Jacobi tidal radius

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 ${\Phi }_h$, andIn the satellite's rotating frame, linearizing the combined host and centrifugal acceleration along the host-satellite line gives $[{\Omega }^2-{\Phi }_h^{\prime \prime }(r_o)]x$. Balancing this outward acceleration against the satellite's inward gravity givesFor an extended satellite, $M_s$ is its enclosed mass at $r_t$, so the equation can be implicit.

Differentiate ${\Phi }_h^{\prime }=G{M}_h/r^2$ and write $\alpha =d\log M_h/d\log r$ at $r_o$. Then ${\Phi }_h^{\prime \prime }=(\alpha -2){\Omega }^2$ andA 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.

For a low-mass singular isothermal sphere satellite on a circular orbit of radius $D$ in a singular isothermal sphere host, ${\Omega }^2=2{\sigma }_h^2/D^2$ and ${\Phi }_h^{\prime \prime }=-2{\sigma }_h^2/D^2$. The radial differential effective acceleration is $({\Omega }^2-{\Phi }_h^{\prime \prime })r$. Balancing it with satellite gravity $2{\sigma }_d^2/r$ gives the displayed Jacobi tidal radius and bound mass $m_t=\sqrt{2}D{\sigma }_d^3/(G{\sigma }_h)$. Equal mean density gives a different order-one approximation, $r_t=D{\sigma }_d/{\sigma }_h$; eccentric orbits require an orbital-history prescription.