<p>There's a fairly full treatment in <a href="https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html" rel="nofollow noreferrer">these lecture notes</a> but it's not really amenable to a summary in a stackexchange answer. However they end up with the following equation for the radius of a circular orbit of a particle with angular momentum $L$ and mass (ie not a photon). </p>

<p>$$GMr^2 - L^2r + 3GML^2 = 0$$</p>

<p>For suitable values of $L$ this has two solutions for $r$ and, one inside the ISCO radius (which is $6GM$ in these units) and one outside. Further analysis shows that the inner orbit is unstable and the outer one stable. See around equations 7.55 for more details. </p>

<p>Photons have an unstable circular orbit at $3GM$ and the event horizon is at $2GM$.</p>

CC BY-SA 4.0
2018-06-30T19:15:46.363Z
2018-06-30T19:15:46.363Z
Steve Linton
<p>If anyone has a derivation of this result, it would be appreciated greatly.</p>

CC BY-SA 4.0
2018-06-30T16:53:00.787Z
These [lecture notes by Mike Guidry](http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch18.pdf) appear to address it on pages 785-790.
Mike G
CC BY-SA 4.0
2022-05-14T07:40:09.187Z
@MikeG no relation?
ProfRob
CC BY-SA 4.0
2022-05-14T11:30:44.720Z
@ProfRob Correct.
Mike G
<orbit><black-hole>
Why is the Innermost Stable Circular Orbit (ISCO) of a non-rotating black-hole 3 times the Schwarzschild radius?
proshiv
# Why is the Innermost Stable Circular Orbit (ISCO) of a non-rotating black-hole 3 times the Schwarzschild radius?

(Asked by: username_0 on 2018-06-30)

If anyone has a derivation of this result, it would be appreciated greatly.

**Tags:** orbit, black-hole
> **Comments:**
> **username_1**: These [lecture notes by Mike Guidry](http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch18.pdf) appear to address it on pages 785-790.  
> **username_2**: @MikeG no relation?  
> **username_1**: @ProfRob Correct.  

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## 1 answers

### Answer by username_3 on 2018-06-30. Score: 1 | Accepted answer

There's a fairly full treatment in [these lecture notes](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html) but it's not really amenable to a summary in a stackexchange answer. However they end up with the following equation for the radius of a circular orbit of a particle with angular momentum $L$ and mass (ie not a photon).

$$GMr^2 - L^2r + 3GML^2 = 0$$

For suitable values of $L$ this has two solutions for $r$ and, one inside the ISCO radius (which is $6GM$ in these units) and one outside. Further analysis shows that the inner orbit is unstable and the outer one stable. See around equations 7.55 for more details.

Photons have an unstable circular orbit at $3GM$ and the event horizon is at $2GM$.


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