This is a follow-up question to the answer given at [What is the exact gravitational force between two masses including relativistic effects?][1]. Unfortunately the author hasn't been online for a few years and therefore does not answer comments any more.

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In the answer given there the differential equation of motion in Schwarzschild coordinates was 

$$\ddot{r} = -\frac{G m}{r^2} + r\dot{\theta}^2 - \frac{ {\color{red} 3} G m}{c^2}\dot{\theta}^2$$

for the radial acceleration and

$$\ddot{\theta} = -\frac{2}{r}\dot{r}\dot{\theta}$$

for the angular acceleration. When I plot the path for an object near the speed of light, with this formula I get a stable orbit at $r_0=2 r_s$:

[![3GM/c² at r=2rs with v=0.999c][2]][2]

But shouldn't that be at $r_0=1.5 r_s$, the [photon sphere][3]? With that formula the orbiting particle would fall into the black hole very quickly, for example, with $v_0=0.999c$ at $r_0=1.6 r_s$:

[![3GM/c² at r=1.6rs with v=0.999c][4]][4]

When I replace the term 3Gm/c² with 2GM/c² so that 

$$\ddot{r} = -\frac{G m}{r^2} + r\dot{\theta}^2 - \frac{ {\color{red} 2} G m}{c^2}\dot{\theta}^2$$

I get the expected result with a stable orbit right at the photon sphere (initial velocity again $v_0=0.999c$):

[![enter image description here][6]][6]

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So my question is: is the formula wrong and the factor 3 needs to be replaced with a factor of 2, or are there different minimum-distances for stable orbits, one for particles, and one for photons? Or did I miss something else? Wikipedia says:

> The radius of the photon sphere, which is also the lower bound *for any stable orbit*, is $1.5 r_s$

so I would expect that also particles with mass should stay in orbit if they are close to the speed of light and slightly above the photon sphere.

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For reproduction of the problem the [Mathematica-code][5] as I believe it to be correct is available (with the factor 2 instead of 3)

  [1]: https://physics.stackexchange.com/q/47884
  [2]: https://i.sstatic.net/Q0DMR.gif
  [3]: http://en.wikipedia.org/wiki/Photon_sphere
  [4]: https://i.sstatic.net/cGIMS.gif
  [5]: http://yukterez.ist.org/schwarzschild,code.txt
  [6]: https://i.sstatic.net/gopWg.gif