DescriptionPrograde and retrograde circular orbital velocity.gif	Deutsch: Prograde und retrograde äquatoriale Kreisbahngeschwindigkeit um ein rotierendes schwarzes Loch. a ist der Spinparameter in Einheiten von J·c/G/M² mit J als dem Drehimpuls des schwarzen Lochs. Die x-Achse ist der Boyer-Lindquist-Radius r, und die y-Achse die lokale 3er-Geschwindigkeit v. Die linke vertikale Hilfslinie bezeichnet den Ereignishorizont (rH) bei r=(1+√(1-a²))GM/c², und die beiden an die Funktionskurven anschließenden vertikalen Hilfslinien die Radien für den pro- (r+) und retrograden (r-) Photonenkreis auf r±=2(1+cos(2arccos(±a))/3)GM/c². Weiterführende Erklärung: Herleitung / Details English: Prograde and retrograde equatorial circular orbit velocity around a rotating black hole. a is the spin parameter in units of J·c/G/M² where J ist the angular momentum of the black hole. The x-axis is the Boyer-Lindquist-Radius r, and the y-axis the local 3-velocity v. The left vertical gridline depicts the event horizon (rH) at r=(1+√(1-a²))GM/c², and the two vertical gridlines connecting to the function curves are the pro- (r+) and retrograde (r-) photon-orbits at r±=2(1+cos(2arccos(±a))/3)GM/c². Further reading: derivation / details
Date	19 July 2017
Source	Own work
Author	Yukterez (Simon Tyran, Vienna)
Other versions

The pro- and retrograde circular orbital velocity is derived by setting

${\displaystyle {\rm {{\dot {p}}_{r}=v_{r}=v_{\theta }=0\ ,\ \ \theta =\pi /2\ ,\ \ v=v_{\phi }}}}$

and solving for ${\displaystyle v_{\phi }}$. This gives the solution

${\displaystyle {\rm {v_{+}^{\circ }={\frac {a^{2}-2a{\sqrt {r}}+r^{2}}{{\sqrt {a^{2}+(r-2)r}}\left(a+r^{3/2}\right)}}}}}$

for the prograde and

${\displaystyle {\rm {v_{-}^{\circ }={\frac {a^{2}+2a{\sqrt {r}}+r^{2}}{{\sqrt {a^{2}+(r-2)r}}\left(a-r^{3/2}\right)}}}}}$

for the retrograde velocity.^[1] For photons with ${\displaystyle v=1,\ \mu =0}$ one gets

${\displaystyle {\rm {r_{+}^{\circ }=2\left(\cos \left({\frac {2}{3}}\cos ^{-1}(-a)\right)+1\right)}}}$

for the prograde photon orbit radius (in Boyer-Lindquist-coordinates), and

${\displaystyle {\rm {r_{-}^{\circ }=2\left(\cos \left({\frac {2}{3}}\cos ^{-1}(+a)\right)+1\right)}}}$

for the retrograde one.^[2] In the Schwarzschild-limit with ${\displaystyle a=0}$ both radii converge to ${\displaystyle {\rm {r^{\circ }=3}}}$.

Units: ${\displaystyle {\rm {G=M=c=1}}}$

Die pro- und retrograde Kreisbahngeschwindigkeit ergibt sich indem ${\displaystyle {\rm {{\dot {p}}_{r}=v_{r}=v_{\theta }=0\ ,\ \ \theta =\pi /2\ ,\ \ v=v_{\phi }}}}$

gesetzt und nach ${\displaystyle v_{\phi }}$ aufgelöst wird. Damit ergibt sich als Lösung

${\displaystyle {\rm {v_{+}^{\circ }={\frac {a^{2}-2a{\sqrt {r}}+r^{2}}{{\sqrt {a^{2}+(r-2)r}}\left(a+r^{3/2}\right)}}}}}$

für die prograde und

${\displaystyle {\rm {v_{-}^{\circ }={\frac {a^{2}+2a{\sqrt {r}}+r^{2}}{{\sqrt {a^{2}+(r-2)r}}\left(a-r^{3/2}\right)}}}}}$

für die retrograde Kreisbahngeschwindigkeit.^[1] Für Photonen mit ${\displaystyle v=1,\ \mu =0}$ ergibt sich daher

${\displaystyle {\rm {r_{+}^{\circ }=2\left(\cos \left({\frac {2}{3}}\cos ^{-1}(-a)\right)+1\right)}}}$

für den prograden Photonenkreisradius (in Boyer-Lindquist-Koordinaten), und

${\displaystyle {\rm {r_{-}^{\circ }=2\left(\cos \left({\frac {2}{3}}\cos ^{-1}(+a)\right)+1\right)}}}$

für den retrograden.^[2] Im Schwarzschild-Limit mit ${\displaystyle a=0}$ fallen beide Radien auf ${\displaystyle {\rm {r^{\circ }=3}}}$.

Einheitssystem: ${\displaystyle {\rm {G=M=c=1}}}$

(* Syntax: Mathematica *)
ClearAll["Global`*"]
vretro[a_,r_]:=(a^2+2 a Sqrt[r]+r^2)/(Sqrt[a^2+(-2+r) r] (a-r^(3/2))); 
vprogr[a_,r_]:=(a^2-2 a Sqrt[r]+r^2)/(Sqrt[a^2+(-2+r) r] (a+r^(3/2)));
rh = 1 + Sqrt[1 - a^2];
r1 = 2 (1 + Cos[2/3 ArcCos[-a]]);
r2 = 2 (1 + Cos[2/3 ArcCos[+a]]);
Do[Print[Rasterize[Grid[{{Show[
Plot[{vprogr[a, r], vretro[a, r]}, {r, 1, 10},
PlotRange->{{1, 10}, {-1, 1}},
GridLines->{{rh, r1, r2}, {}},
Frame->True, ImageSize->500, 
PlotStyle -> {{Blue, Thick}, {Magenta, Thick}}],
Graphics[{LightGray, Rectangle[{1, -1}, {rh, 1}]}]]}, 
{"a"->a}, {"rH"->rh}, {"r+"->r1}, {"r-"->r2}}, 
Alignment -> Left]]], 
{a, 0.0, 1.0, 0.1}]
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1. ↑ ^{a b} Simon Tyran: Kreisbahnen in der Kerr-Raumzeit
2. ↑ ^{a b} Edward Teo: Spherical Photon Orbits Around A Kerr Black Hole

de.wikipedia.org/wiki/Kerr-Metrik