True longitude (source code)

= True longitude
{title2=$\ell_{\rm true}=\Omega+\omega+f$}

True longitude is the conventional orbital angle formed from the <longitude of ascending node>, argument of <periapsis> and <true anomaly>. For a coplanar <Kepler orbit> it is the actual polar longitude of the position vector. For an inclined <orbital plane> the sum of angles is generally not the projected azimuth. The <mean longitude> uses <mean anomaly> $M$ in place of $f$, so $\ell_{\rm true}-\lambda=f-M$. For small <orbital eccentricity>, <Kepler's equation> gives $E=M+e\sin M+O(e^2)$, while the eccentric-to-true anomaly relation gives $f=E+e\sin E+O(e^2)$; hence $f-M=2e\sin M+O(e^2)$. This periodic difference matters when separating a slow <resonant argument> from a rapidly varying geometric angle.