Instability of the 2+2 body problem

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m₁ and m₂ in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm ₁,m ₂ and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.     

On topological stability in the general three-body problem

The confining curves in the general three-body problem are studied; the role of the integralc ² h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m ₃/m ₂ both the positions of the collinear equilibrium points and thec ² h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec ² h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec ² h integral and of the bifurcation values.     

Macroalgal distribution patterns in a shallow, soft-bottom lagoon, with emphasis on the nonnativeGracilaria vermiculophylla andCodium fragile

We determined the distribution of macroalgae in Hog Island Bay, a shallow coastal lagoon in Virginia, USA, seasonally at 12 sites from 1998 to 2000 and at 3 representative sites from 2000 to 2002. We analyzed macroalgal biomass, taxonomic richness, and abundance of two non-native species, the cryptic invaderGracilaria vermiculophylla and the conspicuousCodium fragile, with respect to season, location (mainland, mid lagoon, barrier island sites), and elevation (intertidal, subtidal). Taxonomic richness, total algal biomass, and nonnative biomass peaked in the summer months when temperature and light availability were highest. A few stress tolerant and ephemeral algae dominated the algal assemblage.G. vermiculophylla constituted 74% of the entire algal biomass, was the most abundant alga in all seasons, locations, and elevation levels, and was positively correlated with taxonomic richness and abundance of filamentous species.Ulva curvata, Bryopsis plumosa, andC. fragile accounted for an additional 16% of the algal biomass. There are distinct habitats in Hog Island Bay that can be classified into low diversity-low biomass regions near the mainland and barrier islands and high diversity-high biomass regions in the open mid lagoon, where abundant shells for attachment and intermediate levels of water column nutrients and turbidity likely create better growth conditions. Taxonomic richness and biomass were higher in subtidal than intertidal zones, presumably due to lower desiccation stress. This study provides an example of how a single invasive species can dominate an entire assemblage, both in terms of biomass (being most abundant in all seasons, locations, and tidal levels) and species richness (correlating positively with epiphytic filamentous taxa). By adding hard-substratum structural complexity to a relatively homogenous soft-substratum system,G. vermiculophylla increases substratum availability for attachment and entanglement of other algal species and enhances local diversity. Without widespread and abundantG. vermiculophylla, taxa likePolysiphonia, Ceramium, Bryopsis, Ectocarpus, andChampia would likely be much less common. This study also highlights the importance of using DNA analysis of voucher specimens in monitoring programs to accurately identify cryptic invaders.     

Geochemical pattern classification of recent volcanic products from Mt. Etna, Italy, based on Kohonen maps and fuzzy clustering

We present the application of a classification method based on Kohonen maps and fuzzy clustering to geochemical analyses of volcanic products erupted on Mt. Etna from 1995 to 2005. Based on 13 major and trace elements, the classification allows a new way to visualize distinct compositional features of magma both considering long period as well as single eruptive events, such as in 2001 and 2002–03 flank eruptions. Products of the various vents do not necessarily form homogeneous groups, but show clear trends of chemical evolution with time. Using a convenient color code, the graphical visualization of the results in just a single picture allows the rapid identification of the compositional features of each sample and their comparison with all the products analyzed in the 10-year-long time span. This single picture accounts for the mutual interactions of the 13 components avoiding shortcomings of classical low-dimensional plots where components relevant for the discrimination have to be found in a priori study of many diagrams. On the basis of the synoptic information provided by pattern classification, we identify links between the products of different eruptive vents which deliver a reliable picture of a multifaceted plumbing system, in agreement with geochemical and geophysical evidence reported in literature. The analysis of the 13-dimensional data set using the Kohonen maps and fuzzy clustering simultaneously turned out to be straightforward and easy. Accordingly, the results of this application will be useful also as a contextual data set for new data in future ongoing eruptive episodes.     

Adhesion and collisional release of particles in dense planetary rings

We propose a simple theoretical model for aggregative and fragmentative collisions in Saturn’s dense rings. In this model the ring matter consists of a bimodal size distribution: large (meter sized) boulders and a population of smaller particles (tens of centimeters down to dust). The small particles can adhesively stick to the boulders and can be released as debris in binary collisions of their carriers. To quantify the adhesion force we use the JKR theory (Johnson, K., Kendall, K., Roberts, A. [1971]. Proc. R. Soc. Lond. A 324, 301–313). The rates of release and adsorption of particles are calculated, depending on material parameters, sizes, and plausible velocity dispersions of carriers and debris particles. In steady state we obtain an expression for the amount of free debris relative to the fraction still attached to the carriers. In terms of this conceptually simple model a paucity of subcentimeter particles in Saturn’s rings (French, R.G., Nicholson, P.D. [2000]. Icarus 145, 502–523; Marouf, E. et al. [2008]. Abstracts for “Saturn after Cassini–Huygens” Symposium, Imperial College London, UK, July 28 to August 1, p. 113) can be understood as a consequence of the increasing strength of adhesion (relative to inertial forces) for decreasing particle size. In this case particles smaller than a certain critical radius remain tightly attached to the surfaces of larger boulders, even when the boulders collide at their typical speed. Furthermore, we find that already a mildly increased velocity dispersion of the carrier-particles may significantly enhance the fraction of free debris particles, in this way increasing the optical depth of the system.     

Rapid characterisation of the extremely large landslide threatening the Rules Reservoir (Southern Spain)

Landslides - When an active landslide is first identified in an artificial reservoir, a comprehensive study has to be quickly conducted to analyse the possible hazard that it may represent to such...     

From Jurassic rifting to Cretaceous subduction in NW Iranian Azerbaijan: geochronological and geochemical signals from granitoids

Previous interpretations of a Jurassic subduction in Iran were based on trace element classification diagrams for granitoids, but their reliability is questionable, underscored by modern examples of continental break-up zones such as the Baja California. We present new field observations, bulk rock geochemistry, Sr and Nd isotope analyses and U–Pb zircon geochronology to assess the age and tectonic setting of previously undated intermediate to felsic magmatic rocks cropping out in the Precambrian basement of NW Iranian Azerbaijan. The geochronology revealed an uneven distribution in space and time: Late Jurassic (159–154 Ma) intrusions and dikes are alkaline to calc-alkaline. Their melt source is mantle dominated with a distinct continental contribution disclosed by radiogenic isotopes and abundant inherited zircon cores. Mid-Cretaceous (112–96 Ma) plutonic bodies and associated volcanic rocks occur only to the east of the major Siah Cheshmeh–Khoy Fault. They have geochemical signatures typical of a metasomatized mantle. In consistence with the sedimentation history of the area, our new interpretation attributes the Late Jurassic magmatism to thinning of a continental lithosphere in a rift-related setting. Mid-Cretaceous magmatism was produced by oceanic subduction beneath the Central Iran continent. We interpret the 40-Ma age gap between the two magmatic episodes as the time of opening of the oceanic basin witnessed by the Khoy ophiolite in the study area.     

Calcium Carbonate and Phosphate Reference Materials for Monitoring Bulk and Microanalytical Determination of Sr Isotopes

In situ laser ablation analyses rely on the microanalytical homogeneity of reference materials (RMs) and a similar matrix and mass fraction between unknown samples and RMs to obtain reliable results. Suitable carbonate and phosphate RMs for determination of Sr isotope ratios in such materials are limited. Thus, we determined ⁸⁷Sr/⁸⁶Sr ratios of several carbonate (JCt‐1, JCp‐1, MACS‐1, MACS‐3) and phosphate (MAPS‐4, MAPS‐5, NIST SRM 1400, NIST SRM 1486) international RMs using dissolved samples and two different multi‐collector inductively coupled plasma‐mass spectrometers (MC‐ICP‐MS). Our Sr isotope data are in agreement with published data and have an improved measurement precision for some RMs. For MACS‐1, we present the first ⁸⁷Sr/⁸⁶Sr value. We tested the suitability of these materials for microanalytical analyses by LA‐MC‐ICP‐MS, with two different laser ablation systems: a conventional nanosecond laser and a state‐of‐the‐art femtosecond laser. We investigated the RMs micro‐homogeneity and compared the data with our solution data. Both laser ablation systems yielded identical ⁸⁷Sr/⁸⁶Sr ratios within uncertainty to the solution data for RMs with low interferences of REEs. Therefore, these carbonate and phosphate RMs can be used to achieve accurate and precise results for in situ Sr isotope investigations by LA‐MC‐ICP‐MS of similar materials.     

Thermal expansion on volcanic tuff rocks used as building stones: examples from Mexico

Three cities in the center of Mexico, declared as cultural heritage, were erected mainly of volcanic tuffs as main construction element. Many of the historic buildings of these cities show significant damage and deterioration. One of the causes of these damages can be attributed to a phenomenon poorly studied in volcanic tuffs, the thermal expansion. To understand the response of volcanic tuffs to thermal expansion, as well as to know their thermal expansion coefficient, thermal expansion test was performed on 12 representative tuffs of these localities. In the same way and to know which of their petrographic, petrophysical, mineralogical, and geochemical properties influence this phenomenon, several laboratory studies were carried out to determine which of these plays an important role in the expansion. The results of our tests showed that volcanic tuffs rocks have very varied thermal expansion values, ranging from 0.7 mm/m at their maximum expansion to even contraction in the order of ??0.5 mm/m. Equally varied were the thermal expansion coefficient values ranging from 10.1 to ??7.2 10^?6/K. The most dominant behavior type is the “anisotropic no residual strain behavior”. In general, the behavior of the tuffs in practically all the tests carried out showed a very important heterogeneity in the rocks; however, in spite of the non-homogeneous behavior of the tuffs, three main factors could be identified that play an important role in the thermal expansion of the studied rocks, the chemical composition, e.g., acid volcanic tuff rocks (rhyolitic composition) had greater expansions, while the more basic rocks (basaltic composition) expanded less, the textural and fabrics homogeneity, e.g., the most homogeneous rocks texturally have higher expansion values and the degree of crystallinity, and, e.g., the tuff rocks with the largest amount of glass showed smaller thermal expansions than the tuffs composed of microcrystals.     

Al Huwaysah 010: The most reduced brachinite,so far

Al Huwaysah 010 is an ungrouped achondrite meteorite, recently referred to as a brachinite-like meteorite. This meteorite, showing a fine-grained assemblage of low-Ca pyroxene and opaque phases, is strongly reduced in comparison to other reduced brachinites. The occurrence of some tiny plates of graphite and oldhamite in this meteorite suggests that a partial melt residue has experienced a further reduction process. Olivine, the most abundant phase, is compositionally homogeneous (Fo_83.3) as well as the clinopyroxene (En_45.5Fs_10.8Wo_43.7) and the plagioclase (Ab_69.5). Orthopyroxene (En_85.4Fs_13.9Wo_0.7) also occurs but only in a fine intergrowth. Other accessory phases are Fe metal grains (Ni-free or Cr-bearing Fe-Ni alloy), troilite, chlorapatite, pentlandite (as inclusions in chromite). The sample shows two different closure temperatures: the highest (≈900°C) is determined via the olivine–chromite intercrystalline geothermometer and the lowest temperature (≈520°C) is determined via the pyroxene-based intracrystalline geothermometer. These temperatures may represent, respectively, the closure temperature associated with the formation and a subsequent impact event excavating the sample from the parental body. The visible to near-infrared (VNIR) reflectance spectra of Al Huwaysah 010 exhibit low reflectance, consistent with the presence of darkening components, and weak absorptions indicative of olivine and pyroxene. Comparing the spectral parameters of Al Huwaysah 010 to potential parent bodies characterized by olivine–pyroxene mineralogy, we find that it falls within the field previously attributed to the SIII type asteroids. These results lead us to classify the Al Huwaysah 010 meteorite as the most reduced brachinite, whose VNIR spectral features show strong affinities with those of SIII asteroids.