# gravitational lensing equation

( . − m g y 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. {\displaystyle dl={dz \over c\cos \alpha (z)}} 2 r = ¯ lenstronomy - gravitational lensing software package¶. d θ δ s D , Prediction about the behavior of stars as a gravitational lens in the light curve was the first of the three main predictions of General Relativity. − {\displaystyle {\vec {\hat {\alpha }}}({\vec {\xi }})={\frac {4G}{c^{2}}}\int d^{2}\xi ^{\prime }\int dz\rho ({\vec {\xi }}^{\prime },z){\frac {\vec {b}}{|{\vec {b}}|^{2}}},~{\vec {b}}\equiv {\vec {\xi }}-{\vec {\xi ^{\prime }}}}, where s I 1 + x is small. = [ r / i r   d j ξ M x Φ 2 In gravitational lensing, the image magnification is defined as the image area over the source area. c 4 2 ′ ) ⟨ {\displaystyle {\vec {\hat {\alpha }}}({\vec {\xi }})={\frac {4G}{c^{2}}}\int {\frac {({\vec {\xi }}-{\vec {\xi }}^{\prime })\Sigma ({\vec {\xi }}^{\prime })}{|{\vec {\xi }}-{\vec {\xi }}^{\prime }|^{2}}}d^{2}\xi ^{\prime }}, As shown in the diagram on the right, the difference between the unlensed angular position The book by Bliokh and Minakov  on gravitational lensing is still only available in … o → ( Taken as imaginary and real parts, the real part of the complex ellipticity describes the elongation along the coordinate axes, while the imaginary part describes the elongation at 45° from the axes. 1 2 − 2 Geometrical sensitivity of gravitational lensing . , x The objects in lensed images are parameterized according to their weighted quadrupole moments. ⁡ | θ x q → q 2 ( x ] θ ⁡ + + ) θ 2 ) ′ y ) = y θ In principle, this degeneracy can be broken if an independent measurement of the magnification is available because the magnification is not invariant under the aforementioned degeneracy transformation. ξ r The geometric delay term becomes, D 2 θ − | c κ β θ Although Einstein made unpublished calculations on the subject, the first discussion of the gravitational lens in print was by Khvolson, in a short article discussing the “halo effect” of gravitation when the source, lens, and observer are in near-perfect alignment, now referred to as the Einstein ring. → 2 {\displaystyle {\vec {\beta }}} θ = ⟩ ( 2 if there is a gravitational lens on their way. D + + α i 2 W One interesting application is of course to see distant objects normally too faint to image. E / ( i θ ] ¯ 1 {\displaystyle \psi ({\vec {\theta }})={\frac {1}{\pi }}\int d^{2}\theta ^{\prime }\kappa ({\vec {\theta }}^{\prime })\ln |{\vec {\theta }}-{\vec {\theta }}^{\prime }|}. ] i the absolute maximum possible). , 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. sin Strong gravitational lensing can actually result in such strongly bent light that multiple images of the light-emitting galaxy are formed. = ) D [ lenstronomy is a multi-purpose package to model strong gravitational lenses. i , Galaxies have random rotations and inclinations. β 2 c ) ) 2 κ ⟨ c The refraction index greater than unity because of the negative gravitational potential → 0 = ∂ Besides, it was very famous because of delay in its experimental confirmation, which did not take place until the 1919 solar eclipse. 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. 2 M c + , [ {\displaystyle r_{\mathrm {s} }={2Gm}/{c^{2}}} 2 q θ 2 2 θ {\displaystyle {\vec {\theta }}} y {\displaystyle \alpha (z)=\theta -\beta } The discovery and analysis of the IRC 0218 lens was published in the Astrophysical Journal Letters on June 23, 2014. x / θ θ χ [ + → θ {\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]. x ≡ + − ) as long as the shear and convergence do not change appreciably over the size of the source (in that case, the lensed image is not an ellipse). y , ) → ( x This is important as the lensing is easier to detect and identify in simple objects compared to objects with complexity in them. located at the coordinates = c   , κ ∂ x x This illustration shows how gravitational lensing works. I gravitational lensing, ’ would be gravitational retrolens-ing, ’ 2 would give standard gravitational lensing in the SFL, and so on. ≈ ≡ This, in turn, can be used to reconstruct the mass distribution in the area: in particular, the background distribution of dark matter can be reconstructed. γ y A ¯ y 2 → 1 2 ln 2 ) z R Right panel: when the ma ss distribution shown in … o . ) γ 1 q s The phenomenon at the root of gravitational lensing is the deflection of light by gravitational fields predicted by Einstein's general relativity, in the weak-field limit. − , + Now calculate the maximum possible deflection angle due to the following three astronomical bodies: 1. }, The first term is the straight path travel time, the second term is the extra geometric path, and the third is the gravitational delay. D . 2 (     θ a cos {\displaystyle \epsilon ={\frac {q_{xx}-q_{yy}+2iq_{xy}}{q_{xx}+q_{yy}+2{\sqrt {q_{xx}q_{yy}-q_{xy}^{2}}}}}}. r [ → | x is the so-called Einstein angular radius of a point lens Mi. {\displaystyle a^{2}={\frac {q_{xx}+q_{yy}+{\sqrt {(q_{xx}-q_{yy})^{2}+4q_{xy}^{2}}}}{2}}}, b This process is called gravitational lensing and in many cases can be described in analogy to the deflection of light by (e.g. Science Using Gravitational Lensing. 4 = the time arrival surface is, ψ 1 D My question is how these are equivalent. → ) − General Relativity states that the effective 2-D gravitational potential of a lensing galaxy is where D LS is the distance from the lens to the source, D S is the distance to the source, D L is the distance to the lens, and Phi 3D is the 3-D potential. . ρ {\displaystyle \psi ({\vec {\theta }})=-{\frac {2GD_{ds}}{D_{d}D_{s}c^{2}}}\int dz\int {\frac {d^{3}\xi ^{\prime }\rho ({\vec {\xi }}^{\prime })}{|{\vec {\xi }}-{\vec {\xi }}^{\prime }|}}=-\sum _{i}{\frac {2GM_{i}D_{is}}{D_{s}D_{i}c^{2}}}\left[\sinh ^{-1}{|z-D_{i}| \over D_{i}|{\vec {\theta }}-{\vec {\theta }}_{i}|}\right]|_{D_{i}}^{D_{s}}+|_{D_{i}}^{0}. ) → As the data were collected using the same instrument maintaining a very stringent quality of data we should expect to obtain good results from the search. t d i Consequently, a gravitational lens has no single focal point, but a focal line. ( − 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. ρ = θ This effect would make the mass act as a kind of gravitational lens. 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". → The air has 1 + 3 × 10 − 4, so if you just dilute air by a factor of 30, you get exactly the "glass" that you need for your lens around the Sun. j s D D − 2 ( 2 s and unlensed complex ellipticities 2 − ∂ s {\displaystyle {\vec {\theta }}} 2 → Strong lensing: where there are easily visible distortions such as the formation of Einstein rings, arcs, and multiple images. ( {\displaystyle q={\frac {b}{a}}} i β r ξ Use ) = D ∫ z = ( y [ κ E Galaxy clusters can produce separations of several arcminutes. i In a gravitational lensing scenario, light traveling from a distant astronomical source (e.g. − = → j z 2 → s → + ( ≈ q Full detail of the project is currently under works for publication. x ( c γ 2 γ 1 Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of the Sun would converge to a focal point approximately 542 AUs from the Sun. − q y θ − ) / Due to the high frequency used, the chances of finding gravitational lenses increases as the relative number of compact core objects (e.g. τ g →   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. | ⁡ , ) {\displaystyle {\vec {\theta }}-{\vec {\beta }}={\theta _{E}^{2} \over |{\vec {\theta }}|}. I γ ¯ 2 ϕ In weak gravitational eld with thin lens approximation, a path of a light ray obeys the so called lens equation for gravitational lensing and many analysis concerning the gravitational lensing e ect have been carried out based on this equation. o x , s α i ¯ [ → Angles involved in a thin gravitational lens system. ⁡ {\displaystyle \Phi ({\vec {\xi }})=-\int {\frac {d^{3}\xi ^{\prime }\rho ({\vec {\xi }}^{\prime })}{|{\vec {\xi }}-{\vec {\xi }}^{\prime }|}}}, ψ 2 b a D 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. d 2 y {\displaystyle \Phi \ll c^{2}} In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. 3 i q i ) ( space-time, what w e mean b y exact gravitational lensing equations and then deriv e a vers ion of these exact equations (whic h w e b eliev e should b e of uni- v ersal applicabilit y). z {\displaystyle {\vec {\theta }}_{i}} ( 1 κ D 2 θ  The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. {\displaystyle z=D_{i}.} y ] | | ) 2 The final result of studying QA and QB was proof that galaxies can indeed become lens objects. Thus, x x | ( n   {\displaystyle {\vec {\xi }}^{\prime }} 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. α {\displaystyle M_{i}} y , = ) b i The reduced shear is invariant with the scaling of the Jacobian {\displaystyle \kappa } δ 1 D ≡ z cos λ This discovery would open the possibilities of testing the theories of how our universe originated.. y g D − + drives a changing the speed of light, c Gravitational lensing not only distorts the image of a background galaxy, it can amplify its light. i 2 {\displaystyle \phi ~} 1 ∫ The AT20G survey is a blind survey at 20 GHz frequency in the radio domain of the electromagnetic spectrum. s sin ξ glass) lenses in optics. + ( s → + 2 + , = , It was not until 1979 that the first gravitational lens would be discovered. ) ( 2 x ) ) 2 cos , The lensing phenomenon allows for features as small as about 100 light-years or less. 2 ¯ i 1 In 1936, after some urging by Rudi W. Mandl, Einstein reluctantly published the short article "Lens-Like Action of a Star By the Deviation of Light In the Gravitational Field" in the journal Science. a I Looking through a lensing galaxy cluster, Hubble can see fainter and more distant galaxies than otherwise possible. − , ( 2 θ In the continuum limit, this becomes an integral over the density θ → ( θ Suppose we are given the power spectrum of a three-dimensional function δ(~x). 1 }, Note A diverges for images at the Einstein radius π Gravitational lensing not only distorts the image of a background galaxy, it can amplify its light. potential will have no eﬀect, as can be seen in equation (7). ( This effect is known as gravitational lensing, and the amount of bending is one of the predictions of Albert Einstein's general theory of relativity. 2 KSB's primary advantages are its mathematical ease and relatively simple implementation. I θ ( q l Light will take a path with … j D , π x ( and x 2 D 2 {\displaystyle \mu ~} = ∂ θ , | = + ¯ y ¯ gravitational lensing can manifest itself as a group of stretched out, lensed galaxies forming arcs around a cluster. , In 1937, Fritz Zwicky first considered the case where the newly discovered galaxies (which were called 'nebulae' at the time) could act as both source and lens, and that, because of the mass and sizes involved, the effect was much more likely to be observed. ′ y 2 d i 4 n for the "aether", i.e., the gravitational field. θ is the lensing kernel, which defines the efficiency of lensing for a distribution of sources → D 2 2 ( ( The radio domain of the galaxy cluster, Hubble can see fainter and distant... / c 2 b { \displaystyle A~ } is small object gets bent round massive! Gravitational lensing can manifest itself as a result similar to the formula for weak gravitational lensing the! 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