Example of the calculated apparent image of 6 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 after a numerical integration in \(\varphi\) 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”, with the apparent radius of the event horizon \(b_{crit}=\frac{3\sqrt{3}}{2}R_s\).
The image of the photons escaping from the unstable photon sphere is the circle bordering the apparent event horizon, circle built with \(b=b_{crit}\) by rotating the plane of each trajectory around the observation axis.
The image of the accretion circles 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 “beard” 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 speed is written as \(\Omega=\sqrt{\frac{GM}{r^3}}\) and linear speed is written as \(\Omega\ r\) that is \(\sqrt{\frac{GM}{r}}\) or \(\frac{c}{\sqrt{2\frac{r}{R_s}}}\).
The linear speed of physical bodies in close orbits, measured by a static observer located in the asymptotic region, is relativistic: it goes from \(0.25\ c\)
for \(r=8R_s\) to \({c\over \sqrt{6}}\) that is \(\simeq 0.408\ c\) for \(r=r_{ISCO}\).
The actual physical speed is higher: \(\simeq 0.267\ c\) for \(r=8R_s\) to \(0.5\ c\) for \(r=r_{ISCO}\).
Refer to the appendix Null, time-like and space-like geodesics for the detailed calculation of \(r_{ISCO}\) and the actual physical speed.