Einstein ring
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In observational astronomy a Chwolson ring or Einstein ring is a ring-shaped image on the sky which is caused by gravitational deflection of an intervening object. A distant point source situated exactly behind a galaxy would normally be hidden, but is nevertheless visible because its light bends around the galaxy due to gravitational lensing. An Einstein ring is a special form of a gravitational lens in which source (such as a galaxy) and lens (such as a schwarzschild black hole) are exactly lined up.
The black hole is bending the light of the point source through its gravitational effect. The bending occurs in all directions relative to the lens at a fixed angle, and the source is seen in all directions as a ring. A black hole as a gravitational lens is transparent, because the gravitational pull of a black hole pulls in all other light and it cannot be seen past the event horizon. It is the gravitational field of a black hole, treated as a continuum, in which the lightbending takes place.
Einstein remarked upon this effect in 1936, but thought the chances of such a coalignment were small. The chance observing Einstein rings produced by stars may be low, but the chance of observing those produced by black holes is higher since the angular size of an Einstein ring is proportional to the mass of the lens. Though it is also inversely proportional to the distance at which the ring is observed, it is frequently the case that the greater mass of black hole more than compensates for their being further away than stars. Gravitational lensing is therefore an important tool in cosmology.
Hundreds of gravitational lenses are known nowadays. About half a dozen of them are Einstein rings with diameters up to an arcsec. Most rings have been discovered in the radio range.
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Radius of the Einstein ring
The radius of the Einstein ring is a characteristic angle for gravitational lensing in general. Typical distances between images in gravitational lensing are of the order of the Einstein radius. Assuming all of mass M of the lensing galaxy is concentrated in the center it can be expressed (shown below) in terms of the distance <math>d_L</math> to the lens L, the distance <math>d_S</math> to the source S and the distance between the source and the lens <math>d_{LS}</math> as
- <math>
\theta_E = \left(
\frac{4GM}{c^2}\;\frac{d_{LS}}{d_L d_S}
\right)^{1/2} </math>
- <math>
= \left(
\frac{M}{10^{11.9} M_{O}}
\right)^{1/2}
\left(
\frac{d_L d_S/ d_{LS}}{Gpc}
\right)^{-1/2} arcsec
</math> In the latter form the mass is expressed in solar masses <math>M_{O}</math> and the distances in Gigaparsec (Gpc). The Einstein radius most prominent for a lens typically halfway between the source and the observer.
For a dense cluster with mass <math>M_c \approx 10^{15} M_{O}</math> at a distance of 1 Gigaparsec (1 Gpc) this radius could be as large as 100 arcsec (called macrolensing). For a microlensing event (with masses of order <math>\sim 1 M_{O}</math>) search for at galactic distances (say <math>d\sim 3</math>kpc) the typical Einstein radius would be of order milli-arcseconds. Consequently separate images in microlensing events are difficult to observe.
Deflecting of light by a gravitational field
The bending of light by a gravitational body was predicted by Einstein (1912) a few years before the publication of General Relativity in 1916. For a point mass the deflection can be calculated and is one of the classical tests of general relativity. For small angles <math>\alpha</math> the total deflection by a point mass M is given (see Schwarzschild metric) by
- <math> \alpha = \frac{4G}{c^2}\frac{M}{b}
</math> where b is the distance of nearest approach of the lightbeam to the center of mass and G is the gravitational constant and c is velocity of light. For 1 solar mass and the distance of nearest approach equal to the solar radius, the gravitational bending amounts to 1.75 arcsec.
We can rewrite the bending angle <math>\alpha</math> in terms of the angular distance between the lens and the image. If we see the point of nearest approach b at an angle <math>\theta</math> for the lens L on a distance dL, than (for small angles and the angle expressed in radians) <math>b =\theta d_L</math> and we can express the bending angle <math>\alpha</math> in terms of the observed angle <math>\theta</math> for a point mass M as
- <math>
\alpha(\theta) = \frac{4G}{c^2} \frac{M}{b} =
\frac{4GM}{d_L c^2}\frac{1}{\theta}
</math>
The lens equation
With the geometry given in the figure, one can easily find the expression for the Einstein ring under some simplifying assumpions. Here <math>\theta_S</math> is the angle at which one would see the source without the lens (so not an observable) and <math>\theta_I</math> is the observed angle of the image of the source with respect to the lens and <math>\alpha</math> is the bending angle caused by gravity.
Image:Gravity lens geometry.png One can see in the figure (counting distances in the source plane) that the vertical distance spanned by the angle <math>\theta</math> at a distance <math>d_S</math> is the same as the sum of the two vertical distances <math>\theta_S \;d_{S}</math> plus <math>\alpha \;d_{LS}</math>, so
- <math>
\theta \; d_S = \theta_S\; d_S + \alpha \; d_{LS}
</math> or writing <math>\alpha</math> as
- <math>
\alpha_L(\theta_I) = \frac{d_S}{d_{LS}} (\theta_I - \theta_S)
</math> This is the so-called lens equation. Here <math>\alpha</math> is the bend angle determined by the gravitational field, and <math>\theta_S</math> is the angle with respect to the lens position at which the source would be seen in the absence of the lens and <math>\theta_I</math> is the observed angle of the image.
If we know the mass disstribution (gravitational potential), we know how the bend angle <math>\alpha</math> behaves and we can calculated the positions <math>\theta_I(\theta_S)</math> of the images. 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.
Point masses and the Einstein radius
The light deflections for mass distributions that appear circularly symmetric on the sky can be readily calculated. The formula for <math>\alpha</math> for a point mass M was given above as
- <math>
\alpha(\theta) = \frac{4G}{c^2} \frac{M}{r} =
\frac{4GM}{d_L c^2}\frac{1}{\theta}
</math>
For a point mass the lens equation becomes
- <math>
\theta-\theta_S = \frac{d_{LS}}{d_S d_L}\;
\frac{4GM}{c^2} \;
\frac{1}{\theta}
</math>
For a source right behind the lens, <math>\theta_S=0</math>, the lens equation for a point mass gives a characteristic value for <math>\theta</math> called the Einstein radius <math>\theta_E</math> Putting <math>\theta_S = 0</math> and solving for <math>\theta</math> gives for this characteristic angle
- <math>
\theta_E = \left(
\frac{4GM}{c^2}\;\frac{d_{LS}}{d_L d_S}
\right)^{1/2} </math>
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
- <math>
\theta = \theta_S + \frac{\theta^2_E}{\theta}
</math>
Research papers
- Albert Einstein, "Lens-like Action of a Star by the Deviation of Light in the Gravitational Field", Science 84 506 (1936))
- Jurgen Renn, Tilman Sauer and John Stachel, "The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science paper", Science 275: 184-186 (1997)
The 1997 review paper lists Chwolson's earlier piece as:
- O.Chwolson, Astron. Nachr 221, 329 (1924)
External links
- First detection of an Extrasolar planet with microlensing
- Report by OGLE (the Optical Gravitational Lensing Experiment)
- Press release by JPL/NASA
- Nearly perfect Einstein ring discoveredde:Einsteinring