Duality of osculating developables of projective curves

We give a description of the dual varieties of all developables of osculating linear spaces to a projective curve in terms of the higher order dual varieties of the curve, in arbitrary characteristic. We also determine for these varieties the inseparable degrees of the projections from the conormal varieties onto their dual varieties.

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Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities; we also assume that both planets share the same orbital plane. Initially, we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analysed in more detail using a semi-analytical model. Apart from the well-known quasi-satellite orbits and the classical equilibrium Lagrangian points L(4) and L(5), we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at (delta lambda, delta pi) = (+/- 60 degrees, -/+ 120 degrees), where delta lambda is the difference in mean longitudes and delta pi is the difference in longitudes of pericentre. The position of these anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities and are found for eccentricities as high as similar to 0.7. Finally, we also applied a slow mass variation to one of the planets and analysed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.

Numerical study about natural escape and capture routes by the Moon via Lagrangian points L1 and L2

The lunar sphere of influence, whose radius is some 66,300 km, has regions of stable orbits around the Moon and also regions that contain trajectories which, after spending some time around the Moon, escape and are later recaptured by lunar gravity. Both the escape and the capture occur along the Lagrangian equilibrium points L1 and L2. In this study, we mapped out the region of lunar influence considering the restricted three-body Earth-Moon-particle problem and the four-body Sun-Earth-Moon-particle (probe) problem. We identified the stable trajectories, and the escape and capture trajectories through the L I and L2 in plots of the eccentricity versus the semi-major axis as a function of the time that the energy of the osculating lunar trajectory in the two-body Moon-particle problem remains negative. We also investigated the properties of these routes, giving special attention to the fact that they supply a natural mechanism for performing low-energy transfers between the Earth and the Moon, and can thus be useful on a great number of future missions. (C) 2007 Published by Elsevier Ltd on behalf of COSPAR.

Numerical study of low-cost alternative orbits around the Moon

In this paper, we have investigated a region of direct stable orbits around the Moon, whose stability is related to the H2 Family of periodic orbits and to the quasi-periodic orbits that oscillate around them. The stability criteria adopted was that the path did not escape from the Moon during an integration period of 1000 days (remaining with negative two-body Moon-probe orbital energy during this period). Considering the three-dimensional four-body Sun-Earth-Moon-probe problem, we investigated the evolution of the size of the stability region, taking into account the eccentricity of the Earth's orbit, the eccentricity and inclination of the Moon's orbit, and the solar radiation pressure on the probe. We also investigated the evolution of the region's size and its location by varying the inclination of the probe's initial osculating orbit relative to the Moon's orbital plane between 0 degrees and 180 degrees. The size of the stability region diminishes; nevertheless, it remains significant for 0 <= i <= 25 degrees and 35 degrees <= i <= 45 degrees. The orbits of this region could be useful for missions by space vehicles that must remain in orbit around the Moon for periods of up to 1000 days, requiring low maintenance costs. (c) 2005 COSPAR. Published by Elsevier Ltd. All rights reserved.

Reanálise das defasagens de Prometeu e Pandora

In the present work it is proposed to do a revision on some studies on the dynamics of the Prometheus-Pandora system. In special, those studies that deal with anomalous behaviours observed on its components, identi ed as angular lags in these satellite`s orbits. Initially, it is presented a general description, contextualising the main characteristics of this system. The main publications related to this subject are analised and commented, in chronological order, showing the advances made in the knowledge of such dynamics. An analysis of the initial conditions, used by Goldreich e Rappaport (2003a ,b) e Cruz (2004), obtained through observations made by the Voyager 1 and 2 spacecrafts and by the Hubble space telescope, it is made in order to try to reproduce their results. However, no clear conclusion of the values used were found. The tests addopted in the analysis are from Cruz (2004), which reproduced the results and o ered a new explanation on the origin of the observed angular lags. The addopetd methodology involves the numerical integration of the equations of motion of the system, including the zonal harmonics J2, J4 and J6 of Saturn's gravitational potential. A fundamental consideration in this study is the use of geometric elements instead of osculating elements. It was found the set of initial data that best reproduces the results from Goldreich e Rappaport (2003a, b) and Cruz (2004)

Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Orbital elements of the material surrounding comet 67P/Churyumov-Gerasimenko

Context. We investigate the dust coma within the Hill sphere of comet 67P/Churyumov-Gerasimenko. Aims. We aim to determine osculating orbital elements for individual distinguishable but unresolved slow-moving grains in the vicinity of the nucleus. In addition, we perform photometry and constrain grain sizes. Methods. We performed astrometry and photometry using images acquired by the OSIRIS Wide Angle Camera on the European Space Agency spacecraft Rosetta. Based on these measurements, we employed standard orbit determination and orbit improvement techniques. Results. Orbital elements and effective diameters of four grains were constrained, but we were unable to uniquely determine them. Two of the grains have light curves that indicate grain rotation. Conclusions. The four grains have diameters nominally in the range 0.14-0.50 m. For three of the grains, we found elliptic orbits, which is consistent with a cloud of bound particles around the nucleus. However, hyperbolic escape trajectories cannot be excluded for any of the grains, and for one grain this is the only known option. One grain may have originated from the surface shortly before observation. These results have possible implications for the understanding of the dispersal of the cloud of bound debris around comet nuclei, as well as for understanding the ejection of large grains far from the Sun.

An asymptotic solution for the main problem

In this paper, an analytical solution of the main problem, a satellite only perturbed by the J2 harmonic, is derived with the aid of perturbation theory and by using DROMO variables. The solution, which is valid for circular and elliptic orbits with generic eccentricity and inclination, describes the instantaneous time variation of all orbital elements, that is, the actual values of the osculating elements

Osculation for conic fibrations.

Smooth projective surfaces fibered in conics over a smooth curve are investigated with respect to their k-th osculatory behavior. Due to the bound for the dimension of their osculating spaces they do not differ at all from a general surface for k = 2, while their structure plays a significant role for k >= 3. The dimension of the osculating space at any point is studied taking into account the possible existence of curves of low degree transverse to the fibers, and several examples are discussed to illustrate concretely the various situations arising in this analysis. As an application, a complete description of the osculatory behavior of Castelnuovo surfaces is given. The case k = 3 for del Pezzo surfaces is also discussed, completing the analysis done for k = 2 in a previous paper by the authors (2001). Moreover, for conic fibrations X subset of P-N whose k-th inflectional locus has the expected codimension, a precise description of this locus is provided in terms of Chern classes. In particular, for N = 8, it turns out that either X is hypo-osculating for k = 3, or its third inflectional locus is 1-dimensional

Raw-data to morphometric investigations about the Neogene planktonic foraminifer Globorotalia menardii and related forms from ODP Hole 154-925B (Céara Rise, western tropical Atlantic)

Evolutionary prospection is the study of morphological evolution and speciation in calcareous plankton from selected time-slices and key sites in the world oceans. In this context, the Neogene menardiform globorotalids serve as study objects for morphological speciation in planktic foraminifera. A downcore investigation of test morphology of the lineage of G. menardii-limbata-multicamerata during the past 8 million years was carried out in the western tropical Atlantic ODP Hole 925B. A total of 4669 specimens were measured and analyzed from 38 stratigraphic levels and compared to previous studies from DSDP Sites 502 and 503. Collection of digital images and morphometric measurements from digitized outlines were achieved using a microfossil orientation and imaging robot called AMOR and software, which was especially developed for this purpose. Most attention was given to the evolution of spiral height versus axial length of tests in keel view, but other parameters were investigated as well. The variability of morphological parameters in G. menardii, G. limbata, and G. multicamerata through time are visualized by volume density diagrams. At Hole 925B results show gradual test size increase in G. menardii until about 3.2 Ma. The combination of taxonomic determination in the light microscope with morphometric investigations shows strong morphological overlap and evolutionary continuity from ancestral to extant G. menardii (4-6 chambers in the final whorl) to the descendent but extinct G. limbata (seven chambers in the final whorl) and to G. multicamerata (>=8 chambers in the final whorl). In the morphospace defined by spiral height (dX) and axial length (dY) Globorotalia limbata and G. multicamerata strongly overlap with G. menardii. Distinction of G. limbata from G. menardii is only possible by slight differences in the number of chambers of the final whorl, nuances in spiral convexity, upper keel angles, radii of osculating circles, or by differences in reflectance of their tests. Globorotalia multicamerata can be distinguished from the other two forms by more than eight chambers in the final whorl. It appeared as two stratigraphically separate clusters during the Pliocene. Between 2.88 and 2.3 Ma G. menardii was severely restricted in size and abundance. Thereafter, it showed a rapid and prominent expansion of the upper test size extremes between 2.3 and 1.95 Ma persisting until present. The size-frequency distributions at Hole 925B are surprisingly similar to trends of menardiform globorotalids from Caribbean DSDP Site 502. There, the observations were explained as an adaptation to changes in the upper water column due to the emergence of the Isthmus of Panama. In light of more recent paleontological and geological investigations about the completion of the permanent land connection between North and South America since about 3 Ma the present study gives reason to suspect the sudden test size increase of G. menardii to reflect immigration of extra-large G. menardii from the Indian Ocean or the Pacific. It is hypothesized that during the Late Pliocene dispersal of large G. menardii into the southern to tropical Atlantic occurred during an intermittent episode of intense Agulhas Current leakage around the Cape of Good Hope and from there via warm eddy transport to the tropical Atlantic (Agulhas dispersal hypothesis).