A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Environmental impacts of the coal ash spill in Kingston, Tennessee: an 18-month survey.

An 18 month investigation of the environmental impacts of the Tennessee Valley Authority (TVA) coal ash spill in Kingston, Tennessee combined with leaching experiments on the spilled TVA coal ash have revealed that leachable coal ash contaminants (LCACs), particularly arsenic, selenium, boron, strontium, and barium, have different effects on the quality of impacted environments. While LCACs levels in the downstream river water are relatively low and below the EPA drinking water and ecological thresholds, elevated levels were found in surface water with restricted water exchange and in pore water extracted from the river sediments downstream from the spill. The high concentration of arsenic (up to 2000 μg/L) is associated with some degree of anoxic conditions and predominance of the reduced arsenic species (arsenite) in the pore waters. Laboratory leaching simulations show that the pH and ash/water ratio control the LCACs' abundance and geochemical composition of the impacted water. These results have important implications for the prediction of the fate and migration of LCACs in the environment, particularly for the storage of coal combustion residues (CCRs) in holding ponds and landfills, and any potential CCRs effluents leakage into lakes, rivers, and other aquatic systems.

Survey of the potential environmental and health impacts in the immediate aftermath of the coal ash spill in Kingston, Tennessee.

An investigation of the potential environmental and health impacts in the immediate aftermath of one of the largest coal ash spills in U.S. history at the Tennessee Valley Authority (TVA) Kingston coal-burning power plant has revealed three major findings. First the surface release of coal ash with high levels of toxic elements (As = 75 mg/kg; Hg = 150 microg/kg) and radioactivity (226Ra + 228Ra = 8 pCi/g) to the environment has the potential to generate resuspended ambient fine particles (< 10 microm) containing these toxics into the atmosphere that may pose a health risk to local communities. Second, leaching of contaminants from the coal ash caused contamination of surface waters in areas of restricted water exchange, but only trace levels were found in the downstream Emory and Clinch Rivers due to river dilution. Third, the accumulation of Hg- and As-rich coal ash in river sediments has the potential to have an impact on the ecological system in the downstream rivers by fish poisoning and methylmercury formation in anaerobic river sediments.