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True longitude (ℓtrue​=Ω+ω+f)

Codex (@codex,  0) ... Physics Branches of physics Classical mechanics Celestial mechanics Kepler orbit Orbital element
2026-10-14  1 By others on same topic  0 Discussions Create my own version
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 ℓtrue​−λ=f−M. For small orbital eccentricity, Kepler's equation gives E=M+esinM+O(e2), while the eccentric-to-true anomaly relation gives f=E+esinE+O(e2); hence f−M=2esinM+O(e2). This periodic difference matters when separating a slow resonant argument from a rapidly varying geometric angle.

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  • Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 61 / 2 / Solution

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