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1.
We present a method for computing the probability distribution of microlensed light-curve derivatives both in the case of a static lens with a transverse velocity, and in the case of microlensing that is produced through stellar proper motions. The distributions are closely related in form, and can be considered equivalent after appropriate scaling of the input transverse velocity. The comparison of the distributions in this manner provides a consistent way to consider the relative contribution to microlensing (both large and small fluctuations) of the two classes of motion, a problem that is otherwise an extremely expensive computational exercise. We find that the relative contribution of stellar proper motions to the microlensing rate is independent of the mass function assumed for the microlenses, but is a function of optical depth and shear. We find that stellar proper motions produce a higher overall microlensing rate than a transverse velocity of the same magnitude. This effect becomes more pronounced at higher optical depth. With the introduction of shear, the relative rates of microlensing become dependent on the direction of the transverse velocity. This may have important consequences in the case of quadruply lensed quasars such as Q2237+0305, where the alignment of the shear vector with the source trajectory varies between images.  相似文献   

2.
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   

3.
This paper provides a complete theoretical treatment of the point-mass lens perturbed by constant external shear, often called the Chang–Refsdal lens. We show that simple invariants exist for the products of the (complex) positions of the four images, as well as moment sums of their signed magnifications. The image topographies and equations of the caustics and critical curves are also studied. We derive the fully analytic expressions for pre-caustics, which are the loci of non-critical points that map to the caustics under the lens mapping. They constitute boundaries of the region in the image domain that maps on to the interior of the caustics. The areas under the critical curves, caustics and pre-caustics are all evaluated, which enables us to calculate the mean magnification of the source within the caustics. Additionally, the exact analytic expression for the magnification distribution for the source in the triangular caustics is derived, as well as a useful approximate expression. Finally, we find that the Chang–Refsdal lens with additional convergence greater than unity (the 'overfocusing case') can exhibit third-order critical behaviour, if the 'reduced shear' is exactly equal to     , and that the number of images for N -point masses with non-zero constant shear cannot be greater than  5 N − 1  .  相似文献   

4.
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   

5.
When a source star is gravitationally microlensed by a dark lens, the centroid of the source star image is displaced relative to the position of the unlensed source star, with an elliptical trajectory. Recently, routine astrometric follow-up measurements of these source star image centroid shifts by using high-precision interferometers have been proposed to measure the lens proper motion, which can resolve the lens parameter degeneracy in the photometrically determined Einstein time-scale. When an event is caused by a bright lens, on the other hand, the astrometric shift is affected by the light from the lens, but one cannot identify the existence of the bright lens from the observed trajectory because the resulting trajectory of the bright lens event is also an ellipse. As results, lensing parameters determined from the trajectory differ from those of a dark lens event, causing an incorrect identification of the lens population. In this paper, we show that although the shape and size of the astrometric centroid shift trajectory are changed because of the bright lens, the angular speed of centroid shifts around the apparent position of the unlensed source star is not affected by the lens brightness. Therefore, one can identify the existence of a bright lens and determine its brightness by comparing the lens parameters determined from the 'angular speed curve' with those determined from the trajectory of observed centroid shifts. Once the lens brightness is determined, one can correct for the lens proper motion. As the proposed method provides information about both the lens brightness (dark or bright) and the corrected values of the physical parameters of the lens, one can constrain the nature of massive compact halo objects (MACHOs) significantly better.  相似文献   

6.
在光滑物质分布模型下,临界曲线是强引力透镜系统中像平面上一条放大率为无穷的线,而考虑少量离散质量的微透镜效应后,源平面上的放大率分布会出现复杂的结构,为暗物质成分的探测提供了一种有效途径.模拟临界曲线附近微透镜效应存在临界曲线上放大率无穷大和计算量巨大的困难.要达到所需的模拟精度,直接使用传统的光线追踪算法需要巨大的计算资源.为此发展了一个能实现海量计算的Graphics Processing Unit (GPU)并行方法来模拟临界曲线附近的微引力透镜效应.在型号为NVIDIA Tesla V100S PCIe 32 GB的GPU上,对于需要处理13000多个微透镜天体、发射1013量级光线的模拟,耗时在7000 s左右.在GPU并行的基础上,与直接的光线追踪算法相比,插值近似的引入使计算速度提升约两个数量级.利用该方法生成80个放大率分布图,并从中抽取800条光变曲线,进行了微焦散线数密度和峰值放大率的统计.  相似文献   

7.
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   

8.
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9.
We model the extremely massive and luminous lens galaxy in the Cosmic Horseshoe Einstein ring system J1004+4112, recently discovered in the Sloan Digital Sky Survey. We use the semilinear method of Warren & Dye, which pixelizes the source surface brightness distribution, to invert the Einstein ring for sets of parametrized lens models. Here, the method is refined by exploiting Bayesian inference to optimise adaptive pixelization of the source plane and to choose between three differently parametrized models: a singular isothermal ellipsoid, a power-law model and a Navarro, Frenk & White (NFW) profile. The most probable lens model is the power law with a volume mass density  ρ∝ r −1.96±0.02  and an axis ratio of ∼0.8. The mass within the Einstein ring (i.e. within a cylinder with projected distance of ∼30 kpc from the centre of the lens galaxy) is  (5.02 ± 0.09) × 1012 M ⊙  , and the mass-to-light ratio is ∼30. Even though the lens lies in a group of galaxies, the preferred value of the external shear is almost zero. This makes the Cosmic Horseshoe unique amongst large separation lenses, as almost all the deflection comes from a single, very massive galaxy with little boost from the environment.  相似文献   

10.
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   

11.
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   

12.
We study the anomalous flux ratio which is observed in some four-image lens systems, where the source lies close to a fold caustic. In this case two of the images are close to the critical curve and their flux ratio should be equal to unity, instead in several cases the observed value differs significantly. The most plausible solution is to invoke the presence of substructures, as for instance predicted by the Cold Dark Matter scenario, located near the two images. In particular, we analyze the two fold lens systems PG1115+080 and B1555+375, for which there are not yet satisfactory models which explain the observed anomalous flux ratios. We add to a smooth lens model, which reproduces well the positions of the images but not the anomalous fluxes, one or two substructures described as singular isothermal spheres. For PG1115+080 we consider a smooth model with the influence of the group of galaxies described by a SIS and a substructure with mass ∼105 M ⊙ as well as a smooth model with an external shear and one substructure with mass ∼108 M ⊙. For B1555+375 either a strong external shear or two substructures with mass ∼107 M ⊙ reproduce the data quite well.  相似文献   

13.
We derive a simple semi-analytical approximation for lens equations with an arbitrary radially symmetric mass density ρ( r ), when   r /ξ0≪ 1  and ξ0 is the scalelength of the density profile. At the strong lensing regime, which is mostly constrained by the inner part of the mass density profile, we assume ρ∝ r α.
A dark matter (DM) haloes (GNFW model) are parametrized through a shape parameter α, a concentration parameter c 1 and the total mass M . We apply our semi-analytical model to show how the solutions of the axially symmetric lens equations are degenerated in respect to the parameters α and c 1.
In the case of an asymmetric dual image lens system, similar effective degeneracy is produced when the geometry of the lens is relaxed. Because it is impossible to determine the exact location of the source image, a family of solutions is acquired when the mass of the lens object and location of the observed images are fixed.
Our results indicate that the amount of degeneration is only weakly affected by the asymmetry in the lensing geometry set-up, e.g. the observational effective degeneracy is very close to the true physical degeneracy of the Einstein ring solutions. Basically with high-enough values for the concentration parameter, the degeneracy spawns the whole range for the shape parameter  α=[−2.0, −1.0]  .  相似文献   

14.
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   

15.
In this paper, we investigate the colour changes of gravitational microlensing events caused by the two different mechanisms of differential amplification for a limb-darkened extended source and blending. From this investigation, we find that the colour changes of limb-darkened extended source events (colour curves) have dramatically different characteristics depending on whether the lens transits the source star or not. We show that for a source transit event, the lens proper motion can be determined by simply measuring the turning time of the colour curve instead of fitting the overall colour or light curves. We also find that even for a very small fraction of blended light, the colour changes induced by blending are equivalent to those induced by limb darkening, causing serious distortion in the observed colour curve. Therefore, to obtain useful information about the lens and source star from the colour curve of an event, it will be essential to correct for blending. We discuss various methods of blending correction .  相似文献   

16.
With the steady increase of the sample size of observed microlenses towards the central regions of the Galaxy, the main source of the uncertainty in the lens mass will shift from the simple Poisson noise to the intrinsic non-uniqueness of our dynamical models of the inner Galaxy, particularly the Galactic bar. We use a set of simple self-consistent bar models to investigate how the microlensing event rate varies as a function of axis ratio, bar angle and velocity distribution. The non-uniqueness of the velocity distribution of the bar model adds a significant uncertainty (by about a factor of 1.5) to any prediction of the lens mass. Kinematic data and self-consistent models are critical to lift the non-uniqueness. We discuss the implications of these results for the interpretation of microlensing observations of the Galactic bulge. In particular we show that Freeman bar models scaled to the mass of the Galactic bulge/bar imply a typical lens mass of around 0.8 M⊙, a factor of 3–5 times larger than the value from other models.  相似文献   

17.
The cluster lens Cl 0024+1654 is undoubtedly one of the most beautiful examples of strong gravitational lensing, providing five large images of a single source with well-resolved substructure. Using the information contained in the positions and the shapes of the images, combined with the null space information, a non-parametric technique is used to infer the strong lensing mass map of the central region of this cluster. This yields a strong lensing mass of  1.60 × 1014 M⊙  within a 0.5  arcmin radius around the cluster centre. This mass distribution is then used as a case study of the monopole degeneracy, which may be one of the most important degeneracies in gravitational lensing studies and which is extremely hard to break. We illustrate the monopole degeneracy by adding circularly symmetric density distributions with zero total mass to the original mass map of Cl 0024+1654. These redistribute mass in certain areas of the mass map without affecting the observed images in any way. We show that the monopole degeneracy and the mass-sheet degeneracy together lie at the heart of the discrepancies between different gravitational lens reconstructions that can be found in the literature for a given object, and that many images/sources, with an overall high image density in the lens plane, are required to construct an accurate, high-resolution mass map based on strong lensing data.  相似文献   

18.
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   

19.
Gravitational lensing of a background source by a foreground galaxy lens occasionally produces four images of the source. The cusp and the fold relations impose conditions on the ratios of magnifications of these four-image lenses. In this theoretical investigation, we explore the sensitivity of these relations to the presence of substructure in the lens. Starting with a smooth lens potential, we add varying amounts of substructure, while keeping the source position fixed, and find that the fold relation is a more robust indicator of substructure than the cusp relation for the images. This robustness is independent of the detailed spatial distribution of the substructure, as well as of the ellipticity of the lensing potential and the presence of external shear.  相似文献   

20.
The time-delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. For systems with accurately measured positions and time-delays, the most important uncertainty is related to the mass model used. Simple parametric models like isothermal ellipsoidal mass distributions seem to provide consistent results with a reasonably small scatter when applied to several lens systems. We discuss a family of models with a separable radial power law and an arbitrary angular dependence for the potential   ψ = r β F ( θ )  . Isothermal potentials are a special case of these models with   β =1  . An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H 0 as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to   H 0∝(2- β )/ β   . The effect of external shear is quantified by introducing a 'critical shear' γ c as a measure for the amount of shear that changes the result significantly. The analysis shows that in the general case H 0 and γ c do not depend on the position of the lens galaxy. Spherical lens models with images close to the Einstein radius with fitted external shear differ by a factor of   β /2  from shearless models, leading to   H 0∝2- β   in this case. We discuss these results and compare them with numerical models for a number of real lens systems.  相似文献   

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