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 \(L_5\).     

Glacier changes in the eastern Nyainqêntanglha Range of Tibetan Plateau from 1975 to 2013

Maritime-type glaciers in the eastern Nyainqêntanglha Range, located in the southeastern part of the Tibetan Plateau, are an important water source for downstream residents and ecological systems. To better understand the variability of glaciers in this region, we used the band ratio threshold(TM3/TM5 for the Landsat TM /ETM+ and TM4/TM6 for Landsat OLI) to extract glacier outlines in ~1999 and ~2013. After that, we also generated a series of glacier boundaries and monitored glacier variations in the past 40 years with the help of the Chinese Glacier Inventory data(1975) and Landsat TM, ETM+ and OLI data. The total glacier area decreased by 37.69 ± 2.84% from 1975 to 2013. The annual percentage area change(APAC) was ~1.32% a-1 and ~1.29% a-1 in the periods 1975-1999 and 1999-2013, respectively. According to the lag theory, the reaction time is probably about 10 years and we discuss the variations of temperature and precipitation between 1965 and 2011. Temperature and precipitation increased between 1965 and 2011 at a rate of 0.34°C /10 a and 15.4 mm/10 a, respectively. Extensive meteorological data show that the glacier shrinkage rate over the period may be mainly due to increasing air temperature, while the increasing precipitation partly made up for the mass loss of glacier ice resulting from increasing temperature may also lead to the low APAC between 1999 and 2013. The lag theory suggests that glacier shrinkage may accelerate in the next 10 years. Small glaciers were more sensitive to climate change, and there was a normal distribution between glacier area and elevation. Glaciers shrank in all aspects, and south aspects diminished faster than others.     

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.     

Evaluation of an empirical orthogonal function–based method to downscale soil moisture patterns based on topographical attributes

Soil moisture influences many hydrologic applications including agriculture, land management and flood prediction. Most remote‐sensing methods that estimate soil moisture produce coarse resolution patterns, so methods are required to downscale such patterns to the resolutions required by these applications (e.g. 10‐ to 30‐m grid cells). At such resolutions, topography is known to affect soil moisture patterns. Although methods have been proposed to downscale soil moisture based on topography, they usually require the availability of past high‐resolution soil moisture patterns from the application region. The objective of this article is to determine whether a single topographic‐based downscaling method can be used at multiple locations without relying on detailed local observations. The evaluated downscaling method is developed on the basis of empirical orthogonal function (EOF) analysis of space–time soil moisture data at a reference catchment. The most important EOFs are then estimated from topographic attributes, and the associated expansion coefficients are estimated on the basis of the spatial‐average soil moisture. To test the portability of this EOF‐based method, it is developed separately using four data sets (Tarrawarra, Tarrawarra 2, Cache la Poudre and Satellite Station), and the relationships that are derived from these data sets to estimate the EOFs and expansion coefficients are compared. In addition, each of these downscaling methods is applied not only for the catchment where it was developed but also to the other three catchments. The results suggest that the EOF downscaling method performs well for the location where it is developed, but its performance degrades when applied to other catchments. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress Responses to Rapid Temperature Changes of the Juvenile Sea Cucumber （Apostichopusjaponicus Selenka）

Activities of hexokinase （HK）, pyruvate kinase （PK）, superoxide dismutase （SOD） and catalase （CAT） and Hsp70 level were measured to evaluate the response of the commercially important sea cucumber （Apostichopus japonicus Selenka） to rapid temperature changes in laboratory. Animals were subjected to a higher temperature （from 10 to 20℃） （Tinc treatment） or to a lower temperature （from 20 to 10℃） （Tddec treatment） for 72 h. At 1, 3, 12, 24, 72 h of exposure, animals were removed and prepared for further analysis. Results showed that the effect of acute temperature changes on enzyme activities was significant. In Tinc treatment, activities of SOD and CAT increased immediately. The significant enhancement of SOD and CAT activities suggested that oxidative stress increases significantly when ambient temperature increasing from 10 to 20℃. The up-regulation of Flsp70 in Tinc and Tdec treatments indicated that Hsp70 was a bioindicator of thermal stress in the sea cucumber, and the expression pattern depended on the thermal treatment.     

Geochemical reconstruction of late Holocene drainage and mixing in Kluane Lake,Yukon Territory

The level of Kluane Lake in southwest Yukon Territory, Canada, has fluctuated tens of metres during the late Holocene. Contributions of sediment from different watersheds in the basin over the past 5,000 years were inferred from the elemental geochemistry of Kluane Lake sediment cores. Elements associated with organic material and oxyhydroxides were used to reconstruct redox fluctuations in the hypolimnion of the lake. The data reveal complex relationships between climate and river discharge during the late Holocene. A period of influx of Duke River sediment coincides with a relatively warm climate around 1,300 years BP. Discharge of Slims River into Kluane Lake occurred when Kaskawulsh Glacier advanced to the present drainage divide separating flow to the Pacific Ocean via Kaskawulsh and Alsek rivers from flow to Bering Sea via tributaries of Yukon River. During periods when neither Duke nor Slims river discharged into Kluane Lake, the level of the lake was low and stable thermal stratification developed, with anoxic and eventually euxinic conditions in the hypolimnion.     

CATASTROPHIC ECO-ENVIRONMENTAL CHANGE IN THE SONGNEN PLAIN, NORTHEASTERN CHINA SINCE 1900S

Although the Songnen Plain in the northeastern China was developed relatively late in the temperate zone of the world, its eco-environment has changed greatly. This paper analyzes the changes of land cover and the rates and trends ofdesertification during the past 100 years in the Songnen Plain. According to the macroscopic analysis, we find that the eco-environment in the plain has reached to the threshold of catastrophic change since the 1950s. The Thorn Needle Catastrophic Model was used to determine and validate this conclusion. Human activities, including large-scale construction projects, such as huge dams and dikes, and excessive grazing were the primary factors contributing to regional eco-environmental catastrophe. And irrational reclamation of the wilderness also affected the eco-environmental change. The results reveal the complex human-land interactions.     

Logarithmic spiral trajectories generated by Solar sails

Analytic solutions to continuous thrust-propelled trajectories are available in a few cases only. An interesting case is offered by the logarithmic spiral, that is, a trajectory characterized by a constant flight path angle and a fixed thrust vector direction in an orbital reference frame. The logarithmic spiral is important from a practical point of view, because it may be passively maintained by a Solar sail-based spacecraft. The aim of this paper is to provide a systematic study concerning the possibility of inserting a Solar sail-based spacecraft into a heliocentric logarithmic spiral trajectory without using any impulsive maneuver. The required conditions to be met by the sail in terms of attitude angle, propulsive performance, parking orbit characteristics, and initial position are thoroughly investigated. The closed-form variations of the osculating orbital parameters are analyzed, and the obtained analytical results are used for investigating the phasing maneuver of a Solar sail along an elliptic heliocentric orbit. In this mission scenario, the phasing orbit is composed of two symmetric logarithmic spiral trajectories connected with a coasting arc.