• Home
• Introduction
• Gravitation
• Phenomena
• Lensing
• Hubble views
• Journey

GRAVITATIONAL LENSE

Part III

The first Einstein ring can be seen not only in a high gravity environment, but also in a low gravity environment quite a distance from much larger objects, such as normal stars, galaxies, and clusters of galaxies. In fact, complete first Einstein rings have actually been seen for radio galaxies. A good review of extragalactic measurements of gravitational lens effects is given by Blandford and Narayan.

Another set of Einstein rings is observable when the observer and source are on the same side of the lens. Then, for compact sources such as an ultracompact neutron star, light from behind the observer is able to make a "U-turn" around the neutron star and come back to be visible to the observer. The Einstein ring seen from these light trajectories will be called the second Einstein ring, since it is seen between the first and third Einstein rings, and is brighter than the third but dimmer than the first. The fourth Einstein ring in this set is created when light does a "U-turn" near the photon sphere of the lens, then goes all the way around the lens again near the photon sphere, and finally comes to the observer. Note that there is a critical minimum (or maximum for observers inside the photon sphere) distance for the photon just like in the case of slight deflection, that is given by Eq. (3). There are also an infinite number of higher order Einstein rings of this type. As before, however, these Einstein rings carry relatively little power when compared to the lower order Einstein rings.

It is convenient to also define the zeroth Einstein ring, where light from a source located on the line from the lens through the observer comes directly undeflected to the observer along a radial line (Delta phi = 0). This Einstein "ring" is actually a single point on the observer's sky. It differs from the other Einstein rings in that its angular amplification (of a collinear point source) is not formally divergent.

Note that a single source located precisely on the opposite side of the lens from the observer would create only the first, third, fifth, etc. (i.e. odd numbered) Einstein rings. A single source located on the same side of the lens as the observer would create the zeroth, second, fourth, etc. (i.e. even numbered) Einstein rings.

In general, the position of each set of Einstein rings will be different for each specific source radius from the lens, relative to the observer position. For example, a point source at infinity directly behind the lens from the observer would create a complete set of odd numbered Einstein rings. A point source located a small, finite distance from the lens (but still directly behind the lens) would create a different set of odd numbered Einstein rings. Each set of Einstein rings can thus be labeled by the location of the source sphere. Sources at infinity will be referred to "sky" Einstein rings. For sources on the surface of the lens, the term "surface" Einstein rings will be used. In general, the convention will be taken of labelling each Einstein ring by the name or radius of the source sphere. Mathematically, an Einstein ring will always occur when the total deflection angle due to gravitation Delta phi (Eq. 2) is equal to any integer multiple of pi radians. Note that the Einstein rings are theoretical constructs and would only be visible were a source placed precisely on the observer-lens line, which for any small source is unlikely.

If the angular radius of an opaque lens is larger than the angular radius of the first Einstein ring for the source, then this ring will exist. If the radius of the lens is smaller than the radius of the first Einstein ring but larger than the other Einstein rings, then only the first Einstein ring will exist. If the radius of the lens is small enough so that the lens exhibits a photon sphere, however, an infinite number of Einstein rings exist. This is because a subsequent Einstein ring exists for each revolution of the lens a photon orbit can take, and theoretically, since all of these orbits are contained completely above the photon sphere, it can take an infinite number of them.

It should be noted that the existence of an Einstein ring may depend on the relative positions of the lens, observer, and source, while the existence of the photon sphere or event horizon does not depend on these relative positions. It is possible for the first sky Einstein ring to exist for a given observer looking toward a neutron star lens, but as the observer moves closer to the neutron star the angular size of the surface becomes larger than the angular size of this Einstein ring. For black holes and neutron star's compact enough to have a photon sphere, though, the photon sphere is a real entity - photons do circle there - whether or not an observer is there to see them.

A complete image of the sky is always contained between each two "sky" Einstein rings. Likewise a complete image of the neutron star is always contained between each two "surface" Einstein rings. In general, a single complete image of all the sources on a sphere centered on the lens is visible between each two consecutive Einstein rings of that sphere.