File:Kerr photon orbits with orbital inclination.gif
Kerr_photon_orbits_with_orbital_inclination.gif (758 × 500 pixels, file size: 26.44 MB, MIME type: image/gif, looped, 1,643 frames, 1 min 50 s)
Note: Due to technical limitations, thumbnails of high resolution GIF images such as this one will not be animated. The limit on Wikimedia Commons is width × height × number of frames ≤ 100 million.
Captions
Contents
Summary
[edit]DescriptionKerr photon orbits with orbital inclination.gif |
English: All possible photon-orbits around a black hole rotating with the spin-parameter a=Jc/G/M²=1. The position of photon and ZAMO is shown for t=150GM/c³ coordinate time. Initial position: θ0=π/2, φ0=0.
Deutsch: Alle möglichen Photonenorbits um ein mit dem Spinparameter a=Jc/G/M²=1 rotierendes schwarzes Loch. Gezeigt wird die Position eines Photons und eines ZAMO nach einer Koordinatenzeit von t=150GM/c³. Die Startposition ist auf θ0=π/2, φ0=0. |
Date | |
Source | Own work |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
Display
[edit]
01) a Spin parameter 08) δ local equatorial 15) L Axial angular momentum 22) ω Frame dragging delayed angular velocity of the central mass inclination angle conserved quantity observed at infinty 02) r Boyer-Lindquist radius 09) δ observed equatorial 16) L Poloidial component 23) v Frame dragging local velocity constant for photon orbits inclination angle of the angular momentum equals 1 at the outer ergosurface 03) φ Longitude 10) δ frame drag angle 17) p Radial component 24) Ω Frame dragging observed velocity measured from infinity difference local-observed of the momentum in cartesian coordinates 04) θ Latitude 11) E kinetic energy 18) R Radius 25) v Observed particle velocity 0=northpole, π=southpole local energy of the photon cartesian coordinate in the bookeepers frame of reference 05) ς Grav. time dilation 12) E potential energy 19) x X-axis 26) v Local escape velocity depending on r and θ total-kinetic cartesian coordinate equals 1 at the outer horizon 06) t Coordinate time 13) E total energy 20) y Y-axis 27) v Delayed particle velocity of the distant bookeeper conserved quantity cartesian coordinate differential velocity vs a local ZAMO 07) λ Affine parameter 14) Q Carter constant 21) z Z-axis 28) v Local particle velocity takes the place of τ if μ=0 conserved quantity cartesian coordinate relative velocity vs a local ZAMO
Inclination angle by radius
[edit]For a given a and r and starting from θ0=π/2 the required initial orbital inclination angle δ0 for a photon's circular orbit can be found[1] by setting
and solving for δ0. The real solutions of the polynomial give one possible orbit in the positive poloidial direction, and one other in the opposite z-direction (since the metric is axially symmetric the sign of the coaxial angular momentum can be both). The shorthand terms are:
All photon-orbits have a constant Boyer-Lindquist-radius.[2] [3]
Equations of motion
[edit]All formulas come in natural units:
Coordinate time t by proper time τ (dt/dτ), where τ becomes the affine parameter λ for massless particles:
Radial coordinate time derivative (dr/dτ):
Time derivative of the covariant momentum's r-component (pr/dτ):
Relation to the local velocity:
Latitudinal time derivative (dθ/dτ):
Time derivative of the covariant momentum's θ-component (pθ/dτ):
Relation to the local velocity:
Longitudinal time derivative (dФ/dτ):
Time derivative of the covariant momentum's Ф-component (pФ/dτ):
Carter-constant:
Carter k:
Total energy:
Angular momentum on the Ф-axis:
with the radius of gyration
Frame Dragging angular velocity (dФ/dt):
Gravitational time dilation (dt/dτ):
Local velocity on the r-axis:
Local velocity on the θ-axis:
Local velocity on the Ф-axis:
with the cartesian coordinates:
The observed velocity β is given by:
The local escape velocity is given by the relation:
Shorthand Terms:
de
[edit]Für eine deutschsprachige Version der Bewegungsgleichungen geht es hier entlang
References
[edit]- ↑ Simon Tyran: Kreisbahnen in der Kerr-Raumzeit
- ↑ Stein Leo: Kerr Spherical Photon Orbits
- ↑ Ed Teo: Spherical Photon Orbits around a Kerr Black Hole doi:10.1023/A:1026286607562
- ↑ Pu, Yun, Younsi & Yoon: General-relativistic radiative transfer in Kerr spacetime, p. 2+
- ↑ Janna Levin & Gabe Perez-Giz: A Periodic Table for Black Hole Orbits, p. 30+
- ↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes, p. 5+
- ↑ Janna Levin & Gabe Perez-Giz: The Phase Space Portrait, p. 2+
- ↑ Misner, Thorne & Wheeler (MTW): The Bible archive copy at the Wayback Machine, p. 897+
- ↑ Simon Tyran: Kerr Orbits / Gravitationslinsen
Licensing
[edit]- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
Annotations InfoField | This image is annotated: View the annotations at Commons |
outer ergosphere
horizons
inner ergosphere and ring singularity
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 23:59, 5 November 2022 | 758 × 500 (26.44 MB) | Yukterez (talk | contribs) | the Q was missing a ² | |
02:05, 26 July 2017 | 758 × 500 (26.44 MB) | Yukterez (talk | contribs) | insert short pause between animation of different r and perspectivic rotation for specific r | ||
16:11, 25 July 2017 | 758 × 500 (28.15 MB) | Yukterez (talk | contribs) | 1) added a numerical display in case someone wants to look at the data, 2) perspectivic rotation for some example of closed loop orbits, 3) better quality and more frames for the cost of higher filesize (but since everybody is watching 4k videos in the... | ||
20:02, 23 July 2017 | 758 × 380 (10.15 MB) | Yukterez (talk | contribs) | User created page with UploadWizard |
You cannot overwrite this file.
File usage on Commons
There are no pages that use this file.
File usage on other wikis
The following other wikis use this file:
- Usage on vi.wikipedia.org
Metadata
This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.
Unique ID of original document | xmp.did:ba80e93b-7476-544b-91d3-46625506307b |
---|---|
Software used | Adobe Photoshop 21.0 (Windows) |