Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.     

Using multispectral landsat and sentinel-2 satellite data to investigate vegetation change at Mount St. Helens since the great volcanic eruption in 1980

Long-term analyses of vegetation succession after catastrophic events are of high interest for an improved understanding of succession dynamics. However, in many studies such analyses were restricted to plot-based measurements. Contrarily, spatially continuous observations of succession dynamics over extended areas and time-periods are sparse. Here, we applied a change vector analysis (CVA) to investigate vegetation succession dynamics at Mount St. Helens after the great volcanic eruption in 1980 using Landsat. We additionally applied a supervised random forest classification using Sentinel-2 data to map the currently prevailing vegetation types. Change vector analysis was performed with the normalized difference vegetation index (NDVI) and the urban index (UI) for three subsequent decades after the eruption as well as for the whole observation time between 1984 and 2016. The influence of topography on the current vegetation distribution was examined by comparing altitude, slope angles and aspect values of vegetation classes derived by the random forest classification. Wilcox- Rank-Sum test was applied to test for significant differences between topographic properties of the vegetation classes inside and outside of the areas affected by the eruption. For the full time period, a total area of 516 km² was identified as re-vegetated, whereas the area and magnitude of re-growing vegetation decreased during the three decades and migrated closer to the volcanic crater. Vegetation losses were mainly observed in regions unaffected by the eruption and related mostly to timber harvesting. The vegetation type classification reached a high overall accuracy of approximately 90%. 36 years after the eruption, coniferous and deciduous trees have established at formerly devastated areas dominating with a proportion of 66%, whereas shrubs are more abundant in riparian zones. Sparse vegetation dominates at regions very close to the crater. Elevation was found to have a great influence on the reestablishment and distribution of the vegetation classes within the devastated areas showing in almost all cases significant differences in altitude distribution. Slope was less important for the different classes - only representing significantly higher values for meadows, whereas aspect seems to have no notable influence on the reestablishment of vegetation at Mount St. Helens. We conclude that major vegetation succession dynamics after catastrophic events can be assessed and characterized over large areas from freely available remote sensing data and hence contribute to an improved understanding of succession dynamics.     

Cosmological constraints from the X-ray gas mass fraction in relaxed lensing clusters observed with Chandra

We present precise measurements of the X-ray gas mass fraction for a sample of luminous, relatively relaxed clusters of galaxies observed with the Chandra observatory, for which independent confirmation of the mass results is available from gravitational lensing studies. Parametrizing the total (luminous plus dark matter) mass profiles using the model of Navarro, Frenk & White, we show that the X-ray gas mass fractions in the clusters asymptote towards an approximately constant value at a radius r ₂₅₀₀, where the mean interior density is 2500 times the critical density of the Universe at the redshifts of the clusters. Combining the Chandra results on the X-ray gas mass fraction and its apparent redshift dependence with recent measurements of the mean baryonic matter density in the Universe and the Hubble constant determined from the Hubble Key Project, we obtain a tight constraint on the mean total matter density of the Universe,     , and measure a positive cosmological constant,     . Our results are in good agreement with recent, independent findings based on analyses of anisotropies in the cosmic microwave background radiation, the properties of distant supernovae, and the large-scale distribution of galaxies.