Tidal-gravity coefficient versus rotating Jacobi coefficient

The radial differential gravity of a point-mass host contributes $2GMx/D^3=2{\Omega }^{2}x$ for nearby separation $x$ at orbital radius $D$. A Jacobi tidal radius describes a different calculation: the satellite is on a circular orbit and the equilibrium is evaluated in its rotating frame.

The centrifugal contribution adds another ${\Omega }^{2}x$, making the radial outward coefficient $3{\Omega }^2$ at a stationary point in that frame. Thus comparing self-gravity with $2{\Omega }^{2}x$ and solving the rotating-frame equilibrium with $3{\Omega }^{2}x$ are distinct approximations. The discrepancy is not an inconsistent derivative of the same gravitational field. Orbital geometry and the chosen frame must accompany any quoted tidal-disruption coefficient.