The black disk with radius \(R_s\) represents the event horizon of the black hole.
Photons are emitted from the event horizon (coordinates \(r=R_s\) and \(\varphi=0\)) with an impact parameter value \(b_{crit}\) that moves them to the unstable circular orbit \(r_{crit}={3\over 2}R_s\) (sphere of photons).
After an affine numerical integration, the figure is plotted with Cartesian coordinates \(x=r\cos\varphi\) et \(y=r\sin\varphi\),
considering a hypothetical black hole of the mass of the sun \(M\odot\) that is, \(R_s=2 \ 953\ m\), \(r_{crit}=4\ 430\ m\) and \(b_{crit}=7\ 672.73\ m\).
The value \(b=b_{crit}\) and the emission point \(r=R_s\) et \(\varphi=0\) are input data (Excel spreadsheet or Python script for instance).
This interactive animation is an affine motion which does not represent the time motion of the photon.
For a black hole with a different mass \(M\), the plots are kept by applying the scaling factor
\(\frac{M}{M\odot}\).
Static figure in normalized units (Schwarzschild radius = \(\frac{2GM}{c^2}\)) resulting from a numerical integration in \(\varphi\)