= Jacobi tidal radius
{c}
A <Jacobi tidal radius> estimates the radial saddle distance from a low-<mass> satellite on a <circular orbit> in a smooth spherical host. Let the orbital <radius> be $r_o$, the host potential be $\Phi_h$, and
$$
\Omega^2=\frac{\Phi_h'(r_o)}{r_o}=\frac{GM_h(r_o)}{r_o^3}.
$$
In the satellite's rotating frame, linearizing the combined host and centrifugal <acceleration> along the host-satellite line gives $[\Omega^2-\Phi_h''(r_o)]x$. Balancing this outward <acceleration> against the satellite's inward <gravity> gives
$$
r_t^3=\frac{GM_s}{\Omega^2-\Phi_h''(r_o)}.
$$
For an extended satellite, $M_s$ is its enclosed <mass> at $r_t$, so the equation can be implicit.
Differentiate $\Phi_h'=GM_h/r^2$ and write $\alpha=d\log M_h/d\log r$ at $r_o$. Then $\Phi_h''=(\alpha-2)\Omega^2$ and
$$
r_t=r_o\left[\frac{M_s}{(3-\alpha)M_h(r_o)}\right]^{1/3}.
$$
A finite radial saddle in this leading approximation requires $3-\alpha>0$, as well as $r_t\ll r_o$. For a point-<mass> host, $\alpha=0$ gives the familiar denominator $3M_h$. A flat <galaxy rotation curve> has $\alpha=1$ and instead gives $2M_h$.
Inside an exactly constant-<mass density> core, $\alpha=3$ makes the linear radial stretching coefficient vanish. Dividing by zero is not a prediction of a finite tidal boundary there; higher-order structure or a different model is needed. The <Jacobi tidal radius> describes a rotating-frame equilibrium scale, not a spherical material wall or an exact stripping rule for eccentric, evolving orbits. The distinction from the inertial differential <gravitational field> is explained by the <tidal-gravity coefficient versus rotating Jacobi coefficient>.
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