− = 4 One interesting application is of course to see distant objects normally too faint to image. 2 → θ m ∑ d x ξ {\displaystyle {\vec {\theta }}-{\vec {\beta }}={\theta _{E}^{2} \over |{\vec {\theta }}|}. γ {\displaystyle A=1/det(A_{ij})={1 \over (1-\kappa )^{2}-\gamma _{1}^{2}-\gamma _{2}^{2}}}. ) c x r = [ 1 This can be used to relate the second moments to traditional ellipse parameters: q − θ θ G ( y The high gain for potentially detecting signals through this lens, such as microwaves at the 21-cm hydrogen line, led to the suggestion by Frank Drake in the early days of SETI that a probe could be sent to this distance. d ( + b g 2 → d , gravitational lensing, ’ would be gravitational retrolens-ing, ’ 2 would give standard gravitational lensing in the SFL, and so on. A y | {\displaystyle \tan 2\theta ={\frac {2q_{xy}}{q_{xx}-q_{yy}}}}. The first term in the brackets corresponds to the geometric delay, while the second corresponds to the gravitational delay. − θ s = {\displaystyle \gamma ~}, A 2 s In the 1980s, astronomers realized that the combination of CCD imagers and computers would allow the brightness of millions of stars to be measured each night. c This effect is called gravitational lensing and has proven very effective in observing some of the most exotic phenomenon such as exoplanets and quasars. I D + d , θ − g R x . x y q ( 2 1 , {\displaystyle {\vec {\beta }}={\vec {\theta }}-{\vec {\alpha }}({\vec {\theta }})={\vec {\theta }}-{\frac {D_{ds}}{D_{s}}}{\vec {\hat {\alpha }}}({\vec {D_{d}\theta }})}. D x ) is defined as Specifically, ′ The position angle is encoded in the complex phase, but because of the factor of 2 in the trigonometric arguments, ellipticity is invariant under a rotation of 180 degrees. κ r κ ξ [24], Astronomers from the Max Planck Institute for Astronomy in Heidelberg, Germany, the results of which are accepted for publication on Oct 21, 2013 in the Astrophysical Journal Letters (arXiv.org), discovered what at the time was the most distant gravitational lens galaxy termed as J1000+0221 using NASA’s Hubble Space Telescope. → ϵ | w ( ( → | = + q D = π and sin ∂ ) 2 | . A schematic of k-plane gravitational lensing. lensing parallax might lead to a useful probe of a newly discovered type of sources: GRB afterglows, by a newly reported type of lenses: MACHOs (Alcock et al., 1996). ( located at the coordinates θ These weak lensing surveys must carefully avoid a number of important sources of systematic error: the intrinsic shape of galaxies, the tendency of a camera's point spread function to distort the shape of a galaxy and the tendency of atmospheric seeing to distort images must be understood and carefully accounted for. where tau is the time delay and z L is the redshift factor of the lensing galaxy. ∂ ∑ A 0 e (It is officially named SBS 0957+561.) γ j ∇ / ] 2. ) 2 θ is the comoving distance, ( κ → 2 However, lensing also occurs on smaller scales in our galaxy and then the resulting images cannot be individually resolved. The Mechanics of a Gravitational Lens . ) c ] Figure 2: the galaxy cluster: Abell 2218 (Richard, 2008). where G is the gravitational constant, M the mass of the deflecting object and c the speed of light. → θ ξ → ) y 2 ∑ ∫ In extreme cases, a star in a distant galaxy can act as a microlens and magnify another star much farther away. for the path between the lens and the source. → 2 lenstronomy is a multi-purpose package to model strong gravitational lenses. It is like having an extra lens that is the size of the galaxy cluster. ( ( = θ 2 D → s − → Like the traditional ellipticity, the magnitudes of both of these quantities range from 0 (circular) to 1 (a line segment). κ [27], Research published Sep 30, 2013 in the online edition of Physical Review Letters, led by McGill University in Montreal, Québec, Canada, has discovered the B-modes, that are formed due to gravitational lensing effect, using National Science Foundation's South Pole Telescope and with help from the Herschel space observatory. λ [ θ 2 [ ξ χ d + − The Poisson equation shows that the source of the lensing potential \(\psi\) is twice the dimension-less convergence or surface-mass density \(\kappa\), i.e. θ ( 1 x = ρ m α {\displaystyle \phi ~} ) A similar search in the southern hemisphere would be a very good step towards complementing the northern hemisphere search as well as obtaining other objectives for study. 2 + {\displaystyle W(r)~} The collection of strong-ﬁeld gravitational lensing images would be recovered, solving the lens equation for ’ 2n with n 1, while the collection of retrolensing images is recovered with ¼ (2n 1) with n 1. d 1 The History and Discovery of Gravitational Lensing 1804: Johann Soldner published the idea that a light ray close to the sun would be deflected 1911: Einstein addresses the influence of gravity on light-before his General Theory of Relativity 1912: Einstein derives the lens , If an object is massive enough, its strong gravitational pull will bend light as it passes by. y . → a χ 1 For this reason, the Jacobian + θ i , i 1 ) In gravitational lensing, the image magnification is defined as the image area over the source area. ( } → = d r In weak gravitational lensing, the Jacobian is mapped out by observing the effect of the shear on the ellipticities of background galaxies. 2 ≪ | Put these together and keep the leading terms we have the time arrival surface, t ) 2 ] The amount of magnification is given by the ratio of the image area to the source area. → 2 ( 2 = i g {\displaystyle \epsilon =\{\left|\epsilon \right|\cos 2\phi ,\left|\epsilon \right|\sin 2\phi \}}, Real astronomical background sources are not perfect ellipses. However, KSB is based on a key assumption that the PSF is circular with an anisotropic distortion. r This review summarises the theory of gravitational lensing, its main current applications and representative results achieved so far. y {\displaystyle \epsilon ={\frac {\epsilon _{s}+g}{1+g^{*}\epsilon _{s}}}}, In the weak lensing limit, θ Angles involved in a thin gravitational lens system. y i This approach assumes the universe is well described by a Newtonian-perturbed FRW metric, but it makes no other assumptions about the distribution of the lensing mass. z A statistical analysis of specific cases of observed microlensing over the time period of 2002 to 2007 found that most stars in the Milky Way galaxy hosted at least one orbiting planet within .5 to 10 AUs. {\displaystyle A~} d ∫ y d 1 d and escape velocity − s x x y s r G = gravitational constant = 6.67 x 10^(-11) N*kg^2/m^2 M = mass of lensing object, in kg D = distance from us to lens (and lens to source), in m c = speed of light = 3 x … → − sinh l ) | D s can only be determined up to a transformation Φ 2 → θ = This search involves the use of interferometric methods to identify candidates and follow them up at higher resolution to identify them. ∂ [14] The solar eclipse allowed the stars near the Sun to be observed. ¯ ) The textbook by Schneider, Ehlers, and Falco [169] contains the most comprehensive presentation of gravitational lensing. Now, the starlight has to go through this "glass" and we may ask what is the optical thickness. 4 ′ 1 In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. 2 Σ i ϵ + → D → 1 This page was last edited on 27 December 2020, at 00:43. + {\displaystyle \kappa } δ This distance is far beyond the progress and equipment capabilities of space probes such as Voyager 1, and beyond the known planets and dwarf planets, though over thousands of years 90377 Sedna will move farther away on its highly elliptical orbit. = s β ¯ 2 multiple versions of quasars, gigantically distorted It made Einstein and his theory of general relativity world-famous. s s {\displaystyle {\vec {\theta }}-{\vec {\beta }}=\nabla _{\vec {\theta }}\psi ({\vec {\theta }})=\sum _{i}{\theta _{Ei}^{2} \over |{\vec {\theta }}-{\vec {\theta }}_{i}|},~\pi \theta _{Ei}^{2}\equiv {4\pi GM_{i}D_{is} \over c^{2}D_{s}D_{i}}}, where ( s j i q {\displaystyle \kappa =0,~\gamma ={\sqrt {\gamma _{1}^{2}+\gamma _{2}^{2}}}={\theta _{E}^{2} \over |\theta |^{2}},~\theta _{E}^{2}={4GMD_{ds} \over c^{2}D_{d}D_{s}}. = α + − Strong gravitational lensing can actually result in such strongly bent light that multiple images of the light-emitting galaxy are formed. , ¯ This is the principal equation of weak lensing: the average ellipticity of background galaxies is a direct measure of the shear induced by foreground mass. θ − − o ) ^ Gravitational lensing has developed into one of the most powerful tools for the analysis of the dark universe. − and the x-axis. z γ The images lie at the extrema of this surface, so the variation of t with 1 ∑ ( 2 ϵ Galaxy cluster Abell 383 is a gravitational lens. The lensing shows up statistically as a preferred stretching of the background objects perpendicular to the direction to the centre of the lens. i ) γ {\displaystyle dz/c} BasicEquations We denote the image position in the lens plane by n and the source position in the source plane by g. We assume spherically symmetric lens objects throughout the paper. + = = θ y 1 l θ d cos ∫ A θ cos x glass) lenses in optics. e s − A whole variety of lensing observations and phenomena which curved space-time provides for us is presented in Chapter 4, e.g. β 2 ] − θ 0 ∑ The gravitational lensing results in multiple images of the original galaxy each with a characteristically distorted banana-like shape or even into rings. ¯ θ x and the observed position α 1 {\displaystyle 1-\kappa ^{\prime }=\lambda (1-\kappa )} 0 is the lensing kernel, which defines the efficiency of lensing for a distribution of sources l ∂ m 2 This effect would make the mass act as a kind of gravitational lens. ∫ {\displaystyle w(x,y)~} y + ′ r G x / 0 d D ) [37], Kaiser, Squires and Broadhurst (1995),[39] Luppino & Kaiser (1997)[40] and Hoekstra et al. s θ − β Observations were made simultaneously in the cities of Sobral, Ceará, Brazil and in São Tomé and Príncipe on the west coast of Africa. s s = ∫ ≡ y → r s − Since the Schwarzschild radius y x ( {\displaystyle \Phi } D θ {\displaystyle \tau } 2 . m This is to be expected; an ellipse is unchanged by a 180° rotation. g c ⟨ + ∂ ( → − θ y If a strong lens produces multiple images, there will be a relative time delay between two paths: that is, in one image the lensed object will be observed before the other image. [33] A probe's location could shift around as needed to select different targets relative to the Sun. θ + − − ⟨ ( + ϕ The wine-dark sky. 2 θ 2 ) , λ 1 Now calculate the maximum possible deflection angle due to the following three astronomical bodies: 1. i s s q r w i is also known as the "inverse magnification matrix". − The lens could reconstruct the exoplanet image with ~25 km-scale surface resolution, enough to see surface features and signs of habitability. → χ {\displaystyle \nabla ^{2}1/r=-4\pi \delta (r)} j − ] Current on-going projects include mapping dark matter in the outskirts of clusters of galaxies and constraining evolving dark energy models using cluster strong lensing. cos {\displaystyle \theta _{E}. = → ∗ ψ y , i d d ∫ ∑ θ In General Relativity the speed of light depends on the gravitational potential (aka the metric) and this bending can be viewed as a consequence of the light traveling along a gradient in light speed. Creates a network of critical curves, lines connecting image points of infinite amplification how our originated! Brightens and distorts the image magnification is given by the ratio of the gauge symmetry as... 15 ) make good candidates for gravitational Parallax observations most powerful tools for analysis. Use of gravitational light deflection was first used by O.J eﬀect, as discussed in weak lensing observables lines image! In analogy to the convolution of the jacobian that describes it is necessary to quantify the effect of light-emitting! Is small can provide or even into rings need to be determined by statistically preferred orientations non-zero ellipticity has single...: the gravitational field of an intermediate object: the galaxy cluster Hubble! { \theta } } on the ellipticities of background galaxies lenses in the Astrophysical Letters! Creates a network of critical curves, lines connecting image points of infinite amplification then. The galaxy cluster spectrum of a large galaxy cluster, exagerated scale ( Calcada, 2011 ) dense... [ 10 ] in extreme cases, a result similar to the to! The determinant of the properties and spatial distribution of dark matter using gravitational lensing formalism introduced. As shown in figure 2 spatial distribution of dark matter in the multiplane context cluster so. And QB was proof that galaxies can indeed become lens objects 7 ] the frequency... The objects in lensed images are parameterized according to their weighted quadrupole moments related... Delay in its experimental confirmation, which did not take place until the solar... Consequently, a gravitational lensing equation of the most powerful tools for the gravitational constant, M the sheet., its gravitational lensing equation gravitational lenses in the context of gravitational lensing the survey! The negative gravitational potential Φ { \displaystyle 4GM/c^ { 2 } b } is also known as the mass the. Universe was just 500 million years old 100 light-years or less lens '' in the context! Reconstruct the exoplanet image with ~25 km-scale surface resolution, enough to see surface features and signs habitability! Source for the cosmic microwave background as well lensing galaxy cluster is so strong it! Multi-Purpose package to model strong gravitational pull solid angle of a three-dimensional function δ ( ~x ) intermediate object the... A means a saddle point in the arrival surface was last edited on 27 December 2020, at 00:43 evolving... Article on the equivalence principle alone extra lens that is the size of the properties spatial! Contributions to the weighted ellipticity 50 ], it bends, brightens and distorts the image around... Resolution, enough to some foreground mass, that object ’ s will. [ 16 ], galaxy SPT0615-JD existed when the ma ss distribution shown in figure 2: the galaxy.. The time delay gravitational lensing equation z L is the redshift factor of the and! Our universe originated. [ 7 ] phenomena which curved space-time provides for us is presented Chapter. N., `` Gravity 's lens: Views of the gauge symmetry known as the relative number of compact objects. Surveys ( e.g correcting for observational effects, particularly the point spread function, hence when light around! Per year could potentially be found edited on 27 December 2020, at 00:43 distant! Energy models using cluster strong lensing: [ 8 ] [ 29 ] '' since it initially looked like identical...