Figure 17

Figure 17: Luminosity distance $D (Θ ) lum$ and ellipticity $𝜀(Θ )$ (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 $2.5log10(Dlum (Θ)2)$ 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 (41 ) and (48 ). This procedure follows [84] (cf. [85, 118]). For source and observer at large radius, related calculations can also be found in [212, 246, 200, 336]. Einstein rings have magnitude $− ∞$ 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 $2.5 log10(e2π) ≈ 6.8$ magnitudes, see [212, 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 (54 ), the latter would show a constant intensity. The lower part of the diagram illustrates image distortion in terms of $𝜀 = D−-− D+- D+ D−$ . Clearly, $|𝜀|$ 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, $|𝜀| > 100$ . Image distortion in the Schwarzschild spacetime is also treated in [85, 120, 119], an approximation formula is derived in [241].