The black circle with radius \(R_s\) represents the event horizon of the black hole.

Photons are emitted from the event horizon (coordinates \(r=R_s\) et \(\varphi=\pi\)) 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).

The figure is plotted with Cartesian coordinates \(x=r\cos\varphi\) et \(y=r\sin\varphi\),

considering a 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 a black hole with a different mass \(M\), the plots are kept by applying the scaling factor

\(\frac{M}{M\odot}\).