Notes on Hori Method for Canonical Systems

In this paper a slightly different approach is proposed for the process of determining the functions S _m and H _m ^* of the algorithm of the canonical version of Hori method. This process will be referred to as integration theory of the mth order equation of the method. It will be shown that the ordinary differential equation with an auxiliary parameter t ^* as independent variable, introduced through Hori auxiliary system, can be replaced by a partial differential equation in the time t. In this way, the mth order equation of the algorithm assumes a form very similar to the one of other perturbation methods. In virtue of this new approach of the integration theory for Hori method, Lagrange's variational equations introduced by Sessin are revised. As an example, the Duffing equation is solved through this new approach.     

Geochemistry, petrology and ages of the lunar meteorites Kalahari 008 and 009: New constraints on early lunar evolution

Kalahari 008 and 009 are two lunar meteorites that were found close to each other in Botswana. Kalahari 008 is a typical lunar anorthositic breccia; Kalahari 009 a monomict breccia with basaltic composition and mineralogy. Based on minor and trace elements Kalahari 009 is classified as VLT (very-low-Ti) mare basalt with extremely low contents of incompatible elements, including the REE. The Lu-Hf data define an age of 4286 ± 95 Ma indicating that Kalahari 009 is one of the oldest known basalt samples from the Moon. It provides evidence for lunar basalt volcanism prior to 4.1 Ga (pre-Nectarian) and may represent the first sample from a cryptomare. The very radiogenic initial ¹⁷⁶Hf/¹⁷⁷Hf (εHf = +12.9 ± 4.6), the low REE, Th and Ti concentrations indicate that Kalahari 009 formed from re-melting of mantle material that had undergone strong incompatible trace element depletion early in lunar history. This unusually depleted composition points toward a hitherto unsampled basalt source region for the lunar interior that may represent a new depleted endmember source for low-Ti mare basalt volcanism. Apparently, the Moon became chemically very heterogeneous at an early stage in its history and different cumulate sources are responsible for the diverse mare basalt types.Evidence that Kalahari 008 and 009 may be paired includes the similar fayalite content of their olivine, the identical initial Hf isotope composition, the exceptionally low exposure ages of both rocks and the fact that they were found close to each other. Since cryptomaria are covered by highland ejecta, it is possible that these rocks are from the boundary area, where basalt deposits are covered by highland ejecta. The concentrations of cosmogenic radionuclides and trapped noble gases are unusually low in both rocks, although Kalahari 008 contains slightly higher concentrations. A likely reason for this difference is that Kalahari 008 is a polymict breccia containing a briefly exposed regolith, while Kalahari 009 is a monomict brecciated rock that may never have been at the surface of the Moon.Altogether, the compositions of Kalahari 008 and 009 permit new insight into early lunar evolution, as both meteorites sample lunar reservoirs hitherto unsampled by spacecraft missions. The very low Th and REE content of Kalahari 009 as well as the depletion in Sm and the lack of a KREEP-like signature in Kalahari 008 point to a possible source far from the influence of the Procellarum-KREEP Terrane, possibly the lunar farside.     

Weibull statistical analysis of granite bending strength

Summary  This paper describes and discusses the adequacy of Weibull statistical analysis to analyse the bending strength of granite. The experimental results show that strength variability is related with a specific origin of failure. This conclusion is based on analysing the influence of the surface condition (extrinsic defects) on the bending strength results treated by the Weibull statistics. The conclusions drawn from this study have been validated by analysing the results of the critical flaw dimension estimated by applying the linear elastic fracture mechanics (LEFM) formulae. Results obtained from fractographic examination also have been used to describe the location of the origin of the fracture and understand the distribution of defects; i.e., there is a unimodal distribution of defects (intrinsic defects), despite the fact that some outlier values are normally observed in the fractured surfaces. Correspndence: P. M. Amaral, Department of Materials Engineering, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal     

Carbon stable isotopes in estuarine sediments and their utility as migration markers for nursery studies in the Firth of Forth and Forth Estuary, Scotland

The stable carbon isotope ratios (δ¹³C) of the organic fraction of intertidal sediments in the Forth Estuary and the Firth of Forth, Scotland, were measured to determine if terrestrially derived carbon was present in the estuarine sediments. It was hypothesised that differences in the inputs from marine vs. terrestrial sources to the organic carbon of estuarine and marine sediments, as well as differences in ambient seawater stable oxygen isotope (δ¹⁸O) ratios between the estuary and the Outer Firth, would allow the use of these two stable isotopes as habitat markers for juvenile plaice (Pleuronectes platessa), to allow determination of nursery habitats. Muddy and sandy sediments from the estuary and sandy sediments from the Outer Firth were sampled and δ¹³C measured. Juvenile plaice were caught at two estuarine sites and at two Outer Firth sites and otoliths were removed for δ¹³C and δ¹⁸O analysis. The sandy sediments in the estuary showed a strong gradient of δ¹³C enrichment with distance down the estuary, while the muddy sediments showed a much shallower gradient. δ¹³C and δ¹⁸O measured in the carbonate of juvenile plaice otoliths showed no clear difference between otoliths of fish caught at one of the estuarine sites and at the two Outer Firth sites. However, the isotope ratios of both carbon and oxygen in plaice otoliths from the other estuarine site showed the expected trend of depletion in the heavier isotopes. While the measurements recorded here did not conclusively distinguish between otoliths from juveniles caught in the estuarine site and those caught in the other three sites, they show that stable isotopes have potential to distinguish between estuarine habitats with terrestrial carbon inputs, and coastal marine habitats with predominantly marine carbon inputs.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

Cenozoic tectonic events at the border of the Paraná Basin, São Paulo, Brazil

In the last decade, even in areas that had been considered tectonically stable, a great amount of Cenozoic, including the Quaternary period, structural data have been collected throughout Brazil. The main goal of this study is to describe the Cenozoic structures and tectonic evolution of an area that is located at the border of the Paraná Basin in the state of São Paulo.The research methods consisted of the analysis of: (1) brittle structure data, mainly conjugate fractures and fault slip data; (2) lineaments traced on air photos and TM Landsat and radar images; and (3) a second-order base surface map.The study area, during the Cenozoic, has been affected by five strike–slip tectonic events, which generated mainly strike–slip faults, and secondarily normal and reverse ones. The events were named, from the oldest to the youngest, E1-NE, E2-EW, E3-NW, E4-NS, and E5-NNE; and the maximum principal stresses σ1 strike approximately NE–SW, E–W, NW–SE, N–S, and NNE–SSW, respectively. Event E2-EW seems to have been contemporaneous with the deposition of the Rio Claro Formation, the most important Cenozoic deposit of probable Neogenic age, and also to have controlled the distribution of its deposits. Event E3-NW was the strongest one in the area, as is pointed out by structural data, and the maximum principal stress σ1 of event E5-NNE is partially concordant with the orientation of σH-max of well break-out data in the Paraná Basin, suggesting a Neotectonic activity for this event. Finally, discontinuities parallel and correlated to the directions of strike–slip faults of the Cenozoic events seem to have actively controlled the sculpturing of the relief in the study area.     

Aquatic hydrocarbon distributions in the Seine estuary: Biogenic polyaromatics and n-alkanes

Dissolved and particulate hydrocarbons of biogenic origin were investigated for the first time in surface waters along the Seine River and its estuary. They comprise n-alkanes (n-ALKs) and diagenetic polycyclic aromatic hydrocarbons (PAHs). Samples were collected in three different sections of the estuary: the riverine zone, the mixing zone, and the marine zone. At the river mouth, two mooring stations were used for the collection of samples over tidal cycles. Total particulate n-ALK concentrations ranged from 31 ng 1^?1 to 2,918 ng 1^?1, or 5 μg g^?1 dry ng 1^?1, or 2 μg g^?1 of SM. Concentrations varied with the SM load and could be related to sedimentation and estuarine mixing. The sources of the n-ALKs were different in each zone of the estuary. The dissolved n-ALKs displayed lower concentrations than the particulate phase, varying from 136 ng 1^?1 to 344 ng 1^?1, while biogenic dissolved PAHs were almost absent.