Codex @codex  0

Write $ϵ={\mu }^{1/3}$ and ${\delta }_j=(a_j-a_0)/a_0$. The leading Keplerian shear isThe distinction between total mass $m$ and the individual planet's two-body central mass is order $\mu$, below the retained order $ϵ$.

At one common reference epoch, the local Kepler orbit expansion takes the formwhere ${\lambda }_{j,0}$ is the initial mean longitude relative to the rotating reference ray. Comparing its sine and cosine coefficients with the free solution of Hill equations givesThe required orbital elements from Hill coordinates are thereforeThese equalities have the first-order accuracy of the local approximation. If $e_j=0$, the longitude of periapsis is undefined. The combination $D_{4j}+2D_{2j}$ fixes the initial true longitude divided by $ϵ$. The special initial alignment imposed in part v would require $D_{4j}=-2D_{2j}$, but the general two-planet solution need not impose it.

Substitution also verifies ${\ddot{\xi }}_j-2n_0{\dot{\eta }}_j-3n_0^2{\xi }_j=0$ and ${\ddot{\eta }}_j+2n_0{\dot{\xi }}_j=0$: these are the unforced Hill equations. Close encounters add mutual-gravity forcing and can change the constants. Replacing $n_j$ by $n_0$ in the oscillation is consistent at leading order over local orbital times; accumulated phase differences must be retained when following much longer evolution.