Tidal radius ($r_t$)

The tidal radius bounds material that remains gravitationally attached to a satellite in the differential gravitational field of its host.

For a low-mass satellite of mass $M_s$ on a circular orbit of radius $r_o$ in a spherical host,This local approximation requires $r_t\ll r_o$.

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.

Tidal stripping removes satellite material outside the tidal radius. The resulting mass loss weakens dynamical friction and can greatly delay or prevent a merger.

A tidal tail is an elongated stream of material removed by a differential gravitational field. A satellite generally produces one leading and one trailing tail as stripped material enters nearby orbits with slightly different energy and angular momentum.

Dynamical-friction stalling by tidal stripping occurs when inward motion shrinks a satellite's tidal radius so rapidly that its remaining mass and frictional drag tend to zero before it reaches the host center.