Practical Symplectic Methods with Time Transformation for the Few-Body Problem

The use of the extended phase space and time transformations for constructing efficient symplectic algorithms for the investigation of long term behavior of hierarchical few-body systems is discussed. Numerical experiments suggest that the time-transformed generalized leap-frog, combined with symplectic correctors, is one of the most efficient methods for such studies. Applications extend from perturbed two-body motion to hierarchical many-body systems with large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tidal evolution of hierarchical and inclined systems

We investigate the dynamical evolution of hierarchical three-body systems under the effect of tides, when the ratio of the orbital semi-major axes is small and the mutual inclination is relatively large (greater than 20°). Using the quadrupolar non-restricted approximation for the gravitational interactions and the viscous linear model for tides, we derive the averaged equations of motion in a vectorial formalism which is suitable to model the long-term evolution of a large variety of exoplanetary systems in very eccentric and inclined orbits. In particular, it can be used to derive constraints for stellar spin-orbit misalignment, capture in Cassini states, tidal-Kozai migration, or damping of the mutual inclination. Because our model is valid for the non-restricted problem, it can be used to study systems of identical mass or for the outer restricted problem, such as the evolution of a planet around a binary of stars. Here, we apply our model to various situations in the HD 11964, HD 80606, and HD 98800 systems.     

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In this paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10⁻³ to 10³. We also present more numerical results for short-term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.     

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.     

Eccentricity generation in hierarchical triple systems: the planetary regime

In previous papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. We considered systems with well-separated components and different initial setups (e.g., coplanar and non-coplanar orbits). However, the systems we examined had comparable masses. In the present paper, the validity of some of the formulae derived previously is tested by numerically integrating the full equations of motion for systems with smaller mass ratios (from 10⁻³ to 10³, i.e. systems with Jupiter-sized bodies). There is also discussion about HD 217107 and its planetary companions.     

Eccentricity evolution in hierarchical triple systems with eccentric outer binaries

We develop a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular, while the outer one is eccentric. We consider coplanar systems with well-separated components and comparable masses. The derivation of short-period terms is based on an expansion of the rate of change of the Runge–Lenz vector. Then, the short-period terms are combined with secular terms, obtained by means of canonical perturbation theory. The validity of the theoretical equations is tested by numerical integrations of the full equations of motion.     

Symplectic Tangent map for Planetary Motions

Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in general. This revised version was published online in July 2006 with corrections to the Cover Date.     

Eclipse monitoring of eccentric binary systems

The observation of apsidal motions in eclipsing binaries is a very rewarding area of research which requires only moderate or small telescopes equipped with a photoelectric photometer. Important contributions can be made to the study of stellar internal structure through these observations, as well as the verification of the theory of general relativity using the equations of orbital motion. The main objectives of such an observational effort are described, together with a list of candidate binary systems. An appeal to photometrists with small telescopes around the world is made to observe these eccentric and well-detached eclipsing binaries.     

On the stability of close binaries in hierarchical three-body systems

The Hill-type stability (cf. closure of the zero-velocity curves in the circular restricted three-body problem) of general hierarchical three-body systems is examined analytically in the case where the total mass of the binary is small in comparison to the mass of the external body (e.g. systems of the type Planet-Satellite-Sun, Planet-Planet-Star, etc.). This is compared with results derived by Szebehely, Markellos and Roy in the Planet-Satellite-Sun case of the circular restricted three-body problem. It is demonstrated how the Hill-type stability is affected by the sense of revolution of the binary, i.e. corotational or contrarotational, and the mass ratio within the binary. The effect of the difference in longitudes of the bodies in their orbits is also examined.     

Observational aspect of the three-body problem

Distinguishes three phases in the history of triple-star systems research. The necessity of obtaining some crucial observational data on these systems is also pointed out. An insight into the observational material concerning triple star systems of the hierarchical type and some special properties of this group on the basis of observational data are presented, as well.     

On the Milankovitch orbital elements for perturbed Keplerian motion

We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution.     

A chain regularization method for the few-body problem

A regularization method for integrating the equations of motion of small N-body systems is discussed. We select a chain of interparticle vectors in such a way that the critical interactions requiring regularization are included in the chain. The equations of motion for the chain vectors are subsequently regularized using the KS-variables and a time transformation. The method has been formulated for any number of bodies, but the most important application appears to be in the four-body problem which is therefore discussed in detail.     

Type I and two-gap superconductivity in neutron star magnetism

Neutron star inner cores with several charged baryonic components are likely to be analogues of the two-gap superconductor which is of current interest in condensed-matter physics. Consequently, type I superconductivity is less probable than type II but may nevertheless be present in some intervals of matter density. The intermediate-state structure formed at finite magnetic flux densities after the superconducting transitions is subject to buoyancy, frictional and neutron vortex interaction forces. These are estimated and it is shown that the most important frictional force is that produced by the stable stratification of neutron star matter, the irreversible process being diffusion in the normal, finite magnetic flux density, parts of the structure. The length-scale of the structure, in directions perpendicular to the local magnetic field is of crucial importance. For small scales, the flux comoves with the neutron vortices, as do the proton vortices of a type II superconductor. But for much larger length-scales, flux movement tends to that expected for normal charged Fermi systems.     

Characteristic scales of subclusters in the Virgo area

In this paper, we present a “peak number count method”, based on wavelet transformation, and use it to explore the structure and substructures in the Virgo area. It is found that, besides the two-humped structure, there are also quite a lot of small subclusters with characteristic scales of 180′ and 120′. This result strongly supports the hierarchical clustering model of structures. The wavelet coefficient map clearly shows these substructures and their distributions. Additionally, two new small subclusters are discovered.     

Virgo星系团内次结构的特征尺度

在小波分析的基础上,用峰值计数法考察了Ｖｉｒｇｏ团天区的结构与次结构．在Ｖｉｒｇｏ团典型的双峰结构下面,发现还有１８０’和１２０’两个明显的次结构的特征尺度．该结果有力地支持了等级式的成团模式．与之对应的波长下的小波强度分布清楚地显示了Ｖｉｒｇｏ团丰富的次结构及其分布情况,从中还发现了两个新的较小的次结构。     

等级三星系统的运动学研究

统计分析表明,在目前的观测精度(Imas)下,由于三体效应大多数等级三星系统已不能用经典的双二体运动学模型来描述;即使对于应用需求而言,该模型也不再适用于描述三体视效应较强的等级三星系统.因为在实用星表参考架中不宜采用数据量较大的数值历表来描述子星的运动,所以有必要针对等级三星系统建立尽可能简单实用的运动学模型.借助已有的观测资料和拟合研究结果,给出了6个等级三星系统的质量参数和初始条件,进而根据实用需求讨论了这些系统的运动学描述问题.     

Early formation and evolution of galaxies

I review success and failure of the hierarchical galaxy formation model. In this scenario, the morphology of galaxies is regulated by the mode of gas accretion and intimately linked to discrete accretion events. Some of the common misconceptions about hierarchical clustering are discussed. The need of a self-consistent approach that incorporates the chemical and dynamical evolution on small scalessimultaneously with the cosmological framework of structure formation on large scales is emphasized. This revised version was published online in August 2006 with corrections to the Cover Date.     

Source regions for type II radio bursts

The flares associated with reported type II bursts from 1964–1973 (hereafter abbreviated source-flares) are analyzed with respect to their importance, duration, and heliographic distribution. The source-flares for type II bursts generally are normal to small in area and normal to brilliant in intensity; however, they have slightly longer durations than would be expected from flares with such small areas. Flares associated with type II bursts are distributed uniformly east and west of the central meridian. This implies that type II bursts cannot be narrow-beamed, radially-propagating radio emitters. The latitude of the occurrence of source-flares shows a drift with time that is not significantly different from that of sunspots. The drift rate is a maximum during the period of maximum solar activity. The heliographic distribution of source-flares shows large-scale organization into similarly shaped regions (source-regions) separated by 180° of longitude in each polar hemisphere. The shape of the source-regions differs between the northern and the southern hemispheres. The source-regions exhibit growth and motion characteristics which suggest that their development is due partly to the effects of differential rotation. An analysis of the plage regions which are associated with source-flares shows that relatively few plages ever produce type II bursts, yet certain plages produce type II bursts repeatedly. One active region produced type II bursts on six consecutive disc passages. Since the source-regions for type II bursts are large-scale, persistent solar features that show motion and development related to the solar cycle, an evaluation of the distinctive fluid motion characteristics of these regions appears to be an important step in the determination of specific environmental configurations which can produce solar shock waves.