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GRAVITATIONAL LENSE

Part II

Time induced amplifications result when the observer is at a different r from the lens center than the source. When considering only time induced amplifications, the total bolometric (incorporating all wavelengths) power received will then be changed by an amount A_time = (1-R_S/r_emitted)/(1-R_S/r_observed). For the sources near perfect lens - observer alignment, the angular amplification effects typically dominate over time induced amplification effects. Angular amplifications can be computed from the deflection angle Eq. (2). If a large change in angular position on the observer's sky corresponds with a small change in the angular position at the (unlensed) source location, then the source will appear to be angularly elongated and hence amplified. Similarly a source can be angularly deamplified, but this will be referred to as an angular amplification of less than unity. Angular amplification effects should be computed on the spherical sky of the observer, and so would be given by

Here beta represents the angular distance between the lens and the source on the observer's sky in the absence of the gravitational field of the lens, while alpha represents this distance in the presence of the gravitational field and light deflection. The change in this angular distance, d alpha/d beta can be found by application of Eq. (2). When bending angles due to gravitational effects are small, this reduces to the amplification formulae given by Refsdal and Liebes.

The net angular amplification of all the images of a single source can also be either more or less than the original unlensed flux of the source. This is because the gravitational field does not change the fact that the observer still observes the same total angular area as before: 4 pi steradians. Therefore if the apparent angular area from some sources is greater than without the gravitational field, then there must be other sources with apparent angular area which is lower, to compensate. In practice, only relatively few images from sources that are near the observer - lens line will have angular amplifications very large (A_angular >> 1), while the rest of the sources in the sky will be slightly deamplified (A_angular < ~ 1).

The total flux received by an observer from all the sky can again be either more or less than the original unlensed flux of all the sky. A gravitational field does not create photons - it just redistributes and (red- or) blueshifts them. The observed angular redistribution and the relative time distortions, however, now act in opposite directions. For the background sky, A_angular < 1, because now its angular area, which used to occupy the observer's entire field of view (4 pi steradians) in the absence of gravity, is now less by the amount of the angular size of the photon sphere of the lens. However, the photons from the sky, because of the blueshifting, are relatively more energetic and arriving relatively more often, so that A_time > 1.

In other words, the background sky takes up less of the observer's sky, but the observer receives more photons per unit area, and each photon is of higher energy. Do these effects exactly cancel? No. It turns out that all observers will measure A_total > 1, with the closer the oberver the greater is A_total.

An important observational aspect of visual distortions in a high gravity environment that is discussed more usually in the gravitational lensing literature than in the introductory gravitation texts is called an Einstein ring. Before it was shown that all images must occur in the plane defined by the observer's position, the center of the lens, and the point source. But what if these are all collinear? No plane is then defined. In this case the image of the point source would appear to the observer as an infinitesimally thin ring. This is an Einstein ring. As will be explained, numerous Einstein rings may appear simultaneously, however, and they are also important as invisible dividing lines between sets of images, even when no source is distorted into a ring.

It is not generally appreciated that there can be an infinite number of Einstein rings. In fact, there can be an infinite number of Einstein rings for each set of collinear observer, lens, and source points. The only Einstein ring currently discussed in the literature is the most prominent one that occurs at precise observer - lens - source alignment, where Delta phi = pi. Here light emitted at a specific angle from the source would be slightly deflected by the gravitational field of the lens to reach the observer. Were the source light emitted at a different angle the lens would either not be able to bend the light enough to reach the observer or too much. Since the exact observer, lens, source alignment is symmetric about the line connecting them, this source would be seen as an annular ring. This ring will be referred to as the first Einstein ring. (Later the term Einstein ring will be even additionally labelled by the relative radius of the source.)

Other Einstein rings can be seen angularly closer to the center of the lens. Photons from the third Einstein ring (the second Einstein ring will be defined two paragraphs below) have fully circled the lens once near the photon sphere before coming to the observer. In fact, the path of these photons crosses itself. It is possible for photons to orbit the lens an arbitrarily large number of times before coming to the observer, and each of these orbits corresponds to an Einstein ring. Therefore there are innumerable Einstein rings for this specific observer - lens - source configuration. Each Einstein ring is seen successively closer to the apparent photon sphere position. The more times the photon must circle the neutron star or black hole before reaching the observer, the more precise the direction of its emission must have been emitted to attain this trajectory, the less likely any photon will take this trajectory, the "dimmer" the Einstein ring. For this reason the higher order Einstein rings will usually carry little light when compared to the lower order Einstein rings. In fact, the relative brightness of each Einstein ring decreases exponentially.