Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field

We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations. The connection with the original system is established concluding the existence of various families of invariant 3-tori of it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques.     

SOCIAL THEORY AND THE FUTURE OF URBAN GEOGRAPHY*

Three themes that seem likely to be influential in the near future have recently characterized the work of geographers using social theories as the basis of their explanations of urban phenomena. They are (1) the debates about structure and human agency. (2) the claims of feminists about the analytical significance of the social construction of gender roles, and (3) the demand that we now use our social theories more actively to explain experienced reality. In continuing our use of social theory in urban geography, we must prevent its marginalization as one specific substantive topic; the alternative epistemologies of this type of work can guide new interpretations of every substantive issue in urban geography.     

Cenozoic uplift of Variscan Massifs in the Alpine foreland: Timing and controlling mechanisms

The European Cenozoic Rift System (ECRIS) and associated fault systems transect all Variscan Massifs in the foreland of the Alps. ECRIS was activated during the Eocene in the foreland of the Pyrenees and Alps in response to the build-up of collision-related intraplate stresses. During Oligocene and Neogene times ECRIS evolved by passive rifting under changing stress fields, reflecting end Oligocene consolidation of the Pyrenees and increasing coupling of the Alpine Orogen with its foreland. ECRIS is presently still active, as evidenced by its seismicity and geodetic data.Uplift of the Massif Central and the Rhenish Massif, commencing at the Oligocene–Miocene transition, is mainly attributed to plume-related thermal thinning of the mantle–lithosphere. Mid-Burdigalian uplift of the SW–NE-striking Vosges–Black Forest Arch, that has the geometry of a doubly plunging anticline breached by the Upper Rhine Graben, involved folding of the lithosphere. Late Burdigalian broad uplift of the northern parts of the Bohemian Massif reflects lithospheric buckling whereas late Miocene–Pliocene uplift of its marginal blocks involved transpressional reactivation of pre-existing crustal discontinuities. Crustal extension across ECRIS, amounting to no more than 7 km, was compensated by a finite clockwise rotation of the Paris Basin block, up warping of the Weald–Artois axis and reactivation of the Armorican shear zones. Intermittent, though progressive uplift of the Armorican Massif, commencing in the Miocene, is attributed to transpressional deformation of the lithosphere.Under the present-day NW-directed compressional stress field, that came into evidence during the early Miocene and further intensified during the Pliocene, the Armorican Massif, the Massif Central, the western parts of the Rhenish Massif and the northern parts of the Bohemian Massif continue to rise at rates of up to 1.75 mm/y whilst the Vosges–Black Forest arch is relatively stable.Uplift of the Variscan Massifs and development of ECRIS exerted strong controls on the Neogene evolution of drainage systems in the Alpine foreland.     

Quantum peculiarities of the primordial hydrogen recombination

Some problems of the two-photon decay of highly excited atomic hydrogen and helium levels are considered. The necessary accuracy of calculating the transition probabilities is shown to require abandoning the purely dipole approximation in the theory of atomic hydrogen radiation. The range of applicability of the perturbation theory to this process is discussed. We suggest a new mechanism of the Lyman photon redistribution due to very close locations of the 20P and 19P levels in the hydrogen and deuterium atoms, respectively (hydrogen-deuterium resonance).