# gravitational lensing equation

≈ ∫ ) q ) 1 . s ) α q → → D ¯ θ z {\displaystyle \Phi } θ c 2 ¯ D ψ n ≈ 1 + 10 − 5. x β ( D i 4 1 Most of the gravitational lenses in the past have been discovered accidentally. }, We can also confirm the previously defined reduced deflection angle, θ = = This phenomenon was first mentioned in 1924 by the St. Petersburg physicist Orest Khvolson, and quantified by Albert Einstein in 1936. For a single point lens at the origin we recover the standard result ′ D = γ τ − ( 2 Newton calculated, with the math and physics at his disposal, the equation of the path taken by a light ray past a gravitational mass and he came up with an angle of deflection. x Abell 1689 - actual gravitational lensing effects (Hubble Space Telescope). . 2 d + → {\displaystyle \Phi ~} ) θ ′ ¯ The amount of magnification is given by the ratio of the image area to the source area. y ) A procedure, for application in gravitational lensing using the geodesic deviation equation, is developed and used to determine the magnification of a source when the lens or deflector is modeled by a “thick” Weyl and “thick” Ricci tensor. x ( Several other physicists speculated about gravitational lensing as well, but all reached the same conclusion that it would be nearly impossible to observe.. i − c γ i and convergence ⁡ R I   i i {\displaystyle D_{s}~} This effect would make the mass act as a kind of gravitational lens. − ) Specifically, ψ → = ∑ If such a search is done using well-calibrated and well-parameterized instrument and data, a result similar to the northern survey can be expected. ) Take the Poisson's equation for 3D potential ∂ c c − ) ⁡ , = ¯ Angles involved in a thin gravitational lens system. δ ] ∂ y 1 )   ≈ θ s x − d θ ) θ Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object as had already been supposed by Isaac Newton in 1704 in his Queries No.1 in his book Opticks. D θ ξ ^ t ξ D r q c − g ξ A y s i 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. 1 {\displaystyle \kappa ({\vec {\theta }})={\frac {\Sigma ({\vec {\theta }})}{\Sigma _{cr}}}}, and the critical surface density (not to be confused with the critical density of the universe), Σ , The action of the lensing map is captured by tracing light rays backwards from a subset P of L 1 to the light source plane S. Credits: After [64, p. 199]. For extragalactic lenses, these must be angular diameter distances. 2 {\displaystyle ({\vec {\xi }}^{\prime },z)} ξ ≪ D 1 Here we discuss the basics, followed by some real-world complications and applications. ) − j ) − ∂ = ( β q ϕ d The book by Bliokh and Minakov  on gravitational lensing is still only available in … θ z h Since galaxies are intrinsically elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. For a point mass, the gravitational potential is: ˚= GM r (19) In cartesian coordinates, the impact parameter is b= p x2 + y2 and r= p x2 + y2 + z2 = b2 + z2 The gravitational potential perpendicular to the light path is: r~?˚= (x-axis: @ x˚= GMx r3 y-axis: @ y˚= GMy r3 (20) 5 {\displaystyle a={\frac {R}{1-\kappa -\gamma }}}, b ξ d ′ 2 λ 2 x and the x-axis. + Galaxy clusters can produce separations of several arcminutes. 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. d The amplification is the inverse of the Jacobian, A γ x D , 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. ) τ The results of these surveys are important for cosmological parameter estimation, to better understand and improve upon the Lambda-CDM model, and to provide a consistency check on other cosmological observations. x γ { Σ {\displaystyle t=constant+{D_{d}D_{s} \over D_{ds}c}\tau ,~\tau \equiv \left[{({\vec {\theta }}-{\vec {\beta }})^{2} \over 2}-\psi \right]}, where lenstronomy - gravitational lensing software package¶. can only be determined up to a transformation 2 ∇ | → 2 c ∂ ) and {\displaystyle \chi _{s}~} Gravitational Lensing ENCYCLOPEDIA OF ASTRONOMY AND STROPHYSICS Gravitational Lensing Gravity bends light rays in a way analogous to, but quantitatively different from, the way it bends trajectories of passing particles. y − 2 + b }, In cases there are multiple point lenses plus a smooth background of (dark) particles of surface density ] = → j The theory is correct anyway. = x ′ 2 1 x 1 ,   β m z This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. → 2 {\displaystyle 4GM/c^{2}b} x c ( l → 2 D ϕ i b (1998) prescribed a method to invert the effects of the Point Spread Function (PSF) smearing and shearing, recovering a shear estimator uncontaminated by the systematic distortion of the PSF. x D → {\displaystyle q_{yy}={\frac {\sum (y-{\bar {y}})^{2}w(x-{\bar {x}},y-{\bar {y}})I(x,y)}{\sum w(x-{\bar {x}},y-{\bar {y}})I(x,y)}}}, q κ = D = → − ( e − = → θ , so, χ g q = a −   − is the vector impact parameter of the actual ray path from the infinitesimal mass − ∂ − {\displaystyle {\vec {\theta }}} s z y x 4 x can be written as, α {\displaystyle \delta _{ij}~} → 1 1 γ | θ {\displaystyle \chi =\{\left|\chi \right|\cos 2\phi ,\left|\chi \right|\sin 2\phi \}}, ϵ θ y | s → The gravitational attraction can be viewed as the motion of undisturbed objects in a background curved geometry or alternatively as the response of objects to a force in a flat geometry. → 2 = ) ) ( d ξ → {\displaystyle \mu \propto \lambda ^{-2}} | z Using equation (25), the lens equation becomes : r~ µˆ = (~µ¡ﬂ~) (34) or r~ µ µ1 2 (~µ¡ﬂ~)2 ¡ˆ ¶ = 0 (35) The term in the brackets appears in the physical time delay function for gravitationally lensed images: ¿(~µ; ﬂ~) = ¿ geom +¿grav = 1+ZL c DLDS DLS µ1 2 (~µ¡ﬂ~)2 ¡ˆ(µ) ¶ (36) b δ c G   ( = o = | (See also Calaprice, Alice: Landis, Geoffrey A., “Mission to the Gravitational Focus of the Sun: A Critical Analysis,” paper AIAA-2017-1679, AIAA Science and Technology Forum and Exposition 2017, Grapevine TX, January 9–13, 2017. , ∂ 4 b n 2 and located at the coordinates 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. θ , 2 r Here we used If we can assume that the sources are randomly oriented, their complex ellipticities average to zero, so Strong lenses have been observed in radio and x-ray regimes as well. , and if the deflection is small we can approximate the gravitational potential along the deflected trajectory by the potential along the undeflected trajectory, as in the Born approximation in quantum mechanics. d ξ {\displaystyle \rho ~} → − ( 2 i 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 … → The SH0ES (Supernova, H 0, for the Equation of State of Dark Energy) collaboration has been honing in on an H 0 measurement using so-called standard candles: type Ia supernovae and Cepheid variable stars, whose luminosities are known. + ′ y Due to the high frequency used, the chances of finding gravitational lenses increases as the relative number of compact core objects (e.g. / ) → = → = q cos 2 A naive application of Newtonian gravity can yield exactly half this value, where the light ray is assumed as a massed particle and scattered by the gravitational potential well. In both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from our Galaxy. , , D ∂ , , R 2 the weak gravitational lensing regime, in which sources are singly imaged and weakly distorted. = This formula is identical to the formula for weak gravitational lensing derived using relativistic Newtonian dynamics  without curving spacetime. {\displaystyle 0=\nabla _{\vec {\theta }}\tau ={\vec {\theta }}-{\vec {\beta }}-\nabla _{\vec {\theta }}\psi ({\vec {\theta }})}, which is the lens equation. 2 = I Microlensing: where no distortion in shape can be seen but the amount of light received from a background object changes in time. d − {\displaystyle q_{xx}=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta \,}, q ( s R − y Σ x ( 2 Weak lensing: where the distortions of background sources are much smaller and can only be detected by analyzing large numbers of sources in a statistical way to find coherent distortions of only a few percent. ξ ) x ≪ A schematic of k-plane gravitational lensing. ) The observations were performed in 1919 by Arthur Eddington, Frank Watson Dyson, and their collaborators during the total solar eclipse on May 29. = We can define the complex shear in analogy to the complex ellipticities defined above, γ For this reason, the Jacobian {\displaystyle D_{s}} + ⁡ ) Consequently, a gravitational lens has no single focal point, but a focal line. = ϵ s ( ≈ ϕ [ The first term in the brackets corresponds to the geometric delay, while the second corresponds to the gravitational delay. My question is how these are equivalent. 1 c 2 s 2 {\displaystyle r~} E 0 {\displaystyle \tau } The observer may then see multiple distorted images of the same source; the number and shape of these depending upon the relative positions of the source, lens, and observer, and the shape of the gravitational well of the lensing object. ∫ ( θ i →   This is to be expected; an ellipse is unchanged by a 180° rotation. λ ( Lensing degeneracies and mass substructure 3 Figure 1. θ where G is the gravitational constant, M the mass of the deflecting object and c the speed of light. | However, this effect is more commonly associated with Einstein, who published an article on the subject in 1936.. = → [ 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. 2 ⟩ This effect is purely statistical; the shape of any galaxy will be dominated by its random, unlensed shape, but lensing will produce a spatially coherent distortion of these shapes. l M α ¯ E ∫ d The use of the Australia Telescope 20 GHz (AT20G) Survey data collected using the Australia Telescope Compact Array (ATCA) stands to be such a collection of data. ] , It was not until 1979 that the first gravitational lens would be discovered. {\displaystyle \mu ={\frac {1}{\det A}}={\frac {1}{[(1-\kappa )^{2}-\gamma ^{2}]}}}. → = {\displaystyle \Phi ({\vec {\xi }})=-\int {\frac {d^{3}\xi ^{\prime }\rho ({\vec {\xi }}^{\prime })}{|{\vec {\xi }}-{\vec {\xi }}^{\prime }|}}}, ψ y . Lodge, who remarked that it is "not permissible to say that the solar gravitational field acts like a lens, for it has no focal length". 2 ′ ∑ y d , 4 However, KSB is based on a key assumption that the PSF is circular with an anisotropic distortion. → 2 = Einstein had worked on the same problem in 1911 and came up with the same answer as Newton so as any right-minded person would do, he moved on. c κ β {\displaystyle d^{2}\xi ^{\prime }dz\rho ({\vec {\xi }}^{\prime },z)} c = α the lens equation and magni cation tensor (Section 3) and time delays (Section 4) in the multiplane context. x d → 2 x 0 ρ − = ¯ − i c q Fritz Zwicky posited in 1937 that the effect could allow galaxy clusters to act as gravitational lenses. y → Full detail of the project is currently under works for publication. 2 ψ θ if there is a gravitational lens on their way. θ + → ∫ 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. y Gravitational lens discovered at redshift z = 1.53. | = − The deflection has well-known observable effects, such as multiple images, magnification of images, and time delays for propagation of light along the paths forming different images. G The wine-dark sky. [ θ ¯ 2 ∑ , where θ ) γ ξ ( x By measuring the shapes and orientations of large numbers of distant galaxies, their orientations can be averaged to measure the shear of the lensing field in any region. In weak gravitational lensing, the Jacobian is mapped out by observing the effect of the shear on the ellipticities of background galaxies. d {\displaystyle {\vec {\beta }}={\vec {\theta }}-{\vec {\alpha }}({\vec {\theta }})={\vec {\theta }}-{\frac {D_{ds}}{D_{s}}}{\vec {\hat {\alpha }}}({\vec {D_{d}\theta }})}. | ∂ 2 {\displaystyle \psi ({\vec {\theta }})\approx \sum _{i}{\frac {4GM_{i}D_{is}}{D_{s}D_{i}c^{2}}}\left[\ln \left({|{\vec {\theta }}-{\vec {\theta }}_{i}| \over 2}{D_{i} \over D_{is}}\right)\right]. G And the past regions source for the gravitational lensing and in many cases can be in... Principle alone curved space-time provides for us is presented in Chapter 4 e.g! Very famous because of delay in its experimental confirmation, which did not take until! 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