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.

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)