Solving Kepler’s Equation using Bézier curves

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bézier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10⁻¹³ degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10⁻⁵ degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations. Presented at the 7th Dynamics and Control of Systems and Structures in Space Conference, July 18–22, 2006, Greenwich, England.     

Asteroid 2002NY40 as a source of meteorite-dropping bolides

The existence of asteroidal meteoroid streams capable of producing meteorite-dropping bolides has long being invoked, but evidence is scarce. Recent modelling of previously reported associations suggests that the time-scales to keep the orbital coherence of these streams producing meteorites are too short. We present an unequivocal association between near earth object (NEO) 2002NY40 and at least one bright fireball detected over Finland in 2006 August. Another two additional fireballs recorded from Spain and Finland seem to be related, together producing a fireball-producing stream (β Aquarids). On the basis of historical data, the 2006 finding suggests the existence of a meteoroid complex capable of producing meteorites. Taking into account present time-scales for orbital decoherence, if 2002NY40 has large meteoroids associated with it, such behaviour would be the consequence of a relatively recent asteroidal fragmentation. Supporting our claim, the heliocentric orbits of two recently discovered NEOs, 2004NL8 and 2002NY40, were found to exhibit a good similarity to each other and also to the orbits of the three bolides. The fireball spectra of the two Finish bolides showed that the chemical abundances of these objects are consistent with the main elements found in chondrites. This result is consistent with the probable Low iron, Low metal (LL) chondritic mineralogy of asteroid 2002NY40. Consequently, this asteroid may be delivering LL chondrites to the Earth. Additional fireball reports found in the literature suggest that the associated β Aquarid complex may have been delivering meteorites to the Earth during, at least, the last millennium.     

Convergence in a thermally softened thick crust: Variscan intracontinental tectonics in Iberian plate rocks

The subduction phase in the development of the Variscan Orogen in SW Europe was followed by an extended period of ‘intracontinental’ tectonics. The progressive temperature rise in the hinterland during plate convergence was accompanied by widespread partial melting in the lower crust and the nucleation of kilometric buckle folds and crustal‐scale shear zones in the stronger upper crust. Thermal mechanical weakening in the core of the orogen was contemporaneous with shortening and thickening in the foreland fold‐and‐thrust belt. We evaluate lithospheric strength profiles in the hinterland and foreland based on the metamorphic and structural record for three tectonic stages. We find that lower crustal strength varied in space as well as in time during orogenesis. Strength contrasts between the foreland and the hot hinterland during convergence may have led to the additional indentation of the foreland into the hinterland of the Ibero‐Armorican Arc.     

Negligent killing of scientific concepts: the stationarity case

Abstract

In scientific vocabulary, the term “process” is used to denote change in time. Even a stationary process describes a system changing in time, rather than a static one that keeps a constant state all the time. However, this is often missed, which has led to misuse of the term “nonstationarity” as a synonym of “change”. A simple rule to avoid such misuse is to answer the question: can the change be predicted in deterministic terms? Only if the answer is positive is it legitimate to invoke nonstationarity. In addition, we should have in mind that models are made to simulate the future rather than to describe the past; the past is characterized by observations (data). Usually future changes are not deterministically predictable and thus the models should, on the one hand, be stationary and, on the other hand, describe in stochastic terms the full variability, originating from all agents of change. Even if the past evolution of the process of interest contains changes explainable in deterministic terms (e.g. urbanization), it is better to describe the future conditions in stationary terms, after “stationarizing” the past observations, i.e. adapting them to represent the future conditions.     

A kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers

We consider the problem of predicting the spatial field of particle-size curves (PSCs) from a sample observed at a finite set of locations within an alluvial aquifer near the city of Tübingen, Germany. We interpret PSCs as cumulative distribution functions and their derivatives as probability density functions. We thus (a) embed the available data into an infinite-dimensional Hilbert Space of compositional functions endowed with the Aitchison geometry and (b) develop new geostatistical methods for the analysis of spatially dependent functional compositional data. This approach enables one to provide predictions at unsampled locations for these types of data, which are commonly available in hydrogeological applications, together with a quantification of the associated uncertainty. The proposed functional compositional kriging (FCK) predictor is tested on a one-dimensional application relying on a set of 60 PSCs collected along a 5-m deep borehole at the test site. The quality of FCK predictions of PSCs is evaluated through leave-one-out cross-validation on the available data, smoothed by means of Bernstein Polynomials. A comparison of estimates of hydraulic conductivity obtained via our FCK approach against those rendered by classical kriging of effective particle diameters (i.e., quantiles of the PSCs) is provided. Unlike traditional approaches, our method fully exploits the functional form of PSCs and enables one to project the complete information content embedded in the PSC to unsampled locations in the system.     

Aluminum incorporation in α-PbO2 type TiO2 at pressures up to 20 GPa

Aluminum incorporation into the high pressure polymorph of TiO₂ with the structure of α-PbO₂ has been studied from 10 to 20 GPa and 1300 °C by XRD, high-resolution ²⁷Al MAS-NMR and TEM. Al-doped α-PbO₂ type TiO₂ can be recovered at atmospheric pressure. Al₂O₃ solubility in α-PbO₂ type TiO₂ increases with increasing the synthesis pressure. The α-PbO₂ type TiO₂ polymorph is able to incorporate up to 35 wt.% Al₂O₃ at 13.6 GPa and 1300 °C, being the substitution of Ti⁴⁺ by Al³⁺ on normal octahedral sites the mechanism of solubility. The transition to the higher pressure TiO₂ polymorph with the ZrO₂ baddeleyite structure, Akaogiite, has not been observed in the quenched samples at room pressure. The microstructure of the recovered sample synthesized at 16 GPa and 1300 °C points to the existence of a non-quenchable aluminum titanium oxide phase at these conditions.     

Out‐of‐plane seismic behaviour of rocking masonry walls

The evaluation of the out‐of‐plane behaviour of unreinforced walls is one of the most debated topics in the seismic assessment of existing masonry buildings. The discontinuous nature of masonry and its interaction with the remainder of the building make the dynamic modelling of out‐of‐plane response troublesome. In this paper, the results of a shaking table laboratory campaign on a tuff masonry, natural scale, U‐shaped assemblage (façade adjacent to transverse walls) are presented. The tests, excited by scaled natural accelerograms, replicate the behaviour of external walls in existing masonry buildings, from the beginning of rocking motion to overturning. Two approaches have been developed for modelling the out‐of‐plane seismic behaviour: the discrete element method and an SDOF analytic model. Both approaches are shown to be capable of reproducing the experimental behaviour in terms of maximum rotation and time history dynamic response. Finally, test results and numerical time history simulations have been compared with the Italian seismic code assessment procedures. Copyright © 2011 John Wiley & Sons, Ltd.     

Growth discontinuity in males of the deep-water giant red shrimp Aristaeomorpha foliacea in the Mediterranean Sea

The deep-water giant red shrimp, Aristaeomorpha foliacea (Risso, 1827) (Decapoda, Aristeidae), represents a highly valuable resource for bottom trawling in the Mediterranean Sea. Recent assessments have described both a worsening of the status of the traditionally exploited stocks and low levels of annual-based instantaneous natural mortality (M 0.4–0.7 year⁻¹) in the unexploited stocks. The mortality (M) figures, however, are in contrast with the longevity (T_max) of 3–4-year males and 4–5-year females, estimated from classical length frequency distribution analysis. Reduced growth (after the onset of sexual maturity) and pile-up of older individuals in the larger size classes have been considered to be reasonable explanations for the contrast between M and T_max. We propose that a discontinuity of the growth models could address this contrast. Because a clear discontinuity in sexual maturity is evident only in males, length frequency distribution data for different sets of males collected from the South of Sicily and the Maltese Islands deep bottoms (400–800 m) were fitted with both the classic (c) and double-phased (d) von Bertalanffy growth formula (VBGF). According to the dVBGF, adult males sharply reduce their growth rate after reaching an age between 1.2 and 1.5 year, which corresponds to the estimates of age at sexual maturity (t_m between 1 and 1.5 years). The reduction in growth determines a higher T_max (7.3–9.5 year) and lower M (0.4–0.6 year⁻¹) than previously derived on the basis of cVBGF estimates (T_max 3–4 years and M 1.2 year⁻¹). The dVBGF results suggest that sexual dimorphism and length–sex segregation in giant red shrimps reflect an alternative life history strategy that is based mainly on growth reduction in adult males and it might be adopted for implementing more conservative assessments.