= Laplace radius
{c}
{title2=$a_L=[2J_2(M_p/M_\star)R_p^2a_p^3]^{1/5}$}
The Laplace radius is the satellite distance where the small-tilt <nodal precession> contributions from a planet's axisymmetric quadrupole and the stellar tide have equal magnitudes. In the circular hierarchical limit those rates are $3GM_pJ_2R_p^2/(2na^5)$ and $3GM_\star/(4na_p^3)$. Equating them proves the displayed fifth-root expression. Inside this radius the <planetary quadrupole coefficient> controls the preferred plane; outside it the <double-averaged circular tidal potential> dominates. When equator and planetary orbit are inclined to one another, their balance determines the local <Laplace plane>.
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