their characteristics.     

Structures in the phase space of a four dimensional symplectic map

We numerically investigate the projections of non periodic orbits in a 4-dimensional (4-D) symplectic map composed of two coupled 2-dimensional (2-D) maps. We describe in detail the structures that are produced in different planes of projection and we find how the morphology of the 4-D orbits is influenced by the features of the 2-D maps as the coupling parameter increases. We give an empirical law that describes this influence.     

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.     

Evolutionary and structural properties of mirror star MACHOs

There can exist a hidden sector of the Universe in the form of parallel “mirror” world which has the same particle physics as the observable world and interacts with the latter only gravitationally. Big Bang nucleosynthesis bounds demand that the mirror sector should have a smaller temperature than the ordinary one. This implies that the mirror matter could play a role of dark matter, and in addition its chemical content should be dominated by helium. Here we study the evolutionary and structural properties of the mirror stars which essentially are similar to that of the ordinary stars but with higher helium contents. Being invisible in terms of photons, they could be observed only as MACHOs in the microlensing experiments. Using a numerical code, we compute evolution of stars with large helium abundances (Y = 0.30–0.80) and a wide range of masses, from 0.5 to 10 M. We found that helium dominated mirror star should have much faster evolutionary time (up to a factor 30) than the ordinary star with the same mass. In addition, we show the diagrams of luminosities, effective temperatures, central temperatures and densities, and compute the masses of the He-core at ignition and the minimum mass for carbon ignition, for different chemical compositions. The general conclusion is that mirror stars evolve faster as compared to ordinary ones, and explode earlier as type II supernovae, thus enriching the galactic halo of processed mirror gas with higher metallicity, with implications for MACHO observations and galaxy evolution.     

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.