# Circular orbits and neutron stars

For a black hole with Schwarzschild radius $r$ it is true that there are no circular orbits around the centre of this black hole with radius smaller than $\frac{3}{2} r$.

Is it possible to have neutron star which is not black hole with the property that there is $r>0$ such that there are no circular orbits with radius $<r$ around the centre of this star?

Per Wikipedia here:

https://en.wikipedia.org/wiki/Neutron_star

the radius of a neutron star is approximately 10 km, and the mass is of the order of magnitude of a stellar mass. Assuming about a stellar mass, and using the formula $R_{S} = \frac{2GM}{c^{2}}$, we get the schwarzschild radius of a neutron star as being about 3 km.

Furthermore, stability analysis of orbits around a non-spinning black tells us that the innermost stable circular orbit around a black hole is at $r=6M = 3 R_{s}$, which is really dancing with actually being outside of the surface of the neutron star, so we would actually have to go in and do a careful analysis with exact values to really definitively answer whether you would have GR-unstable orbits outside the surface of the neutron star.

Now, some real-world concerns with this:

1) neutron stars are largely expected to be spinning quite rapidly, and spin will generally bring in the stable orbit from the $6M$ limit for co-rotating orbits and outward for anti-corotating orbits

2) long before GR has made your orbit unstable, I would expect that various interactions with accreting matter fields and with the electromagnetic field of the neutron star would destabilize your orbit. In theory, you can set these to zero, along with the neutron star's atmosphere, but in reality, almost certainly not.