The Hamiltonian transformation in quadratic lie transforms

The algorithm for Hamiltonian transformation in the quadratic perturbation technique of one of the authors admits of various equivalent forms. Using as a criterion the number of inter-term multiplications required for transformation, however, the amount of effort required to obtain the transformed Hamiltonian is not equivalent among these forms. Each is considered in some detail, and general guidelines for the choice of most efficient algorithm to be used in a given problem are provided. Their utility is demonstrated by application to Duffing's equation.     

The formation of the Baptistina family by catastrophic disruption: Porous versus non‐porous parent body

Abstract— In this paper, we present numerical simulations aimed at reproducing the Baptistina family based on its properties estimated by observations. A previous study by Bottke et al. (2007) indicated that this family is probably at the origin of the K/T impactor, is linked to the CM meteorites and was produced by the disruption of a parent body 170 km in size due to the head‐on impact of a projectile 60 km in size at 3 km s^?1. This estimate was based on simulations of fragmentation of non‐porous materials, while the family was assumed to be of C taxonomic type, which is generally interpreted as being formed from a porous body. Using both a model of fragmentation of non‐porous materials, and a model that we developed recently for porous ones, we performed numerical simulations of disruptions aimed at reproducing this family and at analyzing the differences in the outcome between those two models. Our results show that a reasonable match to the estimated size distribution of the real family is produced from the disruption of a porous parent body by the head‐on impact of a projectile 54 km in size at 3 km s^?1. Thus, our simulations with a model consistent with the assumed dark type of the family requires a smaller projectile than previously estimated, but the difference remains small enough to not affect the proposed scenario of this family history. We then find that the break‐up of a porous body leads to different outcomes than the disruption of a non‐porous one. The real properties of the Baptistina family still contain large uncertainties, and it remains possible that its formation did not involve the proposed impact conditions. However, the simulations presented here already show some range of outcomes and once the real properties are better constrained, it will be easy to check whether one of them provides a good match.     

Formation of Asteroid Families by Catastrophic Disruption: Simulations with Fragmentation and Gravitational Reaccumulation

This paper builds on preliminary work in which numerical simulations of the collisional disruption of large asteroids (represented by the Eunomia and Koronis family parent bodies) were performed and which accounted not only for the fragmentation of the solid body through crack propagation, but also for the mutual gravitational interaction of the resulting fragments. It was found that the parent body is first completely shattered at the end of the fragmentation phase, and then subsequent gravitational reaccumulations lead to the formation of an entire family of large and small objects with dynamical properties similar to those of the parent body. In this work, we present new and improved numerical simulations in detail. As before, we use the same numerical procedure, i.e., a 3D SPH hydrocode to compute the fragmentation phase and the parallel N-body code pkdgrav to compute the subsequent gravitational reaccumulation phase. However, this reaccumulation phase is now treated more realistically by using a merging criterion based on energy and angular momentum and by allowing dissipation to occur during fragment collisions. We also extend our previous studies to the as yet unexplored intermediate impact energy regime (represented by the Flora family formation) for which the largest fragment's mass is about half that of the parent body. Finally, we examine the robustness of the results by changing various assumptions, the numerical resolution, and different numerical parameters. We find that in the lowest impact energy regime the more realistic physical approach of reaccumulation leads to results that are statistically identical to those obtained with our previous simplistic approach. Some quantitative changes arise only as the impact energy increases such that higher relative velocities are reached during fragment collisions, but they do not modify the global outcome qualitatively. As a consequence, these new simulations confirm previous main results and still lead to the conclusion that: (1) all large family members must be made of gravitationally reaccumulated fragments; (2) the original fragment size distribution and their orbital dispersion are respectively steeper and smaller than currently observed for the real families, supporting recent studies on subsequent evolution and diffusion of family members; and (3) the formation of satellites around family members is a frequent and natural outcome of collisional processes.     

The composition of carbonaceous chondrite matrix

Matrix compositions of 32 carbonaceous chondrites have been analyzed by an electron microprobe defocussed-beam technique. Except in those chondrites that show evidence of metamorphism, matrices are compositionally similar and have correlation coefficients of +0.96 or greater. Weight per cent Mg/Si in matrices is constant (0.82 ± 0.05) but less than ratios derived from bulk analyses. Matrices in metamorphosed meteorites are Mg-depleted relative to those of other chondrites. Al Rais and Renazzo (anomalous by any classification scheme) have Mg-enriched matrices. Average matrix compositions cluster into chemical subgroups similar to those based on bulk chemical and petrographie criteria [C1, C2, C3(0), C3(V)]. C1 matrices are particularly variable in composition from point to point within the same meteorite, but points within individual breccia clasts appear to be more compositionally uniform. Cl matrices are depleted in Na, S, and Ca relative to solar and C2 matrix values, probably as a result of leaching. Matrix Ca/A1 ratios are highly variable and generally fall below the accepted meteoritic value. The only strong interelement correlation is for Fe, Ni, and S in C2 matrices, suggesting mixing of variable proportions of two components: Mg-rich phyllosilicate and a Ni-bearing chalcophile phase. The amount of magnetite associated with C2 matrix appears to vary systematically with matrix composition. Isotopic, chemical, and mineralogical constraints suggest that matrix, although appreciably altered in some meteorites, is chiefly a solar system condensation product which contains an admixture of unprocessed interstellar dust.     

A Synthetic Haematite Reference Material for LA‐ICP‐MS U‐Pb Geochronology and Application to Iron Oxide‐Cu‐Au Systems

In both nature and synthetic experiments, the common iron oxide haematite (α‐Fe₂O₃) can incorporate significant amounts of U into its crystal structure and retain radiogenic Pb over geological time. Haematite is a ubiquitous component of many ore deposit types and, therefore, represents a valuable hydrothermal mineral geochronometer, allowing direct constraints to be placed on the timing of ore formation and upgrading. However, to date, no suitable natural haematite reference material has been identified. Here, a synthetic haematite U‐Pb reference material (MR‐HFO) is characterised using LA‐ICP‐MS and ID‐TIMS. Centimetre‐scale ‘chips’ of synthesised α‐Fe₂O₃ were randomly microsampled via laser ablation‐extraction and analysed using ID‐TIMS. Reproducible U/Pb and Pb/Pb measurements were obtained across four separate chips (n = 13). Subsequently, an evaluation of the suitability MR‐HFO in constraining U‐Pb data via LA‐ICP‐MS is presented using a selection of natural samples ranging from Cenozoic to Proterozoic in age. The MR‐HFO normalised U‐Pb ratios are more concordant and ages more accurate versus the same LA‐ICP‐MS spot analyses normalised to zircon reference material, when compared with independently acquired ID‐TIMS data from the same natural haematite grains. Results establish MR‐HFO as a suitable reference material for LA‐ICP‐MS haematite U‐Pb geochronology.