Magnification relations for Kerr lensing and testing Cosmic Censorship

A Kerr black hole with mass parameter m and angular momentum parameter a acting as a gravitational lens gives rise to two images in the weak field limit. We study the corresponding magnification relations, namely the signed and absolute magnification sums and the centroid up to post-Newtonian order. We show that there are post-Newtonian corrections to the total absolute magnification and centroid proportional to a/m, which is in contrast to the spherically symmetric case where such corrections vanish. Hence we also propose a new set of lensing observables for the two images involving these corrections, which should allow measuring a/m with gravitational lensing. In fact, the resolution capabilities needed to observe this for the Galactic black hole should in principle be accessible to current and near-future instrumentation. Since a/m >1 indicates a naked singularity, a most interesting application would be a test of the Cosmic Censorship conjecture. The technique used to derive the image properties is based on the degeneracy of the Kerr lens and a suitably displaced Schwarzschild lens at post-Newtonian order. A simple physical explanation for this degeneracy is also given.Comment: 13 pages, version 2: references added, minor changes. To appear in Phys. Rev.

Formalism for testing theories of gravity using lensing by compact objects. III: Braneworld gravity

Braneworld gravity is a model that endows physical space with an extra dimension. In the type II Randall-Sundrum braneworld gravity model, the extra dimension modifies the spacetime geometry around black holes, and changes predictions for the formation and survival of primordial black holes. We develop a comprehensive analytical formalism for far-field black hole lensing in this model, using invariant quantities to compute all geometric optics lensing observables. We then make the first analysis of wave optics in braneworld lensing, working in the semi-classical limit. We show that wave optics offers the only realistic way to observe braneworld effects in black hole lensing. We point out that if primordial braneworld black holes exist, have mass M, and contribute a fraction f of the dark matter, then roughly 3e5 x f (M/1e-18 Msun)^(-1) of them lie within our Solar System. These objects, which we call "attolenses," would produce interference fringes in the energy spectra of gamma-ray bursts at energies ~100 (M/1e-18 Msun)^(-1) MeV (which will soon be accessible with the GLAST satellite). Primordial braneworld black holes spread throughout the universe could produce similar interference effects; the probability for "attolensing" may be non-negligible. If interference fringes were observed, the fringe spacing would yield a simple upper limit on M. Detection of a primordial black hole with M <~ 1e-19 Msun would challenge general relativity and favor the braneworld model. Further work on lensing tests of braneworld gravity must proceed into the physical optics regime, which awaits a description of the full spacetime geometry around braneworld black holes.Comment: 13 pages, 3 figures; accepted in PRD; expanded discussion of prospects for observing attolensing with GLAS

A comparison of approximate gravitational lens equations and a proposal for an improved new one

Keeping the exact general relativistic treatment of light bending as a reference, we compare the accuracy of commonly used approximate lens equations. We conclude that the best approximate lens equation is the Ohanian lens equation, for which we present a new expression in terms of distances between observer, lens and source planes. We also examine a realistic gravitational lensing case, showing that the precision of the Ohanian lens equation might be required for a reliable treatment of gravitational lensing and a correct extraction of the full information about gravitational physics.Comment: 11 pages, 6 figures, to appear on Physical Review

A single parameter representation of hygroscopic growth and cloud condensation nucleus activity – Part 2: Including solubility

The ability of a particle to serve as a cloud condensation nucleus in the atmosphere is determined by its size, hygroscopicity and its solubility in water. Usually size and hygroscopicity alone are sufficient to predict CCN activity. Single parameter representations for hygroscopicity have been shown to successfully model complex, multicomponent particles types. Under the assumption of either complete solubility, or complete insolubility of a component, it is not necessary to explicitly include that component&apos;s solubility into the single parameter framework. This is not the case if sparingly soluble materials are present. In this work we explicitly account for solubility by modifying the single parameter equations. We demonstrate that sensitivity to the actual value of solubility emerges only in the regime of 2&amp;times;10&lt;sup&gt;&amp;minus;1&lt;/sup&gt;–5&amp;times;10&lt;sup&gt;&amp;minus;4&lt;/sup&gt;, where the solubility values are expressed as volume of solute per unit volume of water present in a saturated solution. Compounds that do not fall inside this sparingly soluble envelope can be adequately modeled assuming they are either infinitely soluble in water or completely insoluble

A single parameter representation of hygroscopic growth and cloud condensation nucleus activity &ndash; Part 3: Including surfactant partitioning

Atmospheric particles can serve as cloud condensation nuclei in the atmosphere. The presence of surface active compounds in the particle may affect the critical supersaturation that is required to activate a particle. Modelling surfactants in the context of Köhler theory, however, is difficult because surfactant enrichment at the surface implies that a stable radial concentration gradient must exist in the droplet. In this study, we introduce a hybrid model that accounts for partitioning between the bulk and surface phases in the context of single parameter representations of cloud condensation nucleus activity. The presented formulation incorporates analytical approximations of surfactant partitioning to yield a set of equations that maintain the conceptual and mathematical simplicity of the single parameter framework. The resulting set of equations allows users of the single parameter model to account for surfactant partitioning by applying minor modifications to already existing code

A single parameter representation of hygroscopic growth and cloud condensation nucleus activity

International audienceWe present a method to describe the relationship between dry particle diameter and cloud condensation nuclei (CCN) activity using a single hygroscopicity parameter. Values of the hygroscopicity parameter are between 0.5 and 2 for highly-CCN-active salts such as sodium chloride, between 0.01 and 0.5 for slightly to very hygroscopic organic species, and 0 for nonhygroscopic components. If compositional data are available and if the hygroscopicity parameter of each component is known, a multicomponent hygroscopicity parameter can be computed by weighting component hygroscopicity parameters by their volume fractions in the mixture. In the absence of information on chemical composition, experimental data for complex, multicomponent particles can be fitted to obtain the hygroscopicity parameter. The hygroscopicity parameter can thus also be used to conveniently model the CCN activity of atmospheric particles, including those containing insoluble components. We confirm the general applicability of the hygroscopicity parameter and its mixing rule by applying it to published hygroscopic diameter growth factor and CCN-activation data for single- and multi-component particles

Correlates of School Library Development in Calabar, Nigeria: Implications for Counselling

The study investigated the correlates of school library development in Calabar, Nigeria.  Two hypotheses guided the study.  A thirty item structured questionnaire was developed and validated by experts in Measurement and Evaluation.  This was administered to a total of four hundred and fifty respondents comprising of principals, vice-principals and library staff of the secondary schools in the study area.  Pearson Product Moment Correlation Coefficient was used for data analysis.  The findings revealed that there is a significant relationship between the negative attitude of principals of secondary schools and the school library development.  A significant relationship also existed between funding and school library development in Calabar.  Based on this it was recommended that government should set a compulsory benchmark for every secondary school in the state as a condition towards having a functional, current, well stocked library.  Philanthropic organizations and private investors in the state should be made to promote the development of school library service by presenting books as donations to schools. Keywords: Library Development, Counselling, Principal attitude

Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity

The bending angle of light is a central quantity in the theory of gravitational lensing. We develop an analytical perturbation framework for calculating the bending angle of light rays lensed by a Schwarzschild black hole. Using a perturbation parameter given in terms of the gravitational radius of the black hole and the light ray's impact parameter, we determine an invariant series for the strong-deflection bending angle that extends beyond the standard logarithmic deflection term used in the literature. In the process, we discovered an improvement to the standard logarithmic deflection term. Our perturbation framework is also used to derive as a consistency check, the recently found weak deflection bending angle series. We also reformulate the latter series in terms of a more natural invariant perturbation parameter, one that smoothly transitions between the weak and strong deflection series. We then compare our invariant strong deflection bending-angle series with the numerically integrated exact formal bending angle expression, and find less than 1% discrepancy for light rays as far out as twice the critical impact parameter. The paper concludes by showing that the strong and weak deflection bending angle series together provide an approximation that is within 1% of the exact bending angle value for light rays traversing anywhere between the photon sphere and infinity.Comment: 22 pages, 5 figure