File:Horizon skimming Kerr orbit.gif
Horizon_skimming_Kerr_orbit.gif (758 × 544 pixels, file size: 6.57 MB, MIME type: image/gif, looped, 733 frames, 28 s)
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Summary
[edit]DescriptionHorizon skimming Kerr orbit.gif |
English: Photon orbit below the ergosphere and above the horizon of an extremal Kerr black hole. Spin parameter: a=1, direction of motion: prograde. Initial conditions: radius r=1.448GM/c², polar angle θ=90°, local equatorial inclination angle i=66.1391°, constants of motion: energy E=0.475744hf, axial angular momentum Lz=0.856004GMhf/c³, Carter constant Q=1.75361GMHf/c³. Left: x,z-projection, right: x,y-projection. For a close particle orbit see here. |
Date | |
Source | Own work |
Author | Yukterez (Simon Tyran, Vienna) |
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Display
[edit]
01) Coordinate time (GM/c^3) 11) BL r coordinate (GM/c^2) 21) Radius of gyration (GM/c^2) 31) Observed framedragging rate (c^3/G/M) 02) Affine parameter (GM/c^3) 12) BL φ coordinate (radians) 22) Cartesian radius (GM/c^2) 32) Local framedragging velocity (c) 03) 1st derivative (dt/dτ) 13) BL θ coordinate (radians) 23) BH Irreducible mass (M) 33) Cartesian framedragging velocity (c) 04) Grav. time dilation (dt/dτ) 14) dr/dτ (c) 24) Kinetic energy (hf) 34) Proper velocity (c, dl/dτ) 05) Local energy (dt/dτ, mc^2) 15) dφ/dτ (c^3/G/M) 25) Potential energy (hf) 35) Observed velocity (c, d{x,y,z}/dt) 06) Cartesian radius (GM/c^2) 16) dθ/dτ (c^3/G/M) 26) Total energy (hf) 36) Escape velocity (c) 07) x Axis (GM/c^2) 17) d^2r/dτ^2 (c^6/G/M) 27) Carter constant (GMhf/c^3) 37) Local r velocity (c) 08) y Axis (GM/c^2) 18) d^2φ/dτ^2 (c^6/G^2/M^2) 28) φ angular momentum (GMhf/c^3) 38) Local θ velocity (c) 09) z Axis (GM/c^2) 19) d^2θ/dτ^2 (c^6/G^2/M^2) 29) θ angular momentum (GMhf/c^3) 39) Local φ velocity (c) 10) travelled distance (GM/c^2) 20) Spin parameter (GM^2/c) 30) Radial momentum (hf/c) 40) Total local velocity (c)
Equations of motion
[edit]All formulas come in natural units:
Coordinate time by proper time (dt/dτ):
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 (I is the orbital inclination angel):
Carter k (constant):
Total energy (constant):
Angular momentum on the Ф-axis (constant):
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:
References
[edit]- ↑ 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.
File usage
[edit]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 | |
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current | 23:30, 9 April 2019 | 758 × 544 (6.57 MB) | Yukterez (talk | contribs) | color correction | |
22:57, 20 March 2019 | 758 × 544 (6.72 MB) | Yukterez (talk | contribs) | using an orbit closer to the horizon at r=1.448GM/c² | ||
20:32, 12 July 2017 | 758 × 500 (4.59 MB) | Yukterez (talk | contribs) | The kinetic energy in the display is actually the kinetic energy plus the rest energy | ||
03:43, 9 July 2017 | 758 × 500 (4.62 MB) | Yukterez (talk | contribs) | the units of the linear momentum components need to be mc, not c | ||
05:03, 3 July 2017 | 758 × 500 (4.58 MB) | Yukterez (talk | contribs) | setting loop repeating option to infinity | ||
04:49, 3 July 2017 | 758 × 500 (4.58 MB) | Yukterez (talk | contribs) | chosing a more representative orbit type, expanding numerical display | ||
08:44, 23 June 2017 | 770 × 380 (10.25 MB) | Yukterez (talk | contribs) | User created page with UploadWizard |
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Unique ID of original document | xmp.did:c6defc3c-d347-b247-88d3-f65f48e06b3b |
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Software used | Adobe Photoshop CC 2015 (Windows) |