Lensing by Kerr Black Holes. II: Analytical Study of Quasi-Equatorial Lensing Observables

In this second paper, we develop an analytical theory of quasi-equatorial lensing by Kerr black holes. In this setting we solve perturbatively our general lens equation with displacement given in Paper I, going beyond weak-deflection Kerr lensing to third order in our expansion parameter epsilon, which is the ratio of the angular gravitational radius to the angular Einstein radius. We obtain new formulas and results for the bending angle, image positions, image magnifications, total unsigned magnification, and centroid, all to third order in epsilon and including the displacement. New results on the time delay between images are also given to second order in epsilon, again including displacement. For all lensing observables we show that the displacement begins to appear only at second order in epsilon. When there is no spin, we obtain new results on the lensing observables for Schwarzschild lensing with displacement.Comment: 23 pages; final published versio

The Theory of Caustics and Wavefront Singularities with Physical Applications

This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated caustics and singularities.Comment: Figs. not include

Gravitational lensing in fourth order gravity

Gravitational lensing is investigated in the weak field limit of fourth order gravity in which the Lagrangian of the gravitational field is modified by replacing the Ricci scalar curvature R with an analytical expression $f(R)$. Considering the case of a pointlike lens, we study the behaviour of the deflection angle in the case of power law Lagrangians, i.e. with f(R) = f_0 R^n. In order to investigate possible detectable signatures, the position of the Einstein ring and the solutions of the lens equation are evaluated considering the change with respect to the standard case. Effects on the amplification of the images and the Paczynski curve in microlensing experiments are also estimated.Comment: 10 pages, 3 figures, accepted for publication on Physical Review

Analysis of U.S. Federal Funding Agency Data Sharing Policies

Federal funding agencies in the United States (U.S.) continue to work towards implementing their plans to increase public access to funded research and comply with the 2013 Office of Science and Technology memo Increasing Access to the Results of Federally Funded Scientific Research. In this article we report on an analysis of research data sharing policy documents from 17 U.S. federal funding agencies as of February 2021. Our analysis is guided by two questions: 1.) What do the findings suggest about the current state of and trends in U.S. federal funding agency data sharing requirements? 2.) In what ways are universities, institutions, associations, and researchers affected by and responding to these policies? Over the past five years, policy updates were common among these agencies and several themes have been thoroughly developed in that time; however, uncertainty remains around how funded researchers are expected to satisfy these policy requirements

Image distortion in non perturbative gravitational lensing

We introduce the idea of {\it shape parameters} to describe the shape of the pencil of rays connecting an observer with a source lying on his past lightcone. On the basis of these shape parameters, we discuss a setting of image distortion in a generic (exact) spacetime, in the form of three {\it distortion parameters}. The fundamental tool in our discussion is the use of geodesic deviation fields along a null geodesic to study how source shapes are propagated and distorted on the path to an observer. We illustrate this non-perturbative treatment of image distortion in the case of lensing by a Schwarzschild black hole. We conclude by showing that there is a non-perturbative generalization of the use of Fermat's principle in lensing in the thin-lens approximation.Comment: 22 pages, 6 figures, to appear in Phys. Rev. D (January 2001

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.Comment: 25 pages, 4 figures. Invited review submitted for special issue of General Relativity and Gravitatio

Gravitational lensing by stars with angular momentum

Gravitational lensing by spinning stars, approximated as homogeneous spheres, is discussed in the weak field limit. Dragging of inertial frames, induced by angular momentum of the deflector, breaks spherical symmetry. I examine how the gravito-magnetic field affects image positions, caustics and critical curves. Distortion in microlensing-induced light curves is also considered.Comment: 9 pages, 9 figures; to appear in MNRA

Identifying Lenses with Small-Scale Structure. I. Cusp Lenses

The inability of standard models to explain the flux ratios in many 4-image gravitational lenses has been cited as evidence for significant small-scale structure in lens galaxies. That claim has generally relied on detailed lens modeling, so it is both model dependent and somewhat difficult to interpret. We present a more robust and generic method for identifying lenses with small-scale structure. For a close triplet of images associated with a source near a cusp caustic, the sum of the signed magnifications should approximately vanish. We derive realistic upper bounds on the sum, and argue that lenses with flux ratios that significiantly violate the bounds can be said to have structure in the lens potential on scales smaller than the image separation. Five observed lenses have such flux ratio ``anomalies'': B2045+265, 1RXS J1131-1231, and SDSS J0924+0219 have strong anomalies; B0712+472 has a strong anomaly at optical/near-IR wavelengths and a marginal anomaly at radio wavelengths; and RX J0911+0551 appears to have an anomaly, but this conclusion is subject to uncertainties about octopole modes in early-type galaxies. Analysis of the cusp relation does not identify the known anomaly in B1422+231, so methods that are more sophisticated (and less generic) than the cusp relation may be necessary to uncover flux ratio anomalies in some systems. Although these flux ratio anomalies might represent milli- or micro-lensing, we cannot identify the cause; we can only conclude that the lenses have significant structure in the potential on scales smaller than the separation between the images. Additional arguments must be invoked to specify the nature of this small-scale structure. [Abridged]Comment: significant revisions to extend analysis and strengthen conclusions; low-res version of Fig. 5 here, for high-res version see http://astro.uchicago.edu/~ckeeton/Papers/cuspreln.ps.g

3-D Seismic Interpretation and Volumetric Estimation of “Osaja Field” Niger Delta, Nigeria

3-D seismic interpretation and petrophysical analysis of the Osaja Field, Niger Delta, was carried out with aim of carrying out a detailed structural interpretation, reservoir characterization and volumetric estimation of the field. Four wells were correlated across the field to delineate the lithology and establish the continuity of reservoir sand as well as the general stratigraphy of the area. The petrophysical analysis carried out, revealed two sand units that are hydrocarbon bearing reservoirs (Sand_A and Sand_B).The spatial variation of the reservoirs were studied on a field wide scale using seismic interpretation. Time and depth structural maps generated were used to establish the structural architecture/geometry of the prospect area of the field. The depth structure map revealed NE-SW trending anticlinal structures with F5 and F6 as faults assisted closures to the reservoir. Furthermore, reservoir parameters such as net pay, water saturation porosity, net-to-gross etc, were derived from the integration of seismic and well log data. The structural interpretation on the 3-D seismic data of the study area revealed a total of seven faults ranging from synthetic to antithetic faults. The petrophysical analysis gave the porosity values of the reservoir Sand_A ranging from 18.1 - 20.3% and reservoir Sand_B ranging from 13.1-14.9% across the reservoir. The permeability values of reservoir Sand_A ranging from 63-540md and reservoir Sand_B ranging from 18-80md hence there is decrease in porosity and permeability of the field with depth.The net-to-gross varies from 22.1% to 22.4% in Rerservoir Sand A to between 5.34- 12% for Rerservoir Sand _A while Sw values for the reservoirs ranges from 38-42% in well 2 to about 68.79-96.06% in well 11. The result of original oil in place for all the wells calculated revealed that well 2 has the highest value with 9.3mmbls. These results indicate that the reservoirs under consideration have a poor to fair hydrocarbon (oil) prospect

Asymptotic Expansions and Amplification of a Gravitational Lens Near a Fold Caustic

We propose two methods that enable us to obtain approximate solutions of the lens equation near a fold caustic with an arbitrary degree of accuracy. We obtain "post-linear" corrections to the well-known formula in the linear caustic approximation for the total amplification of two critical images of a point source. In this case, in order to obtain the nontrivial corrections we had to go beyond the approximation orders earlier used by Keeton et al. and to take into account the Taylor expansion of the lens equation near caustic up to the fourth order. Corresponding analytical expressions are derived for the amplification in cases of the Gaussian and power-law extended source models; the amplifications depend on three additional fitting parameters. Conditions of neglecting the correction terms are analysed. The modified formula for the amplification is applied to the fitting of light curves of the Q2237+0305 gravitational lens system in a vicinity of the high amplification events (HAEs). We show that the introduction of some "post-linear" corrections reduces chi^2 by 30% in the case of known HAE on the light curve of image C (1999). These corrections can be important for a precise comparison of different source models with regard for observational data. Key words: gravitational lensing: micro - quasars: individual (Q2237+0305) - gravitational lensing: strong - methods: analyticalComment: 16 pages, 3 figure