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Innermost Stable Circular Orbit (ISCO) ○｜Definition｜1st｜20251119205401-00-⌔

Innermost stable circular orbit - Wikipedia

Innermost stable circular orbit

The innermost stable circular orbit (often called the ISCO) is the smallest marginally stable circular orbit in which a test particle can stably orbit a massive object in general relativity.¹ The location of the ISCO, the ISCO-radius ($r_{\mathrm{isco}}$), depends on the mass and angular momentum (spin) of the central object. The ISCO plays an important role in black hole accretion disks since it marks the inner edge of the disk.

The ISCO should not be confused with the Roche limit, the innermost point where a physical object can orbit before tidal forces break it up. The ISCO is concerned with theoretical test particle s, not real objects. In general terms, the ISCO will be far closer to the central object than the Roche limit.

Printed 2026-06-28.

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Footnotes

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

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Spin (a) ○◂｜untitled_remarks｜1st｜20260516181355-00-●

Remillard & McClintock 2006 ┃ X-Ray Properties… ⚬𓂃

In an astrophysical environment, a BH is completely specified in GR by two numbers, its mass $M$ and its specific angular momentum or spin $a=J/cM$, where $J$ is the BH angular momentum and $c$ is the speed of light. The spin value is conveniently expressed in terms of a dimensionless spin parameter, $a_{\ast }=a/R_{\mathrm{g}}$, where the gravitational radius is $R_{\mathrm{g}}\equiv GM/c^2$. The mass simply supplies a scale, whereas the spin changes the geometry. The value of $a_{\ast }$ lies between 0 for a Schwarzschild hole and 1 for a maximally-rotating Kerr hole. A defining property of a BH is its event horizon, the immaterial surface that bounds the interior region of space-time that cannot communicate with the external universe. The event horizon, the existence of an innermost stable circular orbit (ISCO), and other properties of BHs are discussed in many texts (e.g., Shapiro & Teukolsky 1983; Kato et al. 1998). The radius of the event horizon of a Schwarzschild BH ($a_{\ast }=0$) is $R_{\mathrm{S}}=2R_{\mathrm{g}}=30\mathrm{km}(M/10M_{\odot })$, the ISCO lies at $R_{\mathrm{ISCO}}=6R_{\mathrm{g}}$, and the corresponding maximum orbital frequency is ${\nu }_{\mathrm{ISCO}}=220\mathrm{Hz}(M/10M_{\odot }{)}^{-1}$. For an extreme Kerr BH ($a_{\ast }=1$), the radii of both the event horizon and the ISCO (prograde orbits) are identical, $R_{\mathrm{K}}=R_{\mathrm{ISCO}}=R_{\mathrm{g}}$, and the maximum orbital frequency is ${\nu }_{\mathrm{ISCO}}=1615\mathrm{Hz}(M/10M_{\odot }{)}^{-1}$.

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