Example of calculation of the apparent image of the accretion circles of a black hole
of the mass of the sun, with radii \(3R_s\), \(4R_s\), \(5R_s\), \(6R_s\), \(7R_s\) and \(8R_s\) and with axis inclined from \(5^\circ\) to a longitude of \(45^\circ\) with respect to an observer located at a distance \(10R_s\)
from the center of the black hole.
The image is constructed from images of 120 points of each of the 6 accretion circles 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 \(r=r_{ISCO}\).