Summary

DescriptionOrbit um ein rotierendes schwarzes Loch (Animation).gif	English: Prograde orbit around a black hole spinning with a spin parameter of a=Jc/G/M²=0.9. Initial conditions: local velocity v0=0.4c, cartesian position x0=(√(4981)/10)GM/c², y0=z0=0, orbital inclination angle i0=arctan(5/6)rad=39.8056°, vertical launch angle=0 (which gives the local three-velocity-components vφ0=0.307289c, vθ0=0.256074c, vr0=0) - that corresponds to the conserved quantities total energy E=0.935179mc², axial angular momentum Lz=2.37176GMm/c, Carter constant Q=3.82514GMm/c and the Boyer-Lindquist-coordinates r0=7, θ0=π/2=90°, φ0=0. The observed (shapiro-delayed and frame-dragged) velocity at t0 is u0=0.366484c. Deutsch: Prograder Orbit um ein mit a=Jc/G/M²=0.9 rotierendes schwarzes Loch. Startbedingungen: lokale Geschwindigkeit v0=0.4c, kartesische Position x0=(√(4981)/10)GM/c², y0=z0=0, orbitaler Inklinationswinkel i0=arctan(5/6)rad=39.8056°, vertikaler Abschusswinkel=0 (das ergibt die lokalen Dreiergeschwindigkeitskomponenten vφ0=0.307289c, vθ0=0.256074c, vr0=0) - was den Erhaltungsgrößen Gesamtenergie E=0.935179mc², axialer Drehimpuls Lz=2.37176GMm/c, Carter Konstante Q=3.82514 und den Boyer-Lindquist-Koordinaten r0=7, θ0=π/2=90°, φ0=0 entspricht. Die beobachtete (shapiroverzögerte und geframedragte) Geschwindigkeit bei t0 ist u0=0.366484c.
Date	29 June 2016
Source	http://kerr.yukterez.net
Author	Yukterez (Simon Tyran, Vienna)
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en

01) Coordinate time              08) Axial radius of gyration     15) Axial angular momentum       22) Framedragging delayed angular velocity
02) Proper time                  09) Poloidial radius of gyration 16) Polar angular momentum       23) Framedragging local velocity
03) Total time dilation          10) Radial coefficient           17) Radial momentum              24) Framedragging observed velocity
04) Gravitational time dilation  11) E kinetic                    18) Cartesian radius             25) Observed particle velocity
05) Boyer Lindquist radius       12) Potential energy component   19) Cartesian X-axis             26) Local escape velocity
06) BL Longitude in radians      13) Total particle energy        20) Cartesian Y-axis             27) Delayed particle velocity
07) BL Latitude in radians       14) Carter Constant              21) Cartesian Z-axis             28) Local particle velocity

de

01) Koordinatenzeit              08) Axialer Gyrationsradius      15) Axialer Drehimpuls           22) Framedrag verzögerte Winkelgeschwindigkeit
02) Eigenzeit des Testpartikels  09) Poloidialer Gyrationsradius  16) Polarer Drehimpuls           23) Framedrag lokale Transversalgeschwindigkeit
03) Insgesamte Zeitdilatation    10) Radialer Vorfaktor           17) Radialer Impuls              24) Framedrag beobachtete Transversalgeschwindigkeit
04) Gravitative  Zeitdilatation  11) E kinetisch                  18) Kartesischer Radius          25) Beobachtete Totalgeschwindigkeit
05) Boyer Lindquist Radius       12) Potentielle Energie          19) Kartesische X-Achse          26) Lokale Fluchtgeschwindigkeit
06) BL Längengrad in Radianten   13) Totale Energie               20) Kartesische Y-Achse          27) Verzögerte Geschwindigkeit
07) BL Breitengrad in Radianten  14) Carter Konstante             21) Kartesische Z-Achse          28) Lokale Geschwindigkeit relativ zum ZAMO

Equations / Formeln

en

For an english version of the equations of motions click here

de

Alle Formeln sind in natürlichen Einheiten:

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

Koordinatenzeitableitung nach der Eigenzeit (dt/dτ):

${\displaystyle {\rm {{\dot {t}}={\frac {2\ E\ r\ \left(a^{2}+r^{2}\right)-2\ a\ L_{z}\ r}{\Delta \ \Sigma }}+E={\frac {\varsigma }{\sqrt {1-v^{2}}}}}}}$

Radialkoordinatenableitung (dr/dτ):

${\displaystyle {\rm {{\dot {r}}={\frac {\Delta \ p_{r}}{\Sigma }}}}}$

Radiale Impulskomponentenableitung:

${\displaystyle {\rm {{\dot {p}}_{r}={\frac {(r-1)\left(\mu \ \left(a^{2}+r^{2}\right)-k\right)+2\ E^{2}\ r\left(a^{2}+r^{2}\right)-2\ a\ E\ L_{z}+\Delta \ \mu \ r}{\Delta \ \Sigma }}-{\frac {2\ p_{r}^{2}\ (r-1)}{\Sigma }}}}}$

Zusammenhang mit der lokalen Geschwindigkeit:

${\displaystyle {\rm {p_{r}={\frac {v_{r}}{\sqrt {1+\mu \ v^{2}}}}{\sqrt {\frac {\Sigma }{\Delta }}}}}}$

Breitengradableitung (dθ/dτ):

${\displaystyle {\rm {{\dot {\theta }}={\frac {p_{\theta }}{\Sigma }}}}}$

Drehimpulsableitung auf der θ-Achse (pθ/dτ):

${\displaystyle {\rm {{\dot {p}}_{\theta }={\frac {\sin \theta \ \cos \theta \left({\frac {L_{z}^{2}}{\sin ^{4}\theta }}-a^{2}\left(E^{2}+\mu \right)\right)}{\Sigma }}}}}$

Zusammenhang mit der lokalen Geschwindigkeit:

${\displaystyle {\rm {p_{\theta }={\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1+\mu \ v^{2}}}}}}}$

Längengradableitung (dФ/dτ):

${\displaystyle {\rm {{\dot {\phi }}={\frac {2\ a\ E\ r+L_{z}\ \csc ^{2}\theta \ (\Sigma -2r)}{\Delta \ \Sigma }}}}}$

Drehimpulsableitung auf der Ф-Achse (pФ/dτ):

${\displaystyle {\rm {{\dot {p}}_{\phi }=0}}}$

Erhaltungsgröße Carter-Konstante:

${\displaystyle {\rm {Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}(\mu ^{2}-E^{2})+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)=a^{2}\ (\mu ^{2}-E^{2})\ \sin ^{2}I+L_{z}^{2}\ \tan ^{2}I}}}$

Daraus abgeleitete Erhaltungsgröße:

${\displaystyle {\rm {k=a^{2}\left(E^{2}+\mu \right)+L_{z}^{2}+Q}}}$

Erhaltungsgröße Gesamtenergie:

${\displaystyle {\rm {E={\sqrt {{\frac {(\Sigma -2\ r)\left({\dot {\theta }}^{2}\ \Delta \ \Sigma +{\dot {r}}^{2}\ \Sigma -\Delta \ \mu \right)}{\Delta \ \Sigma }}+{\dot {\varphi }}^{2}\ \Delta \ \sin ^{2}\theta }}={\sqrt {\frac {\Delta \ \Sigma }{(1+\mu \ v^{2})\ \chi }}}+\Omega \ L_{z}}}}$

Erhaltungsgröße Drehimpuls entlang Ф:

${\displaystyle {\rm {L_{z}={\frac {\sin ^{2}\theta \ ({\dot {\phi }}\ \Delta \ \Sigma -2\ a\ E\ r)}{\Sigma -2\ r}}={\frac {v_{\phi }\ {\bar {R}}}{\sqrt {1+\mu \ v^{2}}}}}}}$

mit dem Radius der Gyration

${\displaystyle {\rm {{\bar {R}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }}}$

Frame Dragging Winkelableitung (dФ/dt):

${\displaystyle {\rm {\omega ={\frac {2\ a\ r}{\chi }}}}}$

Gravitative Zeitdilatationskomponente (dt/dτ):

${\displaystyle {\rm {\varsigma ={\sqrt {\frac {\chi }{\Delta \ \Sigma }}}}}}$

Lokale Geschwindigkeit auf der r-Achse:

${\displaystyle {\rm {{\frac {v_{r}}{\sqrt {1+\mu \ v^{2}}}}={\dot {r}}\ {\sqrt {\frac {\Sigma }{\Delta }}}}}}$

Lokale Geschwindigkeit auf der θ-Achse:

${\displaystyle {\rm {{\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1+\mu \ v^{2}}}}={\dot {\theta }}\ \Sigma }}}$

Lokale Geschwindigkeit auf der Ф-Achse:

${\displaystyle {\frac {\rm {v_{\phi }}}{\sqrt {1+\mu \ {\rm {v^{2}}}}}}={\frac {\rm {L_{z}}}{\rm {{\bar {R}}_{\phi }}}}}$

Kartesische Koordinaten:

${\displaystyle {\rm {x={\sqrt {r^{2}+a^{2}}}\sin \theta \ \cos \phi \ ,\ y={\sqrt {r^{2}+a^{2}}}\sin \theta \ \sin \phi \ ,\ z=r\cos \theta \quad }}}$

Beobachtete Geschwindigkeit:

${\displaystyle {\rm {\beta ={\sqrt {(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}}}}}}$

Die radiale Fluchtgeschwindigkeit ergibt sich aus dem Verhältnis:

${\displaystyle {\rm {\varsigma =1/{\sqrt {1-v_{\rm {esc}}^{2}}}\ \to \ v_{\rm {esc}}={\sqrt {\varsigma ^{2}-1}}/\varsigma }}}$

zusammengefasste Terme:

${\displaystyle {\rm {\Sigma =a^{2}\cos ^{2}\theta +r^{2}\ ,\ \ \Delta =a^{2}+r^{2}-2r\ ,\ \ \chi =\left(a^{2}+r^{2}\right)^{2}-a^{2}\ \sin ^{2}\theta \ \Delta }}}$

Quellen:^{[1][2][3][4][5][6]}

Referenzen

1. ↑ Pu, Yun, Younsi & Yoon: General-relativistic radiative transfer in Kerr spacetime, S. 2+
2. ↑ Janna Levin & Gabe Perez-Giz: A Periodic Table for Black Hole Orbits, S. 30+
3. ↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes, S. 5+
4. ↑ Janna Levin & Gabe Perez-Giz: The Phase Space Portrait, S. 2+
5. ↑ Misner, Thorne & Wheeler (MTW): Die Bibel archive copy at the Wayback Machine, S. 897+
6. ↑ Simon Tyran: Kerr Orbits / Gravitationslinsen

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Dateiverwendung in Wikipedia-Artikeln

• de.wikipedia.org/wiki/Kerr-Metrik
• en.wikipedia.org/wiki/Rotating_black_hole