= Tidal-gravity coefficient versus rotating Jacobi coefficient
The radial differential <gravity> of a point-mass host contributes $2GMx/D^3=2\Omega^2x$ 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 $\Omega^2x$, making the radial outward coefficient $3\Omega^2$ at a stationary point in that frame. Thus comparing self-gravity with $2\Omega^2x$ and solving the rotating-frame equilibrium with $3\Omega^2x$ 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.
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