Innermost stable circular orbit

The Innermost stable circular orbit (often called the ISCO) is the smallest orbit in which a test particle can stably orbit a massive object in general relativity.^[1] The location of the ISCO, the ISCO-radius ( $r_{isco}$ ), depends on the angular momentum (spin) of the central object.

Massive particle

For a non-spinning massive object, where the gravitational field can be expressed with the Schwarzschild metric, the ISCO is located at,

r_{isco}={\frac {6\,GM}{c^{2}}}.

As the angular momentum of the central object increases, $r_{isco}$ decreases. Even for a non-spinning object, the ISCO radius is only three times the Schwarzschild radius, suggesting that only black holes have innermost stable circular orbits outside of their surfaces.

Photons

For a photon, the circular orbit occurs at^[2]

r={\frac {3\,GM}{c^{2}}}.

References

↑ Thorne, Misner & Wheeler 1973
↑ Carroll, Sean M. (December 1997). "Lecture Notes on General Relativity: The Schwartzchild Solution and Black Holes". arXiv:gr-qc/9712019 . Retrieved 2017-04-11.

Misner, Charles; Thorne, Kip S.; Wheeler, John (1973). Gravitation. W. H. Freeman and Company. ISBN 0-7167-0344-0.

External links

Relativity

Special
relativity

Background	Principle of relativity Special relativity
Foundations	Frame of reference Speed of light Hyperbolic orthogonality Rapidity Maxwell's equations
Formulation	Galilean relativity Galilean transformation Lorentz transformation
Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks
Spacetime	Light cone World line Spacetime diagram Biquaternions Minkowski space

General
relativity

Background	Introduction Mathematical formulation
Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Minkowski diagram Penrose diagram
Phenomena	Black hole Event horizon Frame-dragging Geodetic effect Lenses Singularity Waves Ladder paradox Twin paradox Two-body problem
Equations	ADM formalism BSSN formalism Einstein field equations Geodesic equation Friedmann equations Linearized gravity Post-Newtonian formalism Raychaudhuri equation Hamilton–Jacobi–Einstein equation Ernst equation
Advanced theories	Brans–Dicke theory Kaluza–Klein Mach's principle Quantum gravity
Solutions	Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub–NUT Milne Robertson–Walker pp-wave van Stockum dust Weyl−Lewis−Papapetrou

Scientists

Einstein field equations:

G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }

and their analytical solution by Ernst equation:

\displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}.

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