= Jacobi truncation of two singular isothermal spheres
{c}
{title2=$r_t=D\sigma_d/(\sqrt2\sigma_h)$}
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 $\Phi_h''=-2\sigma_h^2/D^2$. The radial differential effective acceleration is $(\Omega^2-\Phi_h'')r$. Balancing it with satellite gravity $2\sigma_d^2/r$ gives the displayed <Jacobi tidal radius> and bound mass $m_t=\sqrt2D\sigma_d^3/(G\sigma_h)$. Equal mean density gives a different order-one approximation, $r_t=D\sigma_d/\sigma_h$; eccentric orbits require an orbital-history prescription.
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