Distortions of images of Schwarzschild lensing

K. S. Virbhadra

Phys. Rev. D 106, 064038 – Published 21 September 2022

Abstract

We model the supermassive dark object $M 8 7^{*}$ as a Schwarzschild lens and study the variations in tangential, radial, and total (the product of tangential and radial) magnifications of images (primary, secondary, and relativistic) against the changes in angular source position and the ratio of lens-source to the observer-source distance. Further, we study the behavior of partial derivatives (with respect to the angular source position) of total magnifications of images against the angular source position. Finally, we model supermassive dark objects at centers of 40 galaxies as Schwarzschild lenses and study the variations in tangential, radial, and total magnifications of images against the change in the ratio of mass of the lens to its distance. These studies yield many nonintuitive results which are likely to be significant for next generation Event Horizon Telescope observations. We hypothesize that there exists a distortion parameter such that its signed sum for all images of singular gravitational lensing of a source identically vanishes. We test this with images of Schwarzschild lensing in weak and strong gravitational fields and find that this esthetically appealing hypothesis succeeds with flying colors.

Received 3 April 2022
Accepted 30 August 2022

DOI:https://doi.org/10.1103/PhysRevD.106.064038

Physics Subject Headings (PhySH)

Dark matter Gravitational lenses Quantum aspects of black holes

Astronomical black holes Classical black holes

Numerical relativity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

K. S. Virbhadra ^*

Mathematics Department, Drexel University, 33rd and Market Streets, Philadelphia, Pennsylvania 19104, USA

^*shwetket@yahoo.com

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Issue

Vol. 106, Iss. 6 — 15 September 2022

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Images

Figure 1
First row: the tangential magnification of the primary image $μ_{p t}$ (red dashed), the absolute tangential magnification of the secondary image $| μ_{s t} |$ (green dotted), the radial magnification of the primary image $μ_{p r}$ (red dashed), the radial magnification of the secondary image $μ_{s r}$ (green dotted), the total magnification of the primary image $μ_{p}$ (red dashed), and the absolute total magnification of the secondary image $| μ_{s} |$ (green dotted) are plotted against the angular source position $β$ . Second and third rows: the same three magnifications are plotted against $β$ for the first (see the second row) and the second (see the third row) order relativistic images. The subscripts $1 p$ and $2 p$ stand for the first- and second-order relativistic images on the side of the primary image whereas subscripts $1 s$ and $2 s$ for the first- and second-order relativistic images on the secondary image side. The colors of symbols and corresponding graphs are kept the same for graphs to be identified. The supermassive dark object (SMDO) at the galactic center of M87 is modeled as the Schwarzschild lens, which has mass $M = 6.5 \times 10^{9} M_{⊙}$ and is situated at the distance $D_{d} = 16.8 Mpc$ so that $M / D_{d} \approx 1.84951 \times 10^{- 11}$ . The dimensionless parameter $D = 0.005$ .
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Figure 2
Left: the partial derivatives of total magnifications of primary and secondary images, denoted, respectively, by $\partial_{β} μ_{p}$ (red dashed) and $\partial_{β} | μ_{s} |$ (green dotted) are plotted against the angular source position $β$ . Middle: the partial derivatives of total magnifications of the first order relativistic images on primary and secondary sides, denoted, respectively, by $\partial_{β} μ_{1 p}$ (magenta dashed) and $\partial_{β} | μ_{1 s} |$ (blue dotted) are plotted against $β$ . Right: the partial derivatives of total magnifications of the second-order relativistic images on primary and secondary sides, denoted, respectively, by $\partial_{β} μ_{2 p}$ (brown dashed) and $\partial_{β} | μ_{2 s} |$ (black dotted) are plotted against the angular source position $β$ . The gravitational lens, as well as the parameter $D$ , are the same as for Fig. 1. Angular source positions are expressed in arcsec.
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Figure 3
First row: the tangential magnification of the primary image $μ_{p t}$ (red dashed), the absolute tangential magnification of the secondary image $| μ_{s t} |$ (green dotted), the radial magnification of the primary image $μ_{p r}$ (red dashed), the radial magnification of the secondary image $μ_{s r}$ (green dotted), the total magnification of the primary image $μ_{p}$ (red dashed), and the absolute total magnification of the secondary image $| μ_{s} |$ (green dotted) are plotted against the parameter $D$ . Second and third rows: the same three magnifications are plotted against the parameter $D$ for the first (see the second row) and the second (see the third row) order relativistic images. The subscripts $1 p$ and $2 p$ stand for the first- and second-order relativistic images on the side of the primary image whereas subscripts $1 s$ and $2 s$ for the first- and second-order relativistic images on the secondary image side. The colors of symbols and graphs are kept the same for graphs to be identified. The gravitational lens is the same as for Figs. 1 and 2. The angular source position $β = 1 mas$ .
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Figure 4
First row: the tangential magnification of the primary image $μ_{p t}$ (red dashed), the absolute tangential magnification of the secondary image $| μ_{s t} |$ (green dotted), the radial magnification of the primary image $μ_{p r}$ (red dashed), the radial magnification of the secondary image $μ_{s r}$ (green dotted), the total magnification of the primary image $μ_{p}$ (red dashed), and the absolute total magnification of the secondary image $| μ_{s} |$ (green dotted) are plotted against $M / D_{d}$ (the ratio of the mass of the lens to the lens-observer distance). Second and third rows: the same three magnifications are plotted against $M / D_{d}$ for the first (see the second row) and the second (see the third row) order relativistic images. The subscripts $1 p$ and $2 p$ stand for the first- and second-order relativistic images on the side of the primary image whereas subscripts $1 s$ and $2 s$ for the first- and second-order relativistic images on the secondary image side. The colors of symbols and corresponding graphs are kept the same for graphs to be identified. The angular source position $β = 1 mas$ and $D = 0.005$ .
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Figure 5
Left: the logarithmic distortions of the primary image $δ_{p}$ (red dashed), secondary image $δ_{s}$ (green dotted), the first order relativistic image on primary side $δ_{1 p}$ (magenta dashed), the first order relativistic image on secondary side $δ_{1 s}$ (blue dotted), the second order relativistic image on primary side $δ_{2 p}$ (brown dashed), and the second order relativistic image on secondary side $δ_{2 s}$ (black dotted) are plotted against the angular source position $β$ . The SMDO at the galactic center of M87 is modeled as the Schwarzschild lens, which has mass $M = 6.5 \times 10^{9} M_{⊙}$ and is situated at the distance $D_{d} = 16.8 Mpc$ so that $M / D_{d} \approx 1.84951 \times 10^{- 11}$ . The dimensionless parameter $D = 0.005$ . Middle: the same six quantities (as for the figure on the left) are plotted against the parameter $D$ . The lens is also the same and the angular source position $β = 1 mas$ . Right: the same six quantities (as for the figure on the left) are plotted against $M / D_{d}$ (the ratio of the mass of lens to the lens-observer distance). SMDOs at the centers of 40 galaxies are modeled as Schwarzschild lenses. The angular source position $β = 1 mas$ and $D = 0.005$ . The colors of symbols and graphs are kept the same for graphs to be identified.
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Figure 6
Left: the absolute percentage difference in distortions for the primary-secondary images pair $| P_{p s} |$ , the first order relativistic images pair $| P_{1 p 1 s} |$ , and the second order relativistic images pair $| P_{2 p 2 s} |$ are plotted against the angular source position $β$ . The MDO at the galactic center of M87 is modeled as the Schwarzschild lens, which has mass $M = 6.5 \times 10^{9} M_{⊙}$ and is situated at the distance $D_{d} = 16.8 Mpc$ so that $M / D_{d} \approx 1.84951 \times 10^{- 11}$ . The dimensionless parameter $D = 0.005$ . Middle: the same three quantities (as for the figure on the left) are plotted against the parameter $D$ . The lens is also the same and the angular source position $β = 1 mas$ . Right: the same three quantities (as for the figure on the left) are plotted against $M / D_{d}$ . SMDOs at centers of 40 galaxies are modeled as Schwarzschild lenses. The angular source position $β = 1 mas$ and $D = 0.005$ .
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