Triple approaches in the plane isosceles equal-mass three-body problem

We analyze flyby-type triple approaches in the plane isosceles equal-mass three-body problem and in its vicinity. At the initial time, the central body lies on a straight line between the other two bodies. Triple approaches are described by two parameters: virial coefficient k and parameter $\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }$ , where $\dot r$ is the relative velocity of the extreme bodies and $\dot R$ is the velocity of the central body relative to the center of mass of the extreme bodies. The evolution of the triple system is traceable until the first turn or escape of the central body. The ejection length increases with closeness of the triple approach (parameter k). The longest ejections and escapes occur when the extreme bodies move apart with a low velocity at the time of triple approach. We determined the domain of escapes; it corresponds to close triple approaches (k>0.8) and to μ in the range ?0.2<μ<0.7. For small deviations from the isosceles problem, the evolution does not differ qualitatively from the isosceles case. The domain of escapes decreases with increasing deviations. In general, the ejection length increases for wide approaches and decreases for close approaches.     

4颗卫星情况的几何优化法修正

在GPS定位与导航中,用户可以通过选择至少4颗可见卫星的观测,得到测站位置和站钟差信息,卫星相对于测站的几何关系直接影响到定位和导航的精度,考虑用4颗卫星进行定位,在当前有一定的现实意义,在KihaLra等人工作的基础上对四面体的性质进行了深入讨论,推导了4颗可见卫星构成四面体体积的计算公式,实例分析证明该计算公式更合理、更精确。     

Stochasticity of planetary orbits in double star systems. II

Cosmogonical theories as well as recent observations allow us to expect the existence of numerous exo-planets, including in binaries. Then arises the dynamical problem of stability for planetary orbits in double star systems. Modern computations have shown that many such stable orbits do exist, among which we consider orbits around one component of the binary (called S-type orbits). Within the framework of the elliptic plane restricted three-body problem, the phase space of initial conditions for fictitious S-type planetary orbits is systematically explored, and limits for stability had been previously established for four nearby binaries which components are nearly of solar type. Among stable orbits, found up to distance of their sun of the order of half the binarys periastron distance, nearly-circular ones exist for the three binaries (among the four) having a not too high orbital eccentricity. In the first part of the present paper, we compare these previous results with orbits around a 16 Cyg B-like binarys component with varied eccentricities, and we confirm the existence of stable nearly-circular S-type planetary orbits but for very high binarys eccentricity. It is well-known that chaos may destroy this stability after a very long time (several millions years or more). In a first paper, we had shown that a stable planetary orbit, although chaotic, could keep its stability for more than a billion years (confined chaos). Then, in the second part of the present paper, we investigate the chaotic behaviour of two sets of planetary orbits among the stable ones found around 16 Cyg B-like components in the first part, one set of strongly stable orbits and the other near the limit of stability. Our results show that the stability of the first set is not destroyed when the binarys eccentricity increases even to very high values (0.95), but that the stability of the second set is destroyed as soon as the eccentricity reaches the value 0.8.     

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.     

Long-term evolution of spheroidal galaxies with locally isotropic velocity distribution

The short-term evolution of spheroidal galaxies has been explored by S. Chandrasekhar et al. and G. S. Sunder et al. In this paper, we study their long-term evolution with Laskar's method of frequency analysis. The main new results are as follows: (1) There exists a unique equilibrium, which is spherically symmetric. This equilibrium has a critical linear stability. (2) Generally speaking, the semi-axes exhibit quasiperiodic or nearly quasi-periodic (in a time scale longer than a Hubble time) oscillations around the radius of the above-mentioned equilibrium, so the equilibrium is practically stable. (3) There are cases in which one of the semi-axes tends fast to zero while the other to some finite value. The limit state is generally planar rather than linear, i.e. it is the symmetric semi-axis that tends to zero. This implies that some disk galaxies may have originated from spheroidal pregalaxy material.     

Simulations of spheroidal systems with substructure: trees in fields

We present a hybrid technique of N -body simulation to deal with collisionless stellar systems having an inhomogeneous global structure. We combine a treecode and a self-consistent field code such that each of the codes models a different component of the system being investigated. The treecode is suited to treatment of dynamically cold or clumpy components, which may undergo significant evolution within a dynamically hot system. The hot system is appropriately evolved by the self-consistent field code. This combined code is particularly suited to a number of problems in galactic dynamics. Applications of the code to these problems are briefly discussed.     

Dynamical studies of cluster pairs in the Magellanic Clouds

We performed N -body simulations of star cluster encounters with Hernquist's TREECODE in a CRAY YMP-2E computer under different initial conditions (relative positions and velocities, cluster sizes, masses and concentration degrees). The total number of particles per simulation ranged from 1024 to 20480. These models are compared with a series of isodensity maps of cluster pairs in the Magellanic Clouds. Evidence is found that during the interactions, transient morphological effects such as an expanded halo, isophotal deformation and isophotal twisting can occur as a result of tidal effects and dynamical friction. The simulations also show that different outcomes are possible depending on the initial parameters: (i) long-standing changes of concentration degree can occur after the collision; (ii) one member can disaggregate; or (iii) the pair can coalesce into a single cluster with a distinct structure compared with the original ones. These simulations can reproduce a wide range of morphological structures in observed cluster pairs.     

Linear response of galactic haloes to adiabatic gravitational perturbations

We determine the response of a self-similar isothermal stellar system to small adiabatic gravitational perturbations. For odd spherical harmonics the response is identical to the response of the analogous isothermal fluid system. For even spherical harmonics the response can be regarded as an infinite series of wavetrains in log r , implying alternating compression and rarefaction in equal logarithmic radius intervals. Partly because of the oscillatory nature of the solutions, tidal fields from external sources are not strongly amplified by an intervening isothermal stellar system, except at radii ≲10^−3.5 times the satellite radius; at some radii the stellar system can even screen the external tidal field in a manner analogous to Debye screening. As Weinberg has pointed out, individual resonances in a stellar system can strongly amplify external tidal fields over a limited radial range, but we cannot address this possibility because we examine only adiabatic perturbations. We also discuss the application of our method to the halo response caused by the slow growth of an embedded thin disc.     

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC) ⁿ Hermite scheme for planetary N -body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N -body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N -body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC) ⁿ Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)² Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N  = 100,  m  = 10²⁵ g planetesimal system with the rms eccentricity 〈 e ²〉^1/2 ≲0.03.     

Orbit classification in arbitrary 2D and 3D potentials

A method of classifying generic orbits in arbitrary 2D and 3D potentials is presented. It is based on the concept of spectral dynamics introduced by Binney &38; Spergel that uses the Fourier transform of the time series of each coordinate. The method is tested using a number of potentials previously studied in the literature and is shown to distinguish correctly between regular and irregular orbits, to identify the various families of regular orbits (boxes, loops, tubes, boxlets, etc.), and to recognize the second-rank resonances that bifurcate from them. The method returns the position of the potential centre and, for 2D potentials, the orientation of the principal axes as well, should this be unknown. A further advantage of the method is that it has been encoded in a FORTRAN program that does not require user intervention, except for 'fine tuning' of search parameters that define the numerical limits of the code. The automatic character makes the program suitable for classifying large numbers of orbits.