AN APPLICATION OF THE TOPOLOGICAL DEGREE TO GRAVITATIONAL LENSES

1998 ◽  
Vol 13 (02) ◽  
pp. 83-86 ◽  
Author(s):  
MARCO LOMBARDI
Keyword(s):  
Point Source ◽  
Mass Distribution ◽  
Topological Degree ◽  
General Theorem ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
General Proof ◽  
Special Case

In this letter we provide a new proof of a general theorem on gravitational lenses, first proven by Burke (1981) for the special case of thin lenses. The theorem states that a transparent gravitational lens with non-singular mass distribution produces an odd number of images of a point source. Our general proof shows that the topological degree finds natural and interesting applications in the theory of gravitational lenses.

scholarly journals The number of images of a point source in an N-point gravitational lens

2019 ◽  
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Point Source ◽  
Upper Bound ◽  
Gravitational Lensing ◽  
Gravitational Lens ◽  
Abscissa Axis ◽  
Analysis Theory ◽  
Minimum Number ◽  
Special Case

An important problem of the theory of gravitational lensing is the problem of studying images of a given source in a given lens. A special case of this problem is the problem of the number of images of a point source in a planar N-point gravitational lens. On this issue, several papers have been published. Most of the works are devoted to the upper bound on the number of images. However, there is no work on the lower bound on the number of images. The present work is devoted to this question. The article calculates what the minimum number of images of a point source in an N-point gravitational lens is equal to. Proven the theorem about infimum of a number of point source images in the N-point gravitational lens. Is proved that this limit is being reached. In particular, it is established that a point source has a minimum number of images in the lens if all point masses are equal and located on the abscissa axis. Besides, the source is also on the abscissa axis. Regular and non-regular cases are considered. Using the theorem that was proved in the paper and the previously known results, a classification theorem about the number of images of a point source in an N-point gravitational lens is formulated. The theorem proved in this paper is illustrated by an example of point source images in a binary lens. The point masses in this lens are the same and are located on the abscissa axis symmetrically with regard to the origin of the coordinates. The minimum number of point source images, in this case, is three, and the maximum is five. A point source has a minimum number of images if it is located on the abscissa axis. The paper used methods of mathematical analysis, theory of functions of real variables and algebraic geometry.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

scholarly journals Can We Measure H0 with VLBI Observations of Gravitational Images?

1988 ◽  
Vol 129 ◽  
pp. 207-208
Author(s):  
E. E. Falco ◽  
M. V. Gorenstein ◽  
I. I. Shapiro
Keyword(s):  
Time Delay ◽  
Mass Distribution ◽  
Velocity Dispersion ◽  
Gravitational Lens ◽  
Lens System ◽  
Relative Time ◽  
Surface Mass ◽  
A Value ◽  
Vlbi Observations ◽  
The Difference

We have used the relative positions and magnifications of the A and B images in the gravitational lens system 0957+561, obtained from VLBI observations, to constrain a model for the surface mass distribution of the lens. With measurements of the difference ΔτBA in propagation times associated with A and B (the “relative time delay”) and of the velocity dispersion of the main lensing galaxy, both to be obtained, our model will yield a value for H0 with an uncertainty of ∼ 20% due mainly to uncertainties in our assumptions.

scholarly journals Gaia GraL: Gaia DR2 gravitational lens systems

2018 ◽  
Vol 618 ◽  
pp. A56 ◽  
Author(s):  
C. Ducourant ◽  
O. Wertz ◽  
A. Krone-Martins ◽  
R. Teixeira ◽  
J.-F. Le Campion ◽  
...  
Keyword(s):  
Spatial Resolution ◽  
Space Mission ◽  
Gravitational Lens ◽  
Model Parameters ◽  
Gravitational Lenses ◽  
Bayesian Modelling ◽  
Lens System ◽  
Data Release ◽  
Gaia Dr2 ◽  
Exhaustive List

Context. Thanks to its spatial resolution, the ESA/Gaia space mission offers a unique opportunity to discover new multiply imaged quasars and to study the already known lensed systems at sub-milliarcsecond astrometric precisions. Aims. In this paper, we address the detection of the known multiply imaged quasars from the Gaia Data Release 2 (DR2) and determine the astrometric and photometric properties of the individually detected images found in the Gaia DR2 catalogue. Methods. We have compiled an exhaustive list of quasar gravitational lenses from the literature to search for counterparts in the Gaia DR2. We then analysed the astrometric and photometric properties of these Gaia’s detections. To highlight the tremendous potential of Gaia at the sub-milliarcsecond level we finally performed a simple Bayesian modelling of the well-known gravitational lens system HE0435-1223, using Gaia DR2 and HST astrometry. Results. From 481 known multiply imaged quasars, 206 have at least one image found in the Gaia DR2. Among the 44 known quadruply imaged quasars of the list, 29 have at least one image in the Gaia DR2, 12 of which are fully detected (2MASX J01471020+4630433, HE 0435-1223, SDSS1004+4112, PG1115+080, RXJ1131-1231, 2MASS J11344050-2103230, 2MASS J13102005-1714579, B1422+231, J1606-2333, J1721+8842, WFI2033-4723, WGD2038-4008), eight have three counterparts, eight have two and one has only one. As expected, the modelling of HE0435-1223 shows that the model parameters are significantly better constrained when using Gaia astrometry compared to HST astrometry, in particular the relative positions of the background quasar source and the centroid of the deflector. The Gaia sub-milliarcsecond astrometry also significantly reduces the parameter correlations. Conclusions. Besides providing an up-to-date list of multiply imaged quasars and their detection in the Gaia DR2, this paper shows that more complex modelling scenarios will certainly benefit from Gaia sub-milliarcsecond astrometry.

scholarly journals “Missing image” in gravitational lens systems

1986 ◽  
Vol 119 ◽  
pp. 541-543
Author(s):  
D. Narasimha ◽  
K. Subramanian ◽  
S.M. Chitre
Keyword(s):  
Mass Distribution ◽  
Gravitational Lens

The gravitational lens models involving extended mass distribution generally predict an odd number of images with one of the images close to the centre of the principal lensing galaxy. In all the observed lens systems only an even number of images have been unambigously detected so far. It is demonstrated that the presence of a compact nucleus at the centre of lensing galaxies would dim the “odd” image significantly without affecting the rest of the image configuration.

scholarly journals Gravitational Lensing

1992 ◽  
Vol 9 ◽  
pp. 3-32 ◽  
Author(s):  
Sjur Refsdal ◽  
Jean Surdej
Keyword(s):  
Gravitational Lensing ◽  
Gravitational Lens ◽  
Medium Size ◽  
Gravitational Lenses ◽  
Optical Lens ◽  
Basic Principles ◽  
History Of ◽  
Photometric Monitoring ◽  
A Site ◽  
Near Future

AbstractAtmospheric lensing effects deform our view of distant objects; similarly, without any doubt, gravitational lensing perturbs our view of the distant Universe and affects our physical understanding of various classes of extragalactic objects. We summarize here part of the theoretical and observational evidences supporting these claims.After briefly reviewing the history of gravitational lenses, we recall the basic principles underlying the formation of gravitationally lensed images of distant cosmic sources. We describe a simple optical lens experiment, which was actually shown during the oral discourse, and which accounts for all types of presently known gravitational lens systems.The various optical and radio searches for new gravitational lens systems that are being carried out at major observatories are reviewed. State-of-the-art observations of selected gravitational lens systems, obtained with highly performing ground-based telescopes, are then presented. These include several examples of multiply imaged QSO images, radio rings and giant luminous arcs.Through the modeling of these enigmatic objects, we show how it is possible to weigh the mass of distant lensing galaxies as well as to probe the distribution of luminous and dark matter in the Universe. Among the astrophysical and cosmological interests of observing and studying gravitational lenses, we also discuss the possibility of deriving the value of the Hubble parameter Ho from the measurement of a time delay, and how to determine the size and structure of distant quasars via the observational study of micro-lensing effects.At the end of this paper, we conclude on how to possibly achieve major astro-physical and cosmological goals in the near future by dedicating, on a site with good atmospheric seeing conditions, a medium size (2-3 m) telescope to the photometric monitoring of the multiple images of known and suspected gravitational lens systems.

scholarly journals Gravitational Lenses and VLBI Structures

1984 ◽  
Vol 110 ◽  
pp. 249-250
Author(s):  
K. Subramanian ◽  
D. Narasimha ◽  
S. M. Chitre
Keyword(s):  
Time Delay ◽  
Elliptical Galaxy ◽  
Gravitational Lens ◽  
Gravitational Lenses ◽  
Low Mass Stars ◽  
Effective Time ◽  
Low Mass ◽  
Intensity Fluctuations

The double quasar 00957 + 561 A,B along with bright radio arches and VLBI structures is modelled using a gravitational lens consisting of an elliptical galaxy and a cluster. The effective time-delay between components A and B comes out to be about a year and this enables one to distinguish between intensity fluctuations resulting from intrinsic quasar variations and minilensing by low mass stars.

Imaging reflections in elliptically anisotropic media

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1616-1618 ◽  
Author(s):  
Joe Dellinger ◽  
Francis Muir
Keyword(s):  
Point Source ◽  
Isotropic Medium ◽  
Short Note ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Virtual Image ◽  
Transversely Isotropic Medium ◽  
Sh Waves ◽  
Special Case

In an isotropic medium, waves reflected from a mirror form a virtual image of their source. This property of planar reflectors is generally not true in the presence of anisotropy. In their short note, Blair and Korringa (1987) show that for the special case of SH waves from a point source in a transversely isotropic medium, an aberration‐free image is formed for any orientation of the mirror. While their proof is mathematical, we show the same result in an intuitive, pictorial fashion and in the process discover that although the image is indeed aberration free, it is still distorted.

A Small Axisymmetric Obstacle in the Presence of an Underwater Point-Source Field

1997 ◽  
Vol 05 (03) ◽  
pp. 243-263 ◽  
Author(s):  
A. Charalambopoulos ◽  
G. Dassios ◽  
P. Ergatis
Keyword(s):  
Free Surface ◽  
Point Source ◽  
Pressure Field ◽  
Vertical Line ◽  
Source Field ◽  
Floating Object ◽  
Smooth Objects ◽  
Special Case

A small, acoustically hard and axisymmetric object is placed in a deep homogeneous sea environment with a hard plane bottom. The free surface of the sea is assumed to be soft. The source and the receiver are placed on the same vertical line, far away from the object. Given the positions of the source and the receiver, two problems are solved: the determination of the pressure field at the receiver from the position and the shape of the object, and the determination of the position and the shape of the object from the pressure field at the receiver. The special case of smooth objects generated by the rotation of differentiable curves is studied. We provide results for the case of a floating object and for the case of an object or a boss at the bottom of the sea.

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