About Black Holes

GRAVITATIONAL LENSE

Part I

Part II

Part III

Part IV

Part V

Part IV

If the radius of the lens is small enough so that the lens exhibits a photon sphere, an infinite number of images can be seen of the source, no matter its location. One image of the source comes to the observer relatively undeflected. This image is between the zeroth and first Einstein rings and will be referred to as the primary image. A second image comes around the opposite limb of the lens from the first image, and therefore will appear to the observer 180 degrees around the face of the lens from the first image. This secondary image will always be located between the first and second Einstein rings. A third image comes around the same limb as the first image and is seen even closer to the apparent position of the photon sphere. This image has circled the neutron star or black hole fully once before reaching the observer, and its location is always between the second and third Einstein rings. The photon path for this image (and all higher order images) crosses itself. A fourth image occurs closer to but outside of the same limb as the second image, but has fully circled the lens once in the opposite direction. There is a subsequent image for each revolution of the lens a photon orbits takes, and theoretically it can take an infinite number of them. In practice, these multi-revolution images have little power and would be vanishingly hard to see.

Each set of images contained between successive Einstein rings is converted into "mirror writing" with respect to the images between the previous two Einstein rings. For example, if the source was a book, then the book would be visible with relatively minor distortions in its primary image - between the zeroth and first book Einstein rings. For the secondary image, between the first and second book Einstein rings, the book would appear in mirror writing, but right side up. The mapping of the entire sphere onto the annular ring between the two book Einstein rings would also cause prominent distortions. The third image of the book, between the second and third book Einstein rings, would appear in normal writing again (neither in mirror writing nor inverted), but even more distorted because of the decreased relative angular area between these two book Einstein rings. A discussion of the parity of lensed images for the brightest two images of the point lens (considered here) as well as other gravitational lens types, can be found in Blandford and Kochanek.

Therefore, for a compact enough neutron star, one can see the whole neutron star surface. An observer can see the complete surface of a lens (exactly once) when the first surface Einstein ring is the same angular size as the surface of the lens. (A derivation of the angular size of a sphere of mass M, radius R_* at distance D is given in the Appendix.) When the second surface Einstein ring has equal angular size to the apparent angular size of the lens surface, two complete images of the lens surface are visible.

Any lens which has a first surface Einstein ring is completely incapable of blocking light from any source. These objects cannot "eclipse" anything. This is why a neutron star in a well separated binary system can never block the light of its binary companion.

Less stringently, any lens with a first sky Einstein ring is incapable of blocking light from the background sky. Almost all stars in our galaxy are thus incapable of blocking light from random superpositions of background objects. For example, no supernovae in other galaxies are missed because they are "eclipsed" by a random superposition of a foreground star in the Milky Way Galaxy. Were such a chance superposition to occur (it is very unlikely), the supernova would be greatly amplified by the gravitational field of the intervening star rather than diminished by an "eclipse" effect. With respect to distant sources, these stars are easily compact enough to show a first Einstein ring to a distant observer, and are therefore incapable of blocking the source's light.

Every star in existence, besides the Sun but including even the nearest stars, has a first sky Einstein ring with respect to an earth bound observer. The small angular size of this Einstein ring is below optical resolution, but not below the angular resolution of many radio observations. The gravity of these normal stars is strong enough to bend the background light around them and cause distant sources to be visible to the observer. Almost none of the nearby stars, however, would show a second sky Einstein ring, unless they were a neutron star or black hole. Were the star compact enough to have a photon sphere surrounding it, then, theoretically, an infinite number of sky Einstein rings (and hence sky images) would be visible.