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1.
We study analytically a gravitational lens due to a deformed star, which is modelled by using a monopole and a quadrupole moment. Positions of the images are discussed for a source on the principal axis. We present explicit expressions for the lens equation for this gravitational lens as a single real 10th-order algebraic equation. Furthermore, we compute an expression for the caustics as a discriminant for the polynomial. Another simple parametric representation of the caustics is also presented in a more tractable form. A simple expression for the critical curves is obtained to clarify a topological feature of the critical curves; the curves are simply connected if and only if the distortion is sufficiently large.  相似文献   

2.
The availability of a robust and efficient routine for calculating light curves of a finite source magnified due to bending of its light by the gravitational field of an intervening binary lens is essential for determining the characteristics of planets in such microlensing events, as well as for modelling stellar lens binaries and resolving the brightness profile of the source star. However, the presence of extended caustics, and the fact that the images of the source star cannot be determined analytically while their number depends on the source position (relative to the lens system), makes such a task difficult in general. Combining the advantages of several earlier approaches, an adaptive contouring algorithm is presented, which only relies on a small number of simple rules and operations on the adaptive search grid. By using the parametric representation of critical curves and caustics found by Erdl & Schneider, seed solutions to the adaptive grid are found, which ensures that no images or holes are missed.  相似文献   

3.
We investigate the effect of microlensing on parameters of the images of distant sources seen near the critical curves of complex gravitational lenses, which are represented as a sum of compact structures—microlenses (stars, star-like or planet-like bodies) and diffusely distributed matter (dust and gas clouds etc.). The observation of merging, cross-shaped, annular, or arc-shaped source images is an indication that the images are close to the critical curves of gravitational lenses. Our analysis and numerical solution have allowed us to determine the structures of the critical curves and caustics formed by macro-and microlenses, as well as to estimate the characteristic perturbations introduced by microlenses at their various positions relative to the critical curve of a regular gravitational lens. We show that, the closer are the microlenses to the critical curve, the larger is the discrepancy between our results and those obtained previously with standard (linearized) allowance for the effect of a regular gravitational lens.  相似文献   

4.
Lens models appropriate for representing cusped galaxies and clusters are developed. The analogue of the odd-number theorem for cusped density distributions is given. Density cusps are classified into strong, isothermal or weak, according to their lensing properties. Strong cusps cause multiple imaging for any source position, whereas isothermal and weak cusps give rise to only one image for distant sources. Isothermal cusps always possess a pseudo-caustic. When the source crosses the pseudo-caustic, the number of images changes by unity.   Two families of cusped galaxy and cluster models are examined in detail. The double power-law family has an inner cusp, followed by a transition region and an outer envelope. One member of this family — the isothermal double power-law model — possesses an exceedingly scarce property, namely the lens equation is exactly solvable for any source position. This means that the magnifications, the time delay and the lensing cross-sections are all readily available. The model has a three-dimensional density that is cusped like r −2 at small radii and falls off like r −4 asymptotically. Thus, it provides a very useful representation of the lensing properties of a galaxy or cluster of finite total mass with a flat rotation curve. The second set of models studied is the single power-law family. These are single density cusps of infinite extent. The properties of the critical curves and caustics and the behaviour of the lenses in the presence of external shear are all discussed in some detail.  相似文献   

5.
When the gravitational lensing potential can be approximated by that of a circularly symmetric system affected by weak perturbations, it is found that the shape of the resulting (tangential) caustics is entirely specified by the local azimuthal behaviour of the affecting perturbations. This provides a common mathematical groundwork for understanding problems such as the close–wide  ( d ↔ d −1)  separation degeneracy of binary lens microlensing light curves and the shear–ellipticity degeneracy of quadruple image lens modelling.  相似文献   

6.
Recent observations indicate that many if not all galaxies host massive central black holes. In this paper we explore the influence of black holes on the lensing properties. We model the lens as an isothermal ellipsoid with a finite core radius plus a central black hole. We show that the presence of the black hole substantially changes the critical curves and caustics. If the black hole mass is above a critical value, then it will completely suppress the central images for all source positions. Realistic central black holes are likely to have masses below this critical value. Even in such subcritical cases, the black hole can suppress the central image when the source is inside a zone of influence, which depends on the core radius and black hole mass. In the subcritical cases, an additional image may be created by the black hole in some regions, which for some radio lenses may be detectable with high-resolution and large dynamic range VLBI maps. The presence of central black holes should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.  相似文献   

7.
Images are considered which are produced by a spherically symmetric gravitational lens from a source of small angular size. Simple analytic expressions are derived to describe the position, amplification, shape distortion and time delay for these images provided the mass distribution of the lens corresponds to the King model. Possible types of images are analysed (single and triple images, as well as degenerate double ones at caustics) which may be formed at different positions of the observer relative to the light source and lens. The focal length of galaxies of different types is evaluated on the basis of the obtained results and the conditions are determined at which these galaxies may considerably affect the images of distant sources.  相似文献   

8.
Gravitational lensing provides a unique and powerful probe of the mass distributions of distant galaxies. Four-image lens systems with fold and cusp configurations have two or three bright images near a critical point. Within the framework of singularity theory, we derive analytic relations that are satisfied for a light source that lies a small but finite distance from the astroid caustic of a four-image lens. Using a perturbative expansion of the image positions, we show that the time delay between the close pair of images in a fold lens scales with the cube of the image separation, with a constant of proportionality that depends on a particular third derivative of the lens potential. We also apply our formalism to cusp lenses, where we develop perturbative expressions for the image positions, magnifications and time delays of the images in a cusp triplet. Some of these results were derived previously for a source asymptotically close to a cusp point, but using a simplified form of the lens equation whose validity may be in doubt for sources that lie at astrophysically relevant distances from the caustic. Along with the work of Keeton, Gaudi & Petters, this paper demonstrates that perturbation theory plays an important role in theoretical lensing studies.  相似文献   

9.
The lensing properties of the Plummer model with a central point mass and external shear are derived, including the image multiplicities, critical curves and caustics. This provides a simple model for a flattened galaxy with a central supermassive black hole. For the Plummer model with black hole, the maximum number of images is four, provided the black hole mass is less than an upper bound which is calculated analytically. This introduces a method to constrain black hole masses by counting images, thus applicable at cosmological distance. With shear, the maximum number of images is six and we illustrate the occurrence of an astroid caustic and two metamorphoses.  相似文献   

10.
Several gravitationally lensed quasars are observed with anomalous magnifications in pairs of images that straddle a critical curve. Simple theoretical arguments suggest that the magnification of these images should be approximately equivalent, whereas one image is observed to be significantly demagnified. Microlensing provides a possible explanation for this discrepancy. There are two key parameters when modelling this effect. The first, the fraction of smooth matter in the lens at the image positions, has been explored by Schechter & Wambsganss. They have shown that the anomalous flux ratio observed in the lensed quasar MG 0414+0534 is a priori a factor of 5 more likely if the assumed smooth matter content in the lens model is increased from 0 to 93 per cent. The second parameter, the size of the emission region, is explored in this paper, and shown to be more significant. We find that the broadening of the magnification probability distributions due to smooth matter content is washed out for source sizes that are predicted by standard models for quasars. We apply our model to the anomalous lensed quasar MG 0414+0534, and find a 95 per cent upper limit of  2.62 × 1016  h −1/270 ( M /M⊙)1/2 cm  on the radius of the I -band emission region. The smooth matter percentage in the lens is unconstrained.  相似文献   

11.
This paper makes the first systematic attempt to determine using perturbation theory the positions of images by gravitational lensing due to arbitrary number of coplanar masses without any symmetry on a plane, as a function of lens and source parameters. We present a method of Taylor-series expansion to solve the lens equation under a small mass-ratio approximation. First, we investigate perturbative structures of a single-complex-variable polynomial, which has been commonly used. Perturbative roots are found. Some roots represent positions of lensed images, while the others are unphysical because they do not satisfy the lens equation. This is consistent with a fact that the degree of the polynomial, namely the number of zeros, exceeds the maximum number of lensed images if   N = 3  (or more). The theorem never tells which roots are physical (or unphysical). In this paper, unphysical ones are identified. Secondly, to avoid unphysical roots, we re-examine the lens equation. The advantage of our method is that it allows a systematic iterative analysis. We determine image positions for binary lens systems up to the third order in mass ratios and for arbitrary N point masses up to the second order. This clarifies the dependence on parameters. Thirdly, the number of the images that admit a small mass-ratio limit is less than the maximum number. It is suggested that positions of extra images could not be expressed as Maclaurin series in mass ratios. Magnifications are finally discussed.  相似文献   

12.
High-resolution MERLIN observations of a newly discovered four-image gravitational lens system, B0128+437, are presented. The system was found after a careful re-analysis of the entire CLASS data set. The MERLIN observations resolve four components in a characteristic quadruple-image configuration; the maximum image separation is 542 mas and the total flux density is 48 mJy at 5 GHz. A best-fitting lens model with a singular isothermal ellipsoid results in large errors in the image positions. A significantly improved fit is obtained after the addition of a shear component, suggesting that the lensing system is more complex and may consist of multiple deflectors. The integrated radio spectrum of the background source indicates that it is a gigahertz peaked spectrum source. It may therefore be possible to resolve structure within the radio images with deep VLBI observations and thus to constrain the lensing mass distribution better.  相似文献   

13.
We perform a detailed analysis of the optical gravitational lens ER 0047–2808 imaged with the Wide Field Planetary Camera 2 on the Hubble Space Telescope . Using software specifically designed for the analysis of resolved gravitational lens systems, we focus on how the image alone can constrain the mass distribution in the lens galaxy. We find that the data are of sufficient quality to strongly constrain the lens model with no a priori assumptions about the source. Using a variety of mass models, we find statistically acceptable results for elliptical isothermal-like models with an Einstein radius of 1.17 arcsec. An elliptical power-law model  (Σ∝ R −β)  for the surface mass density favours a slope slightly steeper than isothermal with  β= 1.08 ± 0.03  . Other models including a constant mass-to-light ratio (M/L), pure Navarro, Frenk & White halo and (surprisingly) an isothermal sphere with external shear are ruled out by the data. We find the galaxy light profile can only be fit with a Sérsic plus point-source model. The resulting total  M/L B   contained within the images is  4.7  h 65± 0.3  . In addition, we find the luminous matter is aligned with the total mass distribution within a few degrees. This is the first time a resolved optical gravitational lens image has been quantitatively reproduced using a non-parametric source.
The source, reconstructed by the software, is revealed to have two bright regions, with an unresolved component inside the caustic and a resolved component straddling a fold caustic. The angular size of the entire source is ∼0.1 arcsec and its (unlensed) Lyα flux is  3 × 10−17 erg s−1 cm−2  .  相似文献   

14.
Observations of early-type galaxies, both in the local Universe and in clusters at medium redshifts, suggest that these galaxies often contain discs or disc-like structures. Using the results of Kelson et al. for the incidence of disc-components among the galaxies in the redshift   z = 0.33  cluster CL 1358+62, we investigate the effect of disc structures on the lensing properties of early-type galaxies. Statistical properties, like magnification cross-sections and the expected number of quad (four-image) lens systems, are not affected greatly by the inclusion of discs that contain less than ∼10 per cent of the total stellar mass. However, the properties of individual lens systems are affected. We estimate that 10–30 per cent of all quad-lens systems, with early-type deflector galaxies, would be affected measurably by the presence of disc components. Intriguingly, the image magnification ratios are altered significantly. The amplitude of the predicted change is sufficient to explain the observed magnification ratios in systems like B1422+231 without requiring compact substructure. Furthermore, time-delays between images also change; fitting a bulge-only model to early-type lenses that in fact contain a disc would yield a value of the Hubble constant H 0 that is systematically too low by ∼25 per cent.  相似文献   

15.
Using the ray-bundle method for calculating gravitational lens magnifications, we outline a method by which the magnification probability may be determined specifically in the weak lensing limit for cosmological models obtained from N -body simulations.
16 different models are investigated, which are variations on three broad classes of cold dark matter model: the standard model with  (Ω0, λ 0)=(1.0,0.0)  , the open model with  (Ω0, λ 0)=(0.3,0.0)  and the lambda model, which is a flat model with a cosmological constant  (Ω0, λ 0)=(0.3,0.7)  .
The effects of varying the Hubble parameter, H 0, the power spectrum shape parameter, Γ, and the cluster mass normalization, σ 8, are studied. It is shown that there is no signature of these parameters in the weak lensing magnification distributions. The magnification probability distributions are also shown to be independent of the numerical parameters such as the lens mass and simulation box size in the N -body simulations.  相似文献   

16.
We discuss the classic theorem according to which a gravitational lens always produces at least one image with a magnification greater than unity. This theorem seems to contradict the conservation of total flux from a lensed source. The standard solution to this paradox is based on the exact definition of the reference 'unlensed' situation, in which the lens mass can be either removed or smoothly redistributed.
We calculate magnifications and amplifications (in photon number and energy flux density) for general lensing scenarios not limited to regions close to the optical axis. In this way the formalism is naturally extended from tangential planes for the source and lensed images to complete spheres. We derive the lensing potential theory on the sphere and find that the Poisson equation is modified by an additional source term that is related to the mean density and to the Newtonian potential at the positions of observer and source. This new term generally reduces the magnification, to below unity far from the optical axis, and ensures conservation of the total photon number received on a sphere around the source.
This discussion does not affect the validity of the focusing theorem , in which the unlensed situation is defined to have an unchanged affine distance between source and observer. The focusing theorem does not contradict flux conservation, because the mean total magnification (or amplification) directly corresponds to different areas of the source (or observer) sphere in the lensed and unlensed situation. We argue that a constant affine distance does not define an astronomically meaningful reference.
By exchanging source and observer, we confirm that magnification and amplification differ according to Etherington's reciprocity law, so that surface brightness is no longer strictly conserved. At this level we also have to distinguish between different surface brightness definitions that are based on photon number, photon flux and energy flux.  相似文献   

17.
The recent detection by Limousin et al. of five new strong lensing events dominated by galaxy cluster members in Abell 1689, and outside the critical regime of the cluster itself, offers a way to obtain constraints on the cluster mass distribution in a region inaccessible to standard lensing analysis. In addition, modelling such systems will provide another window on the dark matter haloes of galaxies in very dense environments. Here, it is shown that the boost in image separation due to the external shear and convergence from a smooth cluster component means that more numerous, less massive galaxies have the potential to create multiple images with detectable separations, relative to isolated field galaxies. This comes in addition to a potential increase in their lensing (source plane) cross-section. To gain insight into the factors involved and as a precursor to a numerical study using N -body simulations, a simple analytic model of a cluster at   z = 0.3  lensing background galaxies at   z = 2  is considered here. The fiducial model has cluster members with isothermal density profiles and luminosities L , distributed in a Schechter function (faint-end slope  ν=−1.25  ), related to their velocity dispersions σ via the Faber–Jackson scaling L ∝σ4. Just outside the critical regime of the cluster, the scale of galaxy-dominated image separations is significantly increased. Folding in the fact that less massive galaxies present a lower lensing cross-section, and that the cross-section can itself be enhanced in an external field leads to a factor of a few times more detected events relative to field galaxies. These values will be higher closer to the critical curve. Given that the events in Abell 1689 were detected over a very small region of the cluster where ACS data were available, this motivates the search for such events in other clusters.  相似文献   

18.
We present observations of a new double-image gravitational lens system, ULAS J082016.1+081216, of image separation 2.3 arcsec and high (∼6) flux ratio. The system is selected from the Sloan Digital Sky Survey (SDSS) spectroscopic quasar list using new high-quality images from the UKIRT (United Kingdom Infrared Telescope) Deep Sky Survey (UKIDSS). The lensed quasar has a source redshift of 2.024, and we identify the lens galaxy as a faint red object of redshift  0.803 ± 0.001  . Three other objects from the UKIDSS survey, selected in the same way, were found not to be lens systems. Together with the earlier lens found using this method, the SDSS–UKIDSS lenses have the potential to significantly increase the number of quasar lenses found in SDSS, to extend the survey to higher flux ratios and lower separations, and to give greater completeness which is important for statistical purposes.  相似文献   

19.
We consider small-scale spheroidal clusters of weakly interacting massive particles in our Galaxy as non-compact gravitational microlenses and predict the appearance of caustics in the plane of a lensed source. The crossing of these caustics by a lensed star can produce a large variety of light curves, including some observed in actual microlensing events that have been interpreted as manifestations of binary gravitational lenses. We consider also observable effects during the gravitational microlensing of stars of non-zero angular size with a given brightness distribution across their disks by such an exotic objects as natural wormholes and objects whose space-time environment is described with the NUT metric. We demonstrate that, under certain conditions, the microlensing light curves, chromatic and polarizational effects due to the properties of the lens and the star disk brightness distributions can differ considerably from those observed for a Schwarzschild gravitational lens, so that their analysis can facilitate the identification of such objects.  相似文献   

20.
While the Hubble constant can be derived from observable time delays between images of lensed quasars, the result is often highly sensitive to assumptions and systematic uncertainties in the lensing model. Unlike most previous authors, we put minimal restrictions on the radial profile of the lens and allow for non-elliptical lens potentials. We explore these effects using a broad class of models with a lens potential     which has an unrestricted radial profile but self-similar iso-potential contours defined by     For these potentials, the lens equations can be solved semi-analytically. The axis ratio and position angle of the lens can be determined from the image positions of quadruple gravitational lensed systems directly, independent of the radial profile. We give simple equations for estimating the power-law slope of the lens density directly from the image positions and for estimating the time delay ratios. Our method greatly simplifies the numerics for fitting observations and is fast in exploring the model parameter space. As an illustration, we apply the model to PG1115+080. An entire one-parameter sequence of models fits the observations exactly. We show that the measured image positions and time delays do not uniquely determine the Hubble constant.  相似文献   

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