Complex eccentricity ($z$)

The complex eccentricity packages eccentricity and longitude of periapsis as $z=e{e}^{i\varpi }$, turning linear secular dynamics into a complex linear system.

Choose the fixed planetary pericentre as the real axis, with positive forced amplitude $Z_f$. Then the linear secular equation $\dot{z}=iA(z-Z_f)$ has solution $z=Z_f+(z(0)-Z_f)e^{iAt}$. An initially circular orbit follows $z=Z_f(1-e^{iAt})$, a circle in the complex plane centered at the forced eccentricity $Z_f$, with free eccentricity amplitude $Z_f$. Its maximum orbital eccentricity is $2Z_f$ and its phase-averaged mean orbital eccentricity is $4Z_f/\pi$. The traversal is counterclockwise for $A>0$ and clockwise for $A<0$; at $A=0$ the initial state is stationary. The phase averages assume a nonzero secular frequency.