gravity - What exactly is the microgravity field in orbit?

The ISS and other objects in orbit still experience small acceleration outside from the perfect line of orbit (of the system CM). For instance, two objects in the ISS that are let to be at rest will pass by each other twice as the station makes one orbit because the two items are in separate orbits, and all orbits pass over the same point because they are great circles over the Earth.

My question is how would you actually quantify this? Regarding the ISS, the entire craft could rotate just so it is effectively tidally locked with the Earth. If you assume it does that, then you have one axis along which it is totally acceleration-less relative to the craft. Looking forward along that line, moving to the left or right would create an acceleration back to the line. Moving up or down would also create an acceleration toward the line since they would assume more elliptical orbits. But I'm really curious as to whether there would also be a acceleration parallel to the line of orbit, and I'm also really curious if it would be unstable or stable.

So if the line of orbit is x, down toward the Earth is the negative z, and right of the line of motion is y, then my intuition is that the system is stable on the y-axis, it is stable up-and-down for the z-axis, but movement up the z-axis would create acceleration in the negative x-axis.

One major consequence would be if you left a hammer just outisde the airlock, the stability of these fields would dictate whether it sticks around or leaves. Could we find a simple $(x,y,z)$ equation for the acceleration? I can't find anything that quite answers this, and I my attempts result in more questions than answers.

Answer

For small velocities and displacements around a circular Kepler orbit, the equations of motion are the Hill-Clohessy-Wiltshire equations. They can be exactly solved to give

(The above switches your convention for y and z.)

Your approach of finding the forces in a rotating frame works; there is a derivation here.

The motion is in general a sort of looping that is unstable. A pure-z initial velocity or a pure-y initial velocity or displacement leads to periodic solutions (using your notation), but any other initial displacement or velocity leads to run-away. (However, the run-away is linear in time, not exponential.)

I made some plots of the trajectories here.