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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.



Zoom on a Schwarzschild Black Hole in front of the Milky Way . The first Einstein ring corresponds to the most distorted region of the picture and is clearly depicted by the galactic disc. The zoom then reveals a series of 4 extra rings, increasingly thinner and closer to the black hole shadow. They are easily seen through the multiple images of the galactic disk. Odd rings correspond to points which are behind the black hole (from the observer point of view) and correspond here to the bright yellow region of the galactic disc (close to the galactic center), whereas even rings correspond to images of regions which are behind the observer, which appear bluer since the corresponding part of the galactic disk is dimmer here.

Only black holes can exhibit such multiple rings. The gravitational distortions caused by a star of a galaxy cluster do not allow enough bending of light to produce the extra rings.





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 d_L
to the lens ''L'', the distance d_S to the source
''S'' and the distance between the source and the lens
d_{LS} as
:
heta_E = \left(
rac{4GM}{c^2}\; rac{d_{LS}}{d_L d_S}
ight)^{1/2}
:
= \left(
rac{M}{10^{11.9} M_{O}}
ight)^{1/2}
\left(
rac{d_L d_S/ d_{LS}}{Gpc}
ight)^{-1/2} arcsec

In the latter form the mass is expressed in solar masses
M_{O} 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
M_c \approx 10^{15} M_{O}
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 \sim 1 M_{O})
search for at galactic distances (say d\sim 3kpc)
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 \alpha the total deflection
by a point mass ''M'' is given (see Schwarzschild Metric )
by
: \alpha = rac{4G}{c^2} rac{M}{b}

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 \alpha in terms of the
angular distance between the lens and the image.
If we see the point of nearest approach ''b'' at an
angle heta for the lens ''L'' on a
distance ''d''''L'', than
(for small angles and the angle expressed in radians)
b = heta d_L and we can express the bending angle
\alpha in terms of the observed angle heta
for a point mass ''M'' as
:
\alpha( heta) = rac{4G}{c^2} rac{M}{b} =
rac{4GM}{d_L c^2} rac{1}{ heta}



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 heta_S is the angle at which one would see
the source without the lens (so not an observable) and
heta_I is the observed angle of the image of
the source with respect to the lens and \alpha is the
bending angle caused by gravity.

One can see in the figure (counting distances in the source plane)
that the vertical distance spanned by the angle
heta at a distance d_S is the same as the sum of the two vertical
distances heta_S \;d_{S} plus
\alpha \;d_{LS}, so
:
heta \; d_S = heta_S\; d_S + \alpha \; d_{LS}

or writing \alpha as
:
\alpha_L( heta_I) = rac{d_S}{d_{LS}} ( heta_I - heta_S)

This is the so-called lens equation. Here
\alpha is the bend angle determined
by the gravitational field, and heta_S is
the angle with respect to the lens position
at which the source would be seen in the absence of the lens and
heta_I is the observed angle of the image.

If we know the mass disstribution (gravitational potential), we
know how the bend angle \alpha behaves and
we can calculated the positions heta_I( heta_S)
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 \alpha for a point mass ''M'' was given above as
:
\alpha( heta) = rac{4G}{c^2} rac{M}{r} =
rac{4GM}{d_L c^2} rac{1}{ heta}


For a point mass the lens equation becomes
:
heta- heta_S = rac{d_{LS}}{d_S d_L}\;
rac{4GM}{c^2} \;
rac{1}{ heta}


For a source right behind the lens, heta_S=0,
the lens equation for a point mass
gives a characteristic value for heta
called the Einstein radius heta_E
Putting heta_S = 0 and solving for
heta gives for this characteristic angle
:
heta_E = \left(
rac{4GM}{c^2}\; rac{d_{LS}}{d_L d_S}
ight)^{1/2}

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
:
heta = heta_S + rac{ heta^2_E}{ heta}



RESEARCH PAPERS


The 1997 review paper lists Chwolson's earlier piece as:
  • O.Chwolson, Astron. Nachr 221, 329 (1924)



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