Einstein ring

In observational astronomy an 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 bending. An Einstein ring is a special form of a gravitational lens in which source (such as a quasar) and lens (such as a galaxy) are exactly lined up.

The galaxy is deflecting the light of the point source through its gravitational effect. The deflection occurs in all directions relative to the lens at a fixed angle, and the source is seen in all directions as a ring. A galaxy as a gravitational lens is transparent, because of the negligible size of a star on a galactic scale. It is the gravitational field of the ensemble of stars, 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 alignment of sources may be small, but increases if we observe the universe at larger distances, where the average density is higher. Gravitational lensing therefore is 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.

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 defection 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 d_L, 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.

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 socalled 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>

External links

• First detection of an Extrasolar planet with microlensing
  • Report (http://bulge.princeton.edu/~ogle/ogle3/blg235-53.html) by OGLE (the Optical Gravitational Lensing Experiment)
  • Press release (http://www.jpl.nasa.gov/releases/2004/103.cfm) by JPL/NASA

• Nearly perfect Einstein ring discovered (http://www.universetoday.com/am/publish/perfect_einstein_ring.html)