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     

Otolith shape as a stock discrimination tool for ling (Genypterus blacodes) in the fjords of Chilean Patagonia

ABSTRACT

Genypterus blacodes, in terms of its fishing history and local economic importance, is an emblematic species harvested in Chilean Patagonia (41°00’–57°00’S). Most of the current fisheries and biological knowledge of this species come from the open ocean, whereas information about the species in fjords and inner channels is fragmentary. In 2018, two research surveys targeting G. blacodes were conducted in the fjords and inner channels of Chilean Patagonia. A total of 253 pairs of sagittal otoliths were sampled at three different localities, and their contours were modelled using wavelet analysis as a tool for stock discrimination. Contours were compared using canonical analysis, and classification was performed using linear discriminant and Random Forest analyses. The results indicated that the wavelet method is efficient in modelling otolith contours, and the discriminant analyses showed differences among fishing grounds across the latitudinal gradient, thus confirming the hypothesis that G. blacodes conform to at least two separate stock units in Chilean Patagonia. Fishing grounds that were closer in space showed higher levels of misclassification. The discussion focuses on how environmental variables and the geography of fjords shape stock differences and how this information can be used for the sustainable management of G. blacodes.     

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.     

Reconstruction of total annual rainfall in Andalusia (Southern Spain) during the 16th and 17th centuries from documentary sources

Summary This study describes a reconstruction of rainfall characteristics in Southern Spain from 1501 to 1700 AD, during the beginning of the period called the Little Ice Age. Weather information was taken from original documentary sources (urban annals, local and religious histories, municipal documents, relations, etc.) in the region. A numerical index was established to characterize the rainfall, its characteristics, evolution and geographical distribution. Results were characteristics compared with modern precipitation data and with the results of other studies of historical climate. The general conclusion is that rainfall in Western Andalusia increased from approximately 1550 to 1650 AD. Some perspectives for future research work are outlined.With 6 Figures