GRAVITATION

Details of figure A©

Four trajectories of photons coming from ∞ around a black hole of the mass of the sun (deflections π/2, π, 3π/2 and 2π)

The black circle with radius $R_{s}$ represents the event horizon of the black hole.
Photons arrive from the right ( $x = + \infty$
and $y = b$ ) in the same plane.
Each value of the impact parameter $b$ gives
a different deviation. In dark blue deviation
of $\frac{π}{2}$ , in light blue deviation of $π$ , in green deviation of $\frac{3 π}{2}$ and in yellow deviation of $2 π$ (full revolution).
The figure is plotted with Cartesian coordinates $x = r \cos φ$ and $y = r \sin φ$ , considering a hypothetical black hole of the mass of the sun $M ⊙$

that is, $R_{s} = 2 953 m$ , $b_{c r i t} = 7 672.73 m$ and with $b_{\frac{π}{2}} = 9 107 m$ , $b_{π} = 7 910 m$ , $b_{\frac{3 π}{2}} = 7 720 m$ and $b_{2 π} = 7 682 m$ .
Each value of $b$ can be obtained by choosing the value of the final variation $Δ φ$ = deviation + $π$ (target value in an Excel spreadsheet or input data in a Python script).
For a black hole with a different mass $M$ , the plots are kept by applying the scaling factor
$\frac{M}{M ⊙}$ .

Four trajectories of photons coming from ∞ around a black hole of the mass of the sun (deflections π/2, π, 3π/2 and 2π) on three planes

3D illustration
The black sphere is the event horizon
of the black hole, with radius $R_{s}$
and the light yellow sphere is the sphere
of photons, with radius $\frac{3}{2} R_{s}$ .
The figure is plotted in three dimensions
(90° and 360° deviations in an xy plane,
180° deviation in an xz plane
and 270° deviation in a yz plane).

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