A second-order global solution of the ideal resonance problem

The Ideal Resonance Problem, as formulated in 1966 (Paper I), is defined by the Hamiltonian Following the procedure adopted in the construction of a first-orderglobal solution (Papers II, III, and V), we derive a second-order solution from the von Zeipel-Bohlin recursive algorithm of Paper II. The singularities inherent in the Bohlin expansion in powers of μ have been suppressed by means of theregularizing function of Paper III, and the singularities in the coefficients atAB″=0 have been removed by thenormalization technique of Paper V. As a check, it is shown that the global solution includes asymptotically theclassical solution, expanded in powers ofμ ², and carrying thecritical divisor B′.     

Factors influencing nitrate within a low-gradient agricultural stream

In low-gradient agricultural streams, the proportion of land use devoted to agriculture, the sinuosity of the stream and the time of year influence the concentration of nitrate in the stream waters. Land use influences the source of nitrate and also the morphology of the stream. Greater agricultural land use weakly correlated (r = 0.67) to higher nitrate concentrations. Streams in agricultural areas have been straightened, which decreases the sinuosity. As a stream meanders and becomes more sinuous, the potential for lateral hyporheic flow increases, which can enhance a stream system’s ability to remove nitrate. Logically, higher sinuous streams should remove more nitrate and likely sulfate as there is a greater potential for lateral hyporheic flows. To test this hypothesis, nitrate and sulfate were monitored. Mass fluxes of nitrate along six stream segments with varying sinuosity values were calculated and statistically analyzed to assess if differences in mass fluxes along the segments existed. Along the segments, there are statistically significant differences in the mass fluxes of nitrate [F(5,174) = 4.777; p = 0.001]. Stream segments with higher sinuosity index values exhibited a loss or lower gain in nitrate and sulfate than lower sinuosity index segments. The data suggest that stream segments with high sinuosity indices provide greater stream distance and increased hyporheic interaction within the streambed. Additionally, the more sinuous segments provide for an increase in lateral hyporheic flow beneath meander lobes. These additional hyporheic flows lead to enhanced denitrification in low-gradient agricultural streams. Seasonal differences were also noted. August through October experienced the lower nitrate concentrations as compared to June and July which exhibited the highest nitrate concentrations.     

Three-dimensional digital imaging of environmental data: selection of gridding parameters

Abstract. Three-dimensional imaging is a powerful technique for the visualization and interpretation of environmental data. The success of the process is linked to careful, technically-justifiable selection of variable parameters during the gridding and imaging process. The impacts of various approaches to gridding and possible setting of parameters on the final image and volume calculations were examined by generating alternative images for a very well characterized contaminated site in layered coastal plain sediments. To image properly scattered data collected at close intervals in wells from layered geological media, a higher grid density in the z direction is required along with a weighting factor to emphasize the influence of data in the x and y directions. For steeply-varying contaminant concentration data, the best results were obtained by gridding the log of the property value; an anti-log transformation is carried out to restore property values to the correct value before the visualization file is prepared. The techniques and recommendations made in this article were designed for modelling contaminant values with very steep gradients dispersed in a strongly anisotropic media. These recommendations may not apply directly to other sites but the process of selecting parameters should be similar.     

Olivine or impact melt: Nature of the “Orange” material on Vesta from Dawn

NASA’s Dawn mission observed a great variety of colored terrains on asteroid (4) Vesta during its survey with the Framing Camera (FC). Here we present a detailed study of the orange material on Vesta, which was first observed in color ratio images obtained by the FC and presents a red spectral slope. The orange material deposits can be classified into three types: (a) diffuse ejecta deposited by recent medium-size impact craters (such as Oppia), (b) lobate patches with well-defined edges (nicknamed “pumpkin patches”), and (c) ejecta rays from fresh-looking impact craters. The location of the orange diffuse ejecta from Oppia corresponds to the olivine spot nicknamed “Leslie feature” first identified by Gaffey (Gaffey, M.J. [1997]. Icarus 127, 130–157) from ground-based spectral observations. The distribution of the orange material in the FC mosaic is concentrated on the equatorial region and almost exclusively outside the Rheasilvia basin. Our in-depth analysis of the composition of this material uses complementary observations from FC, the visible and infrared spectrometer (VIR), and the Gamma Ray and Neutron Detector (GRaND). Several possible options for the composition of the orange material are investigated including, cumulate eucrite layer exposed during impact, metal delivered by impactor, olivine–orthopyroxene mixture and impact melt. Based on our analysis, the orange material on Vesta is unlikely to be metal or olivine (originally proposed by Gaffey (Gaffey, M.J. [1997]. Icarus 127, 130–157)). Analysis of the elemental composition of Oppia ejecta blanket with GRaND suggests that its orange material has ∼25% cumulate eucrite component in a howarditic mixture, whereas two other craters with orange material in their ejecta, Octavia and Arruntia, show no sign of cumulate eucrites. Morphology and topography of the orange material in Oppia and Octavia ejecta and orange patches suggests an impact melt origin. A majority of the orange patches appear to be related to the formation of the Rheasilvia basin. Combining the interpretations from the topography, geomorphology, color and spectral parameters, and elemental abundances, the most probable analog for the orange material on Vesta is impact melt.     

Vesta's mineralogical composition as revealed by the visible and infrared spectrometer on Dawn

The Dawn spacecraft mission has provided extensive new and detailed data on Vesta that confirm and strengthen the Vesta–howardite–eucrite–diogenite (HED) meteorite link and the concept that Vesta is differentiated, as derived from earlier telescopic observations. Here, we present results derived by newly calibrated spectra of Vesta. The comparison between data from the Dawn imaging spectrometer—VIR—and the different class of HED meteorites shows that average spectrum of Vesta resembles howardite spectra. Nevertheless, the Vesta spectra at high spatial resolution reveal variations in the distribution of HED‐like mineralogies on the asteroid. The data have been used to derive HED distribution on Vesta, reported in Ammannito et al. (2013), and to compute the average Vestan spectra of the different HED lithologies, reported here. The spectra indicate that, not only are all the different HED lithologies present on Vesta, but also carbonaceous chondritic material, which constitutes the most abundant inclusion type found in howardites, is widespread. However, the hydration feature used to identify carbonaceous chondrite material varies significantly on Vesta, revealing different band shapes. The characteristic of these hydration features cannot be explained solely by infalling of carbonaceous chondrite meteorites and other possible origins must be considered. The relative proportion of HEDs on Vesta's surface is computed, and results show that most of the vestan surface is compatible with eucrite‐rich howardites and/or cumulate or polymict eucrites. A very small percentage of surface is covered by diogenite, and basaltic eucrite terrains are relatively few compared with the abundance of basaltic eucrites in the HED suite. The largest abundance of diogenitic material is found in the Rheasilvia region, a deep basin, where it clearly occurs below a basaltic upper crust. However, diogenite is also found elsewhere; although the depth to diogenite is consistent with one magma ocean model, its lateral extent is not well constrained.     

The problem of small divisors in planetary motion

Small divisors caused by certain linear combinations of frequencies appear in all analytical planetary theories. With the exception of the deep resonance between Neptune and Pluto, they can be removed at the expense of introducing secular and mixed secular terms, limiting the domain in which the solution is valid. Because of them classical solutions are known not to converge uniformly; Poincaré referred to them as asymptotic. The KAM theory shows that if one is far enough from exact commensurability and has small enough planetary masses, expansions exist which will converge to quasi-periodic orbits. Solutions showing very small divisors are excluded from this region of convergence. The question of whether they are intrinsic to the problem or are just manifestations of the method of solution is not settled. Problems with a single commensurabily that can be isolated from the rest of the Hamiltonian may have solutions with no small divisors. The problem of two or more commensurabilities remains unsolved.     

Spectral analyses of solar photospheric fluctuations

The application of the fast-Fourier-transform (FFT) algorithm to calculating one-dimensional and bi-dimensional (temporal and spatial), power and cross-power (coherence and phase) spectra is examined for solar photospheric fluctuations. Alternative methods for smoothing raw spectra, direct averaging (employing various weights) and indirect truncation of the correlation function, are compared, and indirect smoothing is compared with spectra calculated by mean-lagged-product (MLP) methods. Besides providing the raw spectrum, FFT techniques easily allow computing a series of spectra with varying amounts of smoothing. From these spectra a range of satisfactory compromise between resolution and stability can be determined which helps in the interpretation of spectral trends, and in identifying more clearly the existence and significance of spectral features. For bi-dimensional spectra presented as contour plots, this range of satisfactory smoothing can be restricted, particularly when spectral trends must be represented by small-scale contours. Equivalent spectra (i.e. comparable equivalent degrees of freedom) computed or smoothed by different methods have minor, but not negligible, differences. Examination of these differences favors computing of FFT spectra smoothed by averaging for photospheric fluctuations.     

Geomorphic response of an Appalachian Valley and Ridge stream to urbanization

Human‐induced changes to the channel and 18·6 km² catchment of Second Creek, in Knox County, Tennessee (USA), have included deliberate channel realignment, channelization of some reaches in culverts or cement‐lined channels, the addition of coarse particles, and intentional and unintentional changes in catchment hydrology. Field observations and measurements made between 1997 and 2001 showed active adjustment of the stream channel. Channel bank erosion is the dominant adjustment, but aggradation also occurs. One change following urbanization is an increase in bed particle size due to the addition of particles of anthropogenic origin. Such particles constitute 2–21 per cent of particles sampled at eight sites along the stream, and their D₅₀ exceeds the D₅₀ of natural particles at five of the sites. The downstream portion of the catchment has been urbanized for more than 150 years, but urbanizing activity has continued throughout the catchment, occurring not as a discrete perturbation, but as a set of disturbances with varying spatial and temporal scales. Spatial patterns of erosion and deposition in the channel are complex and do not show an upstream–downstream trend. Effective, although unintended, decoupling of the most manipulated reaches has hindered the propagation of changes in channel morphology and channel materials in this urbanized stream system. Copyright © 2006 John Wiley & Sons, Ltd.