On the Chermnykh-Like Problems: I. the Mass Parameter μ = 0.5

Following Papadakis (2005)'s numerical exploration of the Chermnykh's problem, we here study a Chermnykh-like problem motivated by the astrophysical applications. We find that both the equilibrium points and solution curves become quite different from the ones of the classical planar restricted three-body problem. In addition to the usual Lagrangian points, there are new equilibrium points in our system. We also calculate the Lyapunov Exponents for some example orbits. We conclude that it seems there are more chaotic orbits for the system when there is a belt to interact with.     

Efficient Method for Computing with Certainty Periodic Orbits on a Surface of Section

We present an improved method for locating periodic orbits of a dynamical system of arbitrary dimension. The method first employs the characteristic bisection method (CBM) to roughly locate a periodic orbit, followed by the quadratically convergent Newton method to rapidly refine its position. The method is applied to the physically interesting example of the two degrees of freedom photogravitational problem, and shown to surpass the CBM algorithm and Newton's method alone.     

Migration of giant planets in a time-dependent planetesimal accretion disc

In this paper we develop further the model for the migration of planets introduced in Del Popolo et al. We first model the protoplanetary nebula as a time-dependent accretion disc, and find self-similar solutions to the equations of the accretion disc that give us explicit formulae for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc, and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parameter α . We find that planets that are embedded in planetesimal discs, having total mass of  10^-4-0.1 M_⊙  , can migrate inward a large distance for low values of α (e.g.,   α ≃10^-3-10^-2)  and/or large disc mass, and can survive only if the inner disc is truncated or because of tidal interaction with the star. Orbits with larger a are obtained for smaller values of the disc mass and/or for larger values of α . This model may explain several orbital features of the recently discovered giant planets orbiting nearby stars.     

QUANTIFYING SIMILARITY FOR HANDLING INFORMATION IN KNOWLEDGE BASES

When constructing diagnostic systems or using knowledge-based systems,e.g.in analytical chemistry,features of different type and character,represented by numbers,trajectories or linguistic variables suchas intensities or colours,must be considered.To find neighbourhoods or to fill in missing values,thenotion of similarity is of essential importance.The paper presents a new fuzzy-set-theory-based approachto quantifying similarity and provides a system of rules to be implemented into the diagnostic part of theknowledge base to be used.     

Extrasolar planets and the rotation and axisymmetric mass-loss of evolved stars

I examine the implications of the recently found extrasolar planets on the planet-induced axisymmetric mass-loss model for the formation of elliptical planetary nebulae (PNe). This model attributes the low departure from spherical mass-loss of upper asymptotic giant branch (AGB) stars to envelope rotation which results from deposition of orbital angular momentum of the planets. Since about half of all PNe are elliptical, i.e., have low equatorial to polar density contrast, it was predicted that about 50 per cent of all Sun-like stars have Jupiter-like planets around them, i.e., a mass about equal to that of Jupiter, M _J, or more massive. In the light of the new findings that only 5 per cent of Sun-like stars have such planets, and a newly proposed mechanism for axisymmetric mass-loss, the cool magnetic spots model, I revise this prediction. I predict that indeed ∼50 per cent of PN progenitors do have close planets around them, but the planets can have much lower masses, as low as ∼0.01 M _J, in order to spin-up the envelopes of AGB stars efficiently. To support this claim, I follow the angular momentum evolution of single stars with main-sequence mass in the range of 1.3–2.4 M_⊙ , as they evolve to the post-AGB phase. I find that single stars rotate much too slowly to possess any significant non-spherical mass-loss as they reach the upper AGB. It seems, therefore, that planets, in some cases even Earth-like planets, are sufficient to spin-up the envelope of these AGB stars for them to form elliptical PNe. The prediction that on average several such planets orbit each star, as in the Solar system, still holds.     

On the Sitnikov problem

A mapping which reflects the properties of the Sitnikov problem is derived. We study the mapping instead of the original differential equations and discover that there exists a hyperbolic invariant set. The theoretical prediction of the disorder region agrees remarkably with numerical results. We also discuss the LCEs and KS-entropy of the dynamical system.This project is supported by the National Science Foundation of China.     

P–T modelling of the andalusite–kyanite–andalusite sequence and related assemblages in high-Al graphitic pelites. Prograde and retrograde paths in a late kyanite belt in the Variscan Iberia

The exceptional andalusite–kyanite–andalusite sequence occurs in Al‐rich graphitic slates in a narrow pelite belt on the hangingwall of a ductile normal fault in NW Variscan Iberia. Early chiastolite is replaced by Ky–Ms–Pg aggregates, which are overgrown by pleochroic andalusite near granites intruded along the fault. Slates plot in AKFM above the chloritoid‐chlorite tie‐line. Their P–T grids are modelled with Thermocalc v2.7 and the 1998 databases in the NaKFMASH and KFMASH systems. The univariant reaction Ctd + And/Ky = St + Chl + Qtz + H₂O ends at progressively lower pressure as F/FM increases and A/AFM decreases, shrinking the assemblage Cld–Ky–Chl, and opening a chlorite‐free Cld–Ky trivariant field on the low temperature reaction side. This modelling matches the observed absence of chlorite in high F/FM rocks, which is restricted to low pressure in the andalusite stability field. The P–T path deduced from modelling shows a first prograde event in the andalusite field followed by retrogression into the kyanite field, most likely coupled with a slight pressure increase. The final prograde evolution into the andalusite field can be explained by two different prograde paths. Granite intrusion caused the first prograde part of the path with andalusite growth. The subsequent thermal relaxation, together with a_H2O decrease, generated the retrograde andalusite–kyanite transformation, plus chlorite consumption and chloritoid growth. This transformation could have been related to folding in the beginning, and aided later by downthrowing due to normal faulting. Heat supplied by syntectonic granite intrusion explains the isobaric part of the path in the late stages of evolution, causing the prograde andalusite growth after the assemblage St–Ky–Chl. Near postectonic granites, a prograde path with pressure decrease originated the assemblage St–And–Chl.     

西秦岭文县地区关家沟组海底扇沉积体系

对西秦岭南部文县地区关家沟组的成因,学者之间的认识存在较大分歧,长期以来被看作是可与南沱组对比的冰碛层.关家沟组形成于新元古代末期(晚震旦世),沿扬子板块西北边缘分布.根据地层剖面和扇地质体的平面填图研究表明,关家沟组存在两种沉积体系--海底扇沉积体系和盆地沉积体系;海底扇沉积体系中发育大量的各种类型的粒序层理及鲍玛层序;扇体的剖面结构相似于Walker(1979)提出的海底扇反旋回沉积序列,层理类型与Winn和Dott(1979)提出的海底扇的层理类型类似;扇体的平面结构(填图追索圈定)清楚地显示了海底扇砂砾岩与盆地相页岩、硅质岩之间的穿插相变关系.其剖面结构可划分出4个级别的不对称它旋回性韵律,显示出扇体向盆地推进和相对海平面的降低.研究表明关家沟组砂砾岩为海底扇沉积体系,而非冰碛层.     

Model of VLBI observation within the relativistic framework

In this paper we derive the post-newtonian expressions for the VLBI time delay and gravitational delay in the barycentric coordinate system of the solar system. We discuss the effect of the various bodies and their range of action. From the transformation between the barycentric and the geocentric systems we then give the VLBI observational model in the geocentric system. Our final results are given by formula (16) for the gravitational delay and by formula (25) for the VLBI time delay.