On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Petrology of pumices of April 1993 eruption of Lascar (Atacama, Chile)

Lascar Volcano (Atacama, Chile) erupted on 18–20 April 1993. Several sub-Plinian explosions occurred, and some were mushroom-shaped. The highest column rose up to 23 km. Ash clouds crossed South America eastwards. Dacite pumice falls made of blocks and ashes were deposited on the flanks of the volcano as a result of collapsed columns. The pumice contains phenocrysts of plagioclase, enstatite, augite, biotite, magnetite and ilmenite and small crystals of apatite. The 1992 previous andesite dome inside the crater was destroyed. Banded blocks resulting from mingling of the dacitic pumice and andesite from the dome are found in the pumice flow. Both the lava dome and the pumice are representative of the Lascar high-K magma unit. Dacitic pumice is a product of crystal fractionation of the andesitic magma.     

The Intertidal Burrowing Crab <Emphasis Type="Italic">Neohelice</Emphasis> (=<Emphasis Type="Italic">Chasmagnathus</Emphasis>) <Emphasis Type="Italic">granulata</Emphasis> Positively Affects Foraging of Rodents in South Western Atlantic Salt Marshes

The role of positive and indirect interactions is often crucial in communities with intense abiotic stress such as salt marshes. The burrowing crab, Neohelice (=Chasmagnathus) granulata, is the dominant benthic macroinvertebrate of southwest Atlantic marshes (southern Brazil to Northern Argentinean Patagonia), having strong direct and indirect effects on marsh soil and, in consequence, on marsh vegetation and primary consumers. In this work, we investigate if this crab indirectly modifies habitat use by the granivorous rodents, Akodon azarae and Oligoryzomys flavescens, by increasing nutrient availability and thus enhancing seed production by the marsh plant Spartina densiflora. The study was conducted at the Mar Chiquita Coastal Lagoon, Argentina (37°32′ S). Rodent frequencies in S. densiflora were positively correlated with crab densities throughout the low and middle marsh. Additionally, the highest quality of S. densiflora and inflorescence density was recorded at the highest crab densities. Experimental manipulation of crab densities shows that N. granulata indirectly enhances the performance of S. densiflora (e.g., decreased fiber content and C/N ratios) and increases density of seeds. Moreover, N. granulata also facilitates S. densiflora seed availability to rodents by concentrating them in sediment mound at their burrows entrances. Experimental rodent exclusions showed that rodent species used S. densiflora seeds, a variable positively related to crab burrow density. Thus, our results show that N. granulata drives the granivorous rodent distribution and the intensity of seeds–rodent interaction trough facilitative and indirect interactions in marsh community.     

Reliability-based design of R/C water tank structures under seismic action

Based on a reliability level 2 method, a procedure is proposed to design reinforced concrete structures for elevated tanks subjected to seismic action, with a specified probability of failure in a 50-year design life. To evaluate the probability of failure the ultimate limit state is obtained when the top column displacement demanded by the earthquake, a random variable, reaches the allowable displacement, which is here treated as deterministic. The seismic action is characterized probabilistically by the power spectral density function of the ground acceleration, which is obtained from a design spectrum. The strength and ductility of an annular column section of confined reinforced concrete for cyclic loads are evaluated with design aids. Design charts are made for a given tank capacity and specified seismic zone that allow one to choose different combinations of strength, stiffness and ductility for the same tolerable probability of failure. A step by step method is suggested for the design of the annular column section, choosing finally the most convenient design. The advantage of this methodology is shown through a numerical example.