Multipole Gravitational Lensing and High-order Perturbations on the Quadrupole Lens

An arbitrary surface mass density of gravitational lens can be decomposed into multipole components. We simulate the ray-tracing for the multipolar mass distribution of generalized SIS (Singular Isothermal Sphere) model, based on the deflection angles which are analytically calculated. The magnification patterns in the source plane are then derived from inverse shooting technique. As have been found, the caustics of odd mode lenses are composed of two overlapping layers for some lens models. When a point source traverses such kind of overlapping caustics, the image numbers change by \pm 4, rather than \pm 2. There are two kinds of images for the caustics. One is the critical curve and the other is the transition locus. It is found that the image number of the fold is exactly the average value of image numbers on two sides of the fold, while the image number of the cusp is equal to the smaller one. We also focus on the magnification patterns of the quadrupole (m = 2) lenses under the perturbations of m = 3, 4 and 5 mode components, and found that one, two, and three butterfly or swallowtail singularities can be produced respectively. With the increasing intensity of the high-order perturbations, the singularities grow up to bring sixfold image regions. If these perturbations are large enough to let two or three of the butterflies or swallowtails contact, eightfold or tenfold image regions can be produced as well. The possible astronomical applications are discussed.Comment: 24 pages, 6 figure

Magnification relations of quad lenses and applications on Einstein crosses

In this work, we mainly study the magnification relations of quad lens models for cusp, fold and cross configurations. By dividing and ray-tracing in different image regions, we numerically derive the positions and magnifications of the four images for a point source lying inside of the astroid caustic. Then, based on the magnifications, we calculate the signed cusp and fold relations for the singular isothermal elliptical lenses. The signed fold relation map has positive and negative regions, and the positive region is usually larger than the negative region as has been confirmed before. It can also explain that for many observed fold image pairs, the fluxes of the Fermat minimum images are apt to be larger than those of the saddle images. We define a new quantity cross relation which describes the magnification discrepancy between two minimum images and two saddle images. Distance ratio is also defined as the ratio of the distance of two saddle images to that of two minimum images. We calculate the cross relations and distance ratios for nine observed Einstein crosses. In theory, for most of the quad lens models, the cross relations decrease as the distance ratios increase. In observation, the cross relations of the nine samples do not agree with the quad lens models very well, nevertheless, the cross relations of the nine samples do not give obvious evidence for anomalous flux ratio as the cusp and fold types do. Then, we discuss several reasons for the disagreement, and expect good consistencies for more precise observations and better lens models in the future.Comment: 12 pages, 11 figures, accepted for publication in MNRA

Twisted and Nontwisted Bifurcations Induced by Diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling. Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic bifurcations; periodic solutions.Comment: 42 pages in a tar.gz file. Use ``latex2e twisted.tex'' on the tex files. Hard copy of figures available on request from [email protected]

Galaxy alignment on large and small scales

Galaxies are not randomly distributed across the universe but showing different kinds of alignment on different scales. On small scales satellite galaxies have a tendency to distribute along the major axis of the central galaxy, with dependence on galaxy properties that both red satellites and centrals have stronger alignment than their blue counterparts. On large scales, it is found that the major axes of Luminous Red Galaxies (LRGs) have correlation up to 30Mpc/h. Using hydro-dynamical simulation with star formation, we investigate the origin of galaxy alignment on different scales. It is found that most red satellite galaxies stay in the inner region of dark matter halo inside which the shape of central galaxy is well aligned with the dark matter distribution. Red centrals have stronger alignment than blue ones as they live in massive haloes and the central galaxy-halo alignment increases with halo mass. On large scales, the alignment of LRGs is also from the galaxy-halo shape correlation, but with some extent of mis-alignment. The massive haloes have stronger alignment than haloes in filament which connect massive haloes. This is contrary to the naive expectation that cosmic filament is the cause of halo alignment.Comment: 4 pages, 3 figures, To appear in the proceedings of the IAU Symposium 308 "The Zeldovich Universe: Genesis and Growth of the Cosmic Web

Use of hydraulic rating to set environmental flows in the Zhangxi River, China

Ningbo city, China, is a rapidly growing residential and industrial centre, with a current population of 4 million. Its development has required a major water supply expansion programme providing 400,000 m3 of water per day from the upper reaches of the Zhangxi River by means of a cascade of reservoirs. Water resources management is achieved through operation of two major reservoirs, Jiaokou (75 million m3) and Zhougongzhai (93 million m3). Water is released from the reservoirs, via turbines (generating hydropower), for local industry, irrigated agriculture and public supply along the lower reaches of the River and to maintain the river ecosystem. Surveys of local residents along the Zhangxi River showed its important role in aspects of life, social activity, culture and leisure. Analysis of ecological monitoring data demonstrated the diverse nature of fish, plants and invertebrates within the river. Some elements of the ecosystem have a high local economic value to local people. This paper reports an assessment of the environmental flow needed to support key species in the river ecosystem. It employs hydraulic ratings to define sections of the river where flow velocity reaches 0.5 ms-1, required to stimulate spawning of the moonlight fish, an economically important and indicator species in the river. In two out of 6 cross-sections studied, flow releases from the reservoirs meet the needs of fish. The reservoirs reduce flood flows, which may lead to a loss of deep pools that are essential for the fish to survive during winter month