Is there a synchronous orbital height for Phobos?

Question

My assumption is that any body in the solar system would have it's own specific geo-synchronous orbit height. But when I took a look at Phobos and calculated the GEO height based on the simple formula for synchronous orbits I got a height that was way above it's L1/L2 points:

GEO = 12.65km (above the surface) using the following values
ME (mass) = 1.07E+16
T-Rot (rotation period) = 27,552 seconds
R (radius) = 11.26 km

and

L1/L2 = 3.1km (Mars/Phobos)

Since Phobos is tidally locked to Mars I assumed that its rotation period would be equal to a single orbit around Mars (7 h 39.2 min), but perhaps this is flawed?

Is it possible to place an object into a stationary orbit around a small mass like Phobos? Does being tidally locked to Mars cancel out it's rotational force? Or perhaps it's possible to have an L1 point below the GEO height, which would be really cool. Since Phobos is already really cool being subsynchronous to Mars...

Answer 1

No, or at least there isn't a useful synchronous orbital height.

As you pointed out in your question, the mean radius of Phobos is 11.26 km, but if you look closer the sphere of influence of Phobos is only 7.6 km (from here). That means that anything in the neighborhood of Phobos is acted on more strongly by the gravity of Mars than of Phobos, so orbiting Phobos directly doesn't really work (because you're really just orbiting Mars).

That doesn't mean there aren't synchronous orbits available: the Mars-Phobos Lagrange points are fixed relative to Phobos (because Phobos is tidally locked: one orbit around Mars = one rotation).

Answer 2

By definition, the only stationary points in a two-body system are the Langrangian points This applies as Phobos is tidally locked to Mars, and thus the two periods are equivalent. Even L3, L4 and L5 are by definition stationary points even if not close to Phobos.

However that is only the stationary orbits, many more orbits are going to be synchronous orbits.

Lissajous orbits around L1, L2 and L3 are also synchronous, and even if they are numerically improbable to be periodic to an exact degree, that is entirely possible.

More interestingly perhaps are the various Tadpole and Horseshoe orbits connecting L4 and L5 as any such orbit is guaranteed to be synchronous due to its hillbert potential.

A third consideration is that the extremely small mass of Phobos compared with Mars is that any orbit with the same orbital period as Phobos can be considerd a deformed lissajous curve only loosely interacting with Lagrange points. This allows for an infinite family of orbits with two degrees of freedom (eccentricity and inclination) with synchronous orbits of various amplitude.