= Planetary quadrupole coefficient
{title2=$J_2$}
The dimensionless coefficient $J_2$ describes the leading axisymmetric departure of a planet's external <Newtonian gravitational potential> from a monopole: $\Phi=-GM_p/r+[GM_pJ_2R_p^2/r^3]P_2(\cos\vartheta)$, where $\vartheta$ is colatitude from the spin axis. An oblate mass distribution has positive $J_2$. The <disturbing function> is minus the nonspherical potential term. On a circular inclined <Kepler orbit>, $\cos\vartheta=\sin I\sin M$, so <orbit averaging> gives $\overline{\mathcal R}_{J_2}=GM_pJ_2R_p^2a^{-3}(1/2-3\sin^2I/4)$. The exact circular nodal rate is $-3nJ_2(R_p/a)^2\cos I/2$; its small-tilt limit is $-3nJ_2(R_p/a)^2/2$.
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