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| The '''Einstein radius''' is the radius of an [[Einstein ring]], and is a characteristic angle for [[gravitational lens]]ing in general, as typical distances between images in gravitational lensing are of the order of the Einstein radius.
| | Marvella is what you can contact her but it's not the most feminine title out there. Years in the past we moved to North Dakota. To do aerobics is a factor that I'm completely addicted to. Supervising is my occupation.<br><br>my web page: [http://livestudentjobs.co.uk/groups/solid-advice-for-handling-a-candida/ livestudentjobs.co.uk] |
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| == Derivation ==
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| [[Image:Gravitational_lens_geometry.svg|thumb|right|400px|Lens geometry|The geometry of gravitational lenses]]
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| In the following derivation of the Einstein radius, we will assume that all of mass ''M'' of the lensing galaxy ''L'' is concentrated in the center of the galaxy.
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| For a point mass the deflection can be calculated and is one of the classical [[tests of general relativity]]. For small angles ''α''<sub>1</sub> the total deflection by a point mass ''M'' is given (see [[Schwarzschild metric]]) by
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| :<math>\alpha_1 = \frac{4G}{c^2}\frac{M}{b_1}</math>
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| where
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| : ''b''<sub>1</sub> is the [[impact parameter]] (the distance of nearest approach of the lightbeam to the center of mass)
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| : ''G'' is the [[gravitational constant]],
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| : ''c'' is the [[speed of light]].
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| By noting that, for small angles and with the angle expressed in [[radian]]s, the point of nearest approach ''b''<sub>1</sub> at an angle ''θ''<sub>1</sub> for the lens ''L'' on a distance ''d''<sub>L</sub> is given by {{nowrap|''b''<sub>1</sub> {{=}} ''θ''<sub>1</sub> ''d''<sub>L</sub>}}, we can re-express the bending angle ''α''<sub>1</sub> as
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| :<math>\alpha_1(\theta_1) = \frac{4G}{c^2}\frac{M}{\theta_1}\frac{1}{d_{\rm L}}</math> (eq. 1)
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| If we set ''θ''<sub>S</sub> as the angle at which one would see the source without the lens (which is generally not observable), and ''θ''<sub>1</sub> as the observed angle of the image of the source with respect to the lens, then one can see from the geometry of lensing (counting distances in the source plane) that the vertical distance spanned by the angle ''θ''<sub>1</sub> at a distance ''d''<sub>S</sub> is the same as the sum of the two vertical distances {{nowrap|''θ''<sub>S</sub> ''d''<sub>S</sub>}} and {{nowrap|''α''<sub>1</sub> ''d''<sub>LS</sub>}}. This gives the '''[[lens equation]]'''
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| :<math>\theta_1 \; d_{\rm S} = \theta_{\rm S}\; d_{\rm S} + \alpha_1 \; d_{\rm LS}</math>
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| which can be rearranged to give
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| :<math>\alpha_1(\theta_1) = \frac{d_{\rm S}}{d_{\rm LS}} (\theta_1 - \theta_{\rm S})</math> (eq. 2)
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| By setting (eq. 1) equal to (eq. 2), and rearranging, we get
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| :<math>\theta_1-\theta_{\rm S} = \frac{4G}{c^2} \; \frac{M}{\theta_1} \; \frac{d_{\rm LS}}{d_{\rm S} d_{\rm L}}</math>
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| For a source right behind the lens, {{nowrap|''θ''<sub>S</sub> {{=}} 0}}, and the lens equation for a point mass gives a characteristic value for ''θ''<sub>1</sub> that is called the '''Einstein radius''', denoted ''θ''<sub>E</sub>. Putting {{nowrap|''θ''<sub>S</sub> {{=}} 0}} and solving for ''θ''<sub>1</sub> gives
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| :<math>\theta_E = \left(\frac{4GM}{c^2}\;\frac{d_{\rm LS}}{d_{\rm L} d_{\rm S}}\right)^{1/2}</math>
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| The Einstein radius for a point mass provides a convenient linear scale to make dimensionless lensing variables. In terms of the Einstein radius, the lens equation for a point mass becomes
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| :<math>\theta_1 = \theta_{\rm S} + \frac{\theta_E^2}{\theta_1}</math> | |
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| Substituting for the constants gives
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| :<math>\theta_E = \left(\frac{M}{10^{11.09} M_{\bigodot}}\right)^{1/2} \left(\frac{d_{\rm L} d_{\rm S}/ d_{\rm LS}}{\rm{Gpc}}\right)^{-1/2} \rm{arcsec}</math> | |
| In the latter form, the mass is expressed in [[solar mass]]es (M<sub>☉</sub> and the distances in Giga[[parsec]] (Gpc). The Einstein radius most prominent for a lens typically halfway between the source and the observer.
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| For a dense cluster with mass {{nowrap|M<sub>c</sub> ≈ {{val|10|e=15|u=M<sub>☉</sub>}}}} at a distance of 1 Gigaparsec (1 Gpc) this radius could be as large as 100 arcsec (called '''[[macrolensing]]'''). For a [[Gravitational microlensing]] event (with masses of order {{val|1|u=M<sub>☉</sub>}}) search for at galactic distances (say {{nowrap|''d'' ~ {{val|3|u=kpc}}}}), the typical Einstein radius would be of order milli-arcseconds. Consequently separate images in microlensing events are impossible to observe with current techniques.
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| Likewise, for the ''lower'' ray of light reaching the observer from below the lens, we have
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| :<math>\theta_2 \; d_{\rm S} = - \; \theta_{\rm S}\; d_{\rm S} + \alpha_2 \; d_{\rm LS}</math>
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| and
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| :<math>\theta_2 + \theta_{\rm S} = \frac{4G}{c^2} \; \frac{M}{\theta_2} \; \frac{d_{\rm LS}}{d_{\rm S} d_{\rm L}}</math>
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| and thus
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| :<math>\theta_2 = - \; \theta_{\rm S} + \frac{\theta_E^2}{\theta_2}</math>
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| The argument above can be extended for lenses which have a distributed mass, rather than a point mass, by using a different expression for the bend angle α. The positions ''θ''<sub>I</sub>(''θ''<sub>S</sub>) of the images can then be calculated. For small deflections this mapping is one-to-one and consists of distortions of the observed positions which are invertible. This is called '''[[weak lensing]]'''. For large deflections one can have multiple images and a non-invertible mapping: this is called '''[[strong lensing]]'''. Note that in order for a distributed mass to result in an Einstein ring, it must be axially symmetric.
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| == References ==
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| * {{cite journal | first = O | last = Chwolson | title = Über eine mögliche Form fiktiver Doppelsterne | journal = Astronomische Nachrichten | volume = 221 | issue = 20 | pages = 329–330 | year = 1924 | bibcode = 1924AN....221..329C | doi = 10.1002/asna.19242212003}} (The first paper to propose rings)
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| * {{cite journal | first = Albert | last = Einstein | authorlink = Albert Einstein | title = Lens-like Action of a Star by the Deviation of Light in the Gravitational Field | journal = Science | volume = 84 | pages = 506–507 | year = 1936 | doi = 10.1126/science.84.2188.506 | pmid = 17769014 | issue = 2188 | jstor=1663250|bibcode = 1936Sci....84..506E }} (The famous Einstein Ring paper)
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| * {{cite journal | first = Jurgen | last = Renn | coauthors = Tilman Sauer and John Stachel | title =The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science paper | journal = Science | volume = 275 | pages = 184–186 | year = 1997 | doi = 10.1126/science.275.5297.184 | pmid =8985006 | issue = 5297|bibcode = 1997Sci...275..184R }}
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| == See also ==
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| * [[Gravitational lens]]
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| * [[Einstein ring]]
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| {{Einstein}}
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| [[Category:Albert Einstein|Radius]]
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| [[Category:Gravitational lensing]]
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Marvella is what you can contact her but it's not the most feminine title out there. Years in the past we moved to North Dakota. To do aerobics is a factor that I'm completely addicted to. Supervising is my occupation.
my web page: livestudentjobs.co.uk