On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Sources of dissolved REE in mountainous streams draining granitic rocks, Sierras Pampeanas (Córdoba, Argentina)

Stream waters draining granitic terrains from the highest part (850 to 2200 m a.s.l.) of Sierras Pampeanas (Córdoba, Argentina, ∼32°S, ∼65°W) were sampled in order to define sources and distribution of dissolved rare earth elements (REE), and to describe the geochemical processes that govern their mobility. The contribution of the regional granite to the dissolved REE pool in stream water is limited due to the physical conditions predominating in the area (i.e., steep slopes and semiarid climate). Therefore, precipitation is considered a seasonally significant source controlling REE concentration in stream water. Dissolved REE concentrations are inversely correlated with monthly precipitation and rainfall frequency. During the rainy season (i.e., the austral summer) REE concentrations in stream water are lower than during the dry season (i.e., austral winter). Such low concentrations reflect the balance between the REE input from precipitation and their removal by adsorption. In contrast, during the dry season, the longer residence time of water within fractures and colluvium determines an increased REE concentration in the base flow. Lower pH values also contribute to raise REE concentration through desorption from mineral surfaces.     

Testing links between river patterns and in-channel characteristics using MRPP and ANOVA

This study tests the assumption that the characteristics of channels within multiple channel rivers are different from those of single channel rivers. Some river restoration approaches propose radical transformation of river patterns, from multiple to single channels, based on the link between river patterns and their in-channel characteristics. Determining the links between river patterns and their in-channel characteristics is complicated by differences in geology, history, climate and discharge among rivers. Furthermore, multiple channel rivers are composed of a mosaic of channel types with a range of in-channel characteristics. This study minimizes these problems by analysing a single river containing neighbouring single and multiple channel patterns with little change in discharge downstream, and by analysing all channel types. The study addressed two objectives: to determine the hydraulic geometry, energy, and sediment mobility characteristics of neighbouring single and multiple channel river patterns, and to test for statistical differences in these characteristics between patterns. The Renous River shows a wandering pattern for 11.5 km, with multiple channels around semipermanent islands and abandoned channels in the flood plain. The river displays a single channel river pattern where channels are confined by their valley walls, upstream and downstream of wandering. The analysis was conducted at three scales. First, the confined single channel and wandering multiple channel patterns were compared (pattern scale). Second, the confined channel pattern was compared to single and multiple channel sections within the wandering pattern (section scale). Third, all channel types were compared (channel type scale). Multi response permutation procedure (MRPP) and analysis of variance (ANOVA) were used to analyze differences between channels. Difference tests found no simple discrimination between the single and multiple channel river patterns of the Renous River. Tests between the single confined and multiple wandering channel patterns found few differences in the in-channel variables. The tests did find differences between multiple channel sections within the wandering pattern and confined single channels; however, a greater number of differences were found between multiple channel and single channel sections within the wandering pattern, highlighting the variability within the wandering pattern. Two groups emerged when all channel types were tested for differences: perennial main-channels containing the thalweg, and ephemeral side-channels. Therefore, side-channels define the in-channel characteristics of wandering rivers because few differences were found among main-channels in either pattern. This analysis suggests that all channel types, not just main-channels, should be investigated to obtain a complete picture of a river pattern prior to any restoration efforts. Engineers must exercise caution when applying the link between river patterns and in-channel characteristics to river restoration efforts.     

P- and S-wave anisotropy of a synthetic sandstone with controlled crack geometryl1

A method is reported for the production of synthetic porous sandstones containing cracks of known dimensions and geometry with respect to the matrix. A synthetic sandstone was manufactured from Sand cemented with an epoxy glue. The cracks of known geometry were introduced into the material in the manufacturing stage, by emplacing thin metallic discs in the Sand-epoxy matrix. These discs were chemically leached out of the consolidated porous sandstone. Acoustic anisotropy. and shear-wave splitting were observed in the synthetic sandstones. For the dry sample the observed angular dependence of the P- and S- wave velocities (at 100 kHz) compares well, qualitatively, with the theoretical models of Hudson and of Thomsen. Quantitatively, however, the experimental data fits Hudson's model better. For the case of a saturated sample the experimental results are in excellent agreement with Thomsen's model. Hudson's model, on the other hand, predicts a different angular dependence for P-waves. This demonstrates that the concept of fluid transfer between cracks and the ambient porosity can be a significant process. The results reported here are from the first successful experiment in which the theoretical models were tested on a porous material containing a known crack geometry.     

Distinct metamorphic evolution of alternating silica-saturated and silica-deficient microdomains within garnet in ultrahigh-temperature granulites: An example from Sri Lanka

Here we report the occurrence of garnet porphyroblasts that have overgrown alternating silica-saturated and silica deficient microdomains via different mineral reactions. The samples were collected from ultrahigh-temperature (UHT) metapelites in the Highland Complex, Sri Lanka. In some of the metapelites, garnet crystals have cores formed via a dehydration reaction, which had taken place at silica-saturated microdomains and mantle to rim areas formed via a dehydration reaction at silica-deficient microdomains. In contrast, some other garnets in the same rock cores had formed via a dehydration reaction which occurred at silica-deficient microdomains while mantle to rim areas formed via a dehydration reaction at silica-saturated microdomains. Based on the textural observations, we conclude that the studied garnets have grown across different effective bulk compositional microdomains during the prograde evolution. These microdomains could represent heterogeneous compositional layers (paleobedding/laminations) in the precursor sediments or differentiated crenulation cleavages that existed during prograde metamorphism. UHT metamorphism associated with strong ductile deformation, metamorphic differentiation and crystallization of locally produced melt may have obliterated the evidence for such microdomains in the matrix. The lack of significant compositional zoning in garnet probably due to self-diffusion during UHT metamorphism had left mineral inclusions as the sole evidence for earlier microdomains with contrasting chemistry.     

Biogeochemical records of paleoenvironmental changes in Nainital Lake,Kumaun Himalayas,India

Rapid urbanization and increased tourism around Nainital Lake in the Kumaun Himalayan region in north India has raised concerns about sediment and water pollution. Lead-210 dated sediment cores from the lake represent ~95 years of accumulation and yield a mean sedimentation rate of ~4.7 mm year⁻¹. Total organic carbon (TOC), percent N and S and their atomic C/N and C/S ratios, stable isotopes (δ¹³C, δ¹⁵N, and δ³⁴S), and specific biomarkers (n-alkanes and pigments) were measured in the core. Organic matter is primarily derived from in-lake algal production and TOC flux varies from 1.0 to 3.5 g m⁻² year⁻¹. Sediments are anoxic (Eh −328 to −187 mV) and have low (0.10–0.30 g m⁻² year⁻¹) N, but high (0.37–1.0 g m⁻² year⁻¹) S flux. Shifts in δ¹³C, δ¹⁵N, and δ³⁴S suggest in-lake microbial processes dominated by denitrification and sulfate reduction. The sediments are dominated by short-chain hydrocarbons with low Carbon Preference Index values. The pigments indicate a gradual shift to cyanobacterial domination of the phytoplankton community in recent years. Despite an increase in external input of nutrients, the trophic state of the lake has remained largely unchanged, and the perceived human-induced impacts are limited.     

The neotectonics and geodynamics of the Lower Oka Region (East European Craton)

The neotectonic structures of the Lower Oka (Nizhneokskii) Region formed under different geodynamic conditions. This is attested by the morphology, orientation, internal structure, and jointing of the structures. The Oka-Tsna arc formed under the effect of tension from an inner source on the one hand and stress from the Alpian belt on the other hand. The latitudinally-oriented structures of the northwestern slope of the Tokmovo arc emerged as a result of uplift and widening. Both types of structure are combined within the limits of the Oka-Murom trough, which is a geodynamically active zone.     

The formation of small cones in Isidis Planitia,Mars through mobilization of pyroclastic surge deposits

We investigate the distinctive distribution and pattern of subkilometer-sized cones in Isidis Planitia, and their relationship to the material that fills the basin. We observe that: (1) the cones show uniform size, spacing, and morphology across the entire basin; (2) they have large central depressions (relative to cone basal diameters) that are generally filled in and commonly show layering; (3) cone chains form highly organized spatial patterns; cones are also found in isolation and in dense fields; (4) many adjacent chains are parallel, and evenly spaced, creating a pattern that mimics lava flows that have experienced compressional folding; (5) no flows are associated with the cones, even where summit crater rims are breached; (6) the cones are at least temporally related to basin fill; (7) the basin fill material is largely fine-grained, but with locally indurated layers at shallow depth; blocks ejected by small craters from these locally indurated layers are commonly dark in color; and (8) relationships between cones along a chain show no systematic temporal formation sequence along the chain. We conclude that the basin was likely inundated by one or more hot, partially welded pyroclastic flows; devolatilization of these flows remobilized volcaniclastic material to form the cones without associated lava flows. The volume of gas required to account for the observed number of cones is low and does not require the presence of ground water or ice.