Figure 17

Figure 17: Luminosity distance D (Q) lum and ellipticity e(Q) (image distortion) in the Schwarzschild spacetime. The data are the same as in Figures 15 and 16. If point sources of equal bolometric luminosity are distributed at r = rS, the plotted function 2.5 log10(Dlum(Q)2) gives their magnitude on the observer’s sky, modulo an additive constant m0. For the calculation of Dlum one needs D+ and D- (see Figure 16), and the general relations (41View Equation) and (48View Equation). This procedure follows [84] (cf. [85, 119]). For source and observer at large radius, related calculations can also be found in [211, 246, 201, 337]. Einstein rings have magnitude - oo in the ray-optical treatment. For a light source not on the axis, the image of order i + 2 is fainter than the image of order i by 2.5 log10(e2p) ~~ 6.8 magnitudes, see [211, 246]. (This is strictly true in the “strong-field limit”, or “strong-bending limit”, which is explained in the caption of Figure 15.) The above picture is similar to Figure 6 in [246]. Note that it refers to point sources and not to a radiating spherical surface r = rS of constant surface brightness; by Equation (54View Equation), the latter would show a constant intensity. The lower part of the diagram illustrates image distortion in terms of e = D--- D+- D+ D-. Clearly, |e| is infinite at each Einstein ring. The double-logarithmic representation shows that beyond the second Einstein ring all images are extremely elongated in the tangential direction, |e|> 100. Image distortion in the Schwarzschild spacetime is also treated in [85, 121, 120], an approximation formula is derived in [240].