Example of the calculated apparent image of the accretion circles of a black hole with radii \(3R_s\), \(4R_s\), \(5R_s\), \(6R_s\), \(7R_s\) and \(8R_s\) for an observer located at distance \(10R_s\) from the center of the black hole, with an elevation of \(5^\circ\) and the azimuth \(135^\circ\).
Photon shootings from each of 120 points of each accretion circle are performed in the plane defined by the point, the center of the black hole and the location of the observer, by varying the impact parameter \(b\), and the trajectories selected are those passing through the location of the observer.
The image is then built from the images corresponding to the trajectories for each point and images of order \(\geq\) 1 (one revolution or more around the black hole) are not shown.
Unlike previous figures A to F, the black hole is represented here by its « shadow »,
and the apparent radius of the event horizon is \(b_{crit}=\frac{3\sqrt{3}}{2}R_s\).
The image can be split into two superimposed images:
the « hat » for photons emitted by accretion circles and passing « above » the black hole
(\(\frac{d\varphi}{dt}>0)\),
the « hair » and « necklace » for photons emitted by accretion circles and passing « below » the black hole (\(\frac{d\varphi}{dt}<0)\).
As accretion circles are composed of materials, the orbit \(r=3R_s\) is the last stable circular orbit or « Innermost Stable Circular Orbit – ISCO » below which matter will be absorbed
into the event horizon1.
For a static observer located in the asymptotic region, angular velocity is written as \(\Omega=\sqrt{\frac{GM}{r^3}}\) and linear velocity is written as \(\Omega\ r\) that is \(\sqrt{\frac{GM}{r}}\) or \(\frac{c}{\sqrt{2\frac{r}{R_s}}}\).
The linear velocity of material bodies in close orbits is relativistic: it goes from \(0.25\ c\)
for \(r=8R_s\) to \({c\over \sqrt{6}}\) that is \(\simeq 0.408\ c\) for \(r=r_{ISCO}\).