Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Review of the neotectonics of the Eastern Turkish–Armenian Plateau by geomorphic analysis of digital elevation model imagery

Digital elevation model (DEM) images provide synoptic views of the Earth’s surface allowing the analysis of landforms of still active tectonic and volcanic structures at regional scale. A DEM at 250 m pixel size constitutes regional scale data particularly efficient to investigate the late Miocene–Quaternary deformation of the Eastern Turkish–Armenian Plateau in the Arabian–Eurasian area of convergence. Geomorphic analysis of the DEM image associated with review of fault-plane solutions of earthquakes show that faults are mostly strike-slip with small vertical component. Here we show that the orientations of the tectonic and volcanic structures fit with a tectonic regime characterized by N–S shortening and E–W lengthening, consistent with westward escape of Anatolia perpendicular to the direction of the Arabia–Eurasia shortening. The uniform uplift of the plateau, the predominance of strike-slip faulting, the lack of major thrusts and the occurrence of normal faults do not support a model of going-on crustal thickening due to intracontinental convergence. On the contrary, our observations can be better interpreted in terms of lithospheric thinning and mantle upwelling related to gravity escape of Anatolia.     

Evolution of the stress and strain fields in the Eastern Cordillera,Colombia

This work integrates stress data from Global Positioning System measurements and earthquake focal mechanism solutions, with new borehole breakout and natural fracture system data to better understand the complex interactions between the major tectonic plates in northwestern South America and to examine how the stress regime in the Eastern Cordillera and the Llanos foothills in Colombia has evolved through time. The dataset was used to generate an integrated stress map of the northern Andes and to propose a model for stress evolution in the Eastern Cordillera. In the Cordillera, the primary present-day maximum principal stress direction is WNW–ESE to NW–SE, and is in the direction of maximum shortening in the mountain range. There is also a secondary maximum principal stress direction that is E–W to ENE–WSW, which is associated with the northeastward “escape” of the North Andean block, relative to stable South America. In the Cupiagua hydrocarbon field, located in the Llanos foothills, the dominant NNE–SSW fractures are produced by the Panama arc–North Andes collision and range-normal compression. However, less well developed asymmetrical fractures oriented E–W to WSW–ENE and NNW–SSE are also present, and may be related to pre-folding stresses in the foreland basin of the Central Cordillera or to present-day shear associated with the northeastward “escape” of the north Andean block. Our study results suggest that an important driver for orogenic deformation and changes in the stress field at obliquely convergent subduction zone boundaries is the arrival of thickened crust, such as island arcs and aseismic ridges, at the trench.     

中国蛤蜊Mactra chinensis Philippi稚贝运动行为学初步研究

为从行为学角度解释中国蛤蜊(Mactra chinensis Philippi)的跑滩现象, 研究了规格、温度对其跳跃行为以及底质、流速和周期性干露对其潜沙行为的影响。结果表明, 中国蛤蜊运动能力与其规格和水温有关, 3 mm稚贝不具备跳跃能力, 5 mm以上具备跳跃能力; 跳跃频率和高度随水温的升高呈上升趋势。中国蛤蜊稚贝潜沙比例均随时间推移逐渐上升; 各时间节点中国蛤蜊稚贝在细沙底质的潜沙比例最高, 中沙底质次之, 粗沙底质最低。施加微弱水流后, 中国蛤蜊稚贝潜沙速度和比例显著增加; 流速增大到一定程度后部分稚贝被水流冲走, 潜沙比例随之下降。周期性干露会导致中国蛤蜊稚贝从底质中爬出, 干露时间越长, 爬出比例越高。干露后的稚贝重新放置于海水中, 部分稚贝会在水面漂浮一段时间后再下落至水底, 漂浮比例随干露时间的增加而增加。本研究从行为学角度初步查明了中国蛤蜊跑滩现象, 为解决中国蛤蜊防跑滩问题提供了思路。     

Empirical test of the spectral invariants theory using imaging spectroscopy data from a coniferous forest

The spectral invariants theory presents an alternative approach for modeling canopy scattering in remote sensing applications. The theory is particularly appealing in the case of coniferous forests, which typically display grouped structures and require computationally intensive calculation to account for the geometric arrangement of their canopies. However, the validity of the spectral invariants theory should be tested with empirical data sets from different vegetation types. In this paper, we evaluate a method to retrieve two canopy spectral invariants, the recollision probability and the escape factor, for a coniferous forest using imaging spectroscopy data from multiangular CHRIS PROBA and NADIR-view AISA Eagle sensors. Our results indicated that in coniferous canopies the spectral invariants theory performs well in the near infrared spectral range. In the visible range, on the other hand, the spectral invariants theory may not be useful. Secondly, our study suggested that retrieval of the escape factor could be used as a new method to describe the BRDF of a canopy.     

Mars atmospheric escape and evolution; interaction with the solar wind

This tutorial deals with the question of atmospheric escape on Mars. After a brief introduction describing the general context of Mars escape studies, we will present in Section 2 a simplified theory of thermal escape, of both Jeans and hydrodynamic types. The phenomenon of hydrodynamic escape, still hypothetical and not proved to have ever existed on terrestrial planets, will be treated with the help of two well known examples: (i) the isotopic fractionation of xenon in Mars and Earth atmospheres, (ii) the paradox of missing oxygen in Venus atmosphere. In Section 3, a simplified approach of non-thermal escape will be developed, treating in a specific way the different kinds of escape (photochemical escape, ion sputtering, ion escape and ionospheric outflow). As a matter of illustration, some calculations of the relative contributions of these mechanisms, and of their time evolutions, will be given, and the magnitude of the total amount of atmosphere lost by non-thermal escape will be estimated. Section 4 will present the state of knowledge concerning the constraints derived from Mars isotopic geochemistry in terms of past escape and evolution. Finally, a few conclusions, which are more interrogations, will be proposed.     

Evolution of the General Solution of the Restricted Problem Covering Symmetric and Escape Solutions

The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10⁻³. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions.     

2010年玉树地震地表破裂带典型破裂样式及其构造意义

野外调查表明,青海玉树M_S7.1地震发生在青藏高原中部甘孜-玉树断裂的玉树段上,在玉树县结古镇至隆宝镇之间产生了一系列包括剪切破裂、张剪切破裂、压剪切破裂、张性破裂及其不连续岩桥区出现的鼓包或陷落坑(拉分盆地)、高寒地区特有的冰裂缝等地表破裂单元,它们斜列组合成整体走向约300°、长约65 km、最大同震左旋位移2.4 m的地表破裂带,具有变形局部化的基本特征.玉树地震地表破裂带整体上可划分为长约15 km的结隆次级地表破裂带和长约31 km的结古次级地表破裂带,两者呈左阶羽列,其间无地表破裂段长约17 km,对应于M_W6.4和M_W6.9两个次级地震事件.地表破裂类型、基本组合特征等显示出甘孜-玉树断裂两盘块体的运动方式以纯剪切的左旋走滑为主,从一个方面反映了青藏高原物质存在着向东的逃逸和挤出现象.     

Spatial and temporal relationships between compression, strike-slip and extension in the Central Venezuelan Andes: Clues for Plio-Quaternary tectonic escape

The geometry of tectonic structures, attributed to the Neogene–Quaternary time interval, is described in the active setting of the Venezuelan Andes. Our methodology is based on the analysis of radar satellite and Digital Elevation Model imagery, complemented by structural fieldwork and the compilation of seismotectonic data to make a structural analysis on a regional scale. Radar images provide first class data for morphostructural analysis in areas of dense vegetation and frequent cloud covering, like the Venezuelan Andes. We focused our analysis in the Burbusay–Río Momboy and Boconó faults corner located in the central part of the belt.We have described three stages of deformation during the Neogene–Quaternary. The first one, Mio-Pliocene in age, is a NW–SE compression responsible for the uplift of the Venezuelan Andes. The second tectonic stage corresponds to a strike-slip regime of deformation marked by shearing along the Boconó, Burbusay and Valera faults, which separates two triangular wedges in the larger Trujillo block. This strike-slip faulting-dominated compressional-extensional tectonic regime allowed the Trujillo crustal block to move towards the NE. Wrenching has therefore started at some point between the Pliocene and the Quaternary. These two tectonic events are consistent with ongoing strain partitioning in the Venezuelan Andes. The third stage corresponds to extensional deformation limited to the Trujillo block and is still active today. Extension is associated with the motion of crustal blocks moving relative to each other, probably above the upper-lower crust boundary. Such extensional deformation can be understood considering that the crust extends and stretches at the same time as it moves towards the NE. The combination of both horizontal lateral motion and extension is characteristic of a tectonic escape process. The northeastward escape of the Trujillo block, which belongs to the larger North Andes block, occurs as a result of the combination of the NW–SE intracontinental convergence between the South-American plate and the Maracaibo block, and the presence to the north of the Caribbean oceanic plate considered as a free boundary. We have showed that the kinematics of the Caribbean plate offers not only a favorable environment, but may also be considered as the driving force of the tectonic escape of the North Andes block.     

A quantitative characterization of the seasonal Lagrangian circulation of the Gulf of California from a three-dimensional numerical model

A set of four indices that quantify Lagrangian properties of the Gulf of California seasonal circulation were implemented from outputs of a three-dimensional numerical model. From trajectories of particles seeded over the entire Gulf, we calculated for 12 one-month periods the following indices: net and total distance traveled by the particles, the number of particles that are found within an area centered on the release positions after one month, and time taken by particles to escape from a 50-km-radius circle. These indices can be used for studies on transport of inert properties and passive planktonic organisms such as eggs and early-stage larvae; their use is illustrated for typical summer and winter conditions in the Gulf of California. These indices show the potential for connecting areas separated by a few hundreds of km along the eastern side of the Gulf, due to the strong seasonal up-gulf and down-gulf current. In the Northern Gulf, large displacements occur at the borders of the basin-wide seasonally reversing eddy that dominates the large-scale circulation (cyclonic in summer, anticyclonic in winter). On the other hand, the potential for self-recruitment areas is found as particles can be trapped for longer than one month within these eddies, as well as in smaller ones in the Northern Gulf, and near the coast of the peninsular side of the Southern Gulf, where current speeds are slow and many small capes and islands are present.