Multiplication operator

Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular, and free of perturbations, and if its inclination were zero.

Description

Both the mean longitude and the true longitude of the body in the orbit described above would change at a constant rate over time. But real orbits are eccentric and so depart from circularity, at least slightly, even if presumed to be free from any perturbations. These unperturbed eccentric orbits are called Keplerian ellipses, and in them the progress of the orbiting body in true longitude does not change at a constant rate over time. So the mean longitude is an abstracted quantity for Keplerian orbits, still proportional to the time, but now only indirectly related to the position of the orbiting body: the difference between the mean longitude and the true longitude is usually called the equation of the center. In such an elliptical orbit, the only times when the mean longitude is equal to the true longitude are the times when the orbiting body passes through periapsis (or pericenter) and apoapsis (or apocenter).

In an orbit that is undergoing perturbations, an osculating orbit together with its (elliptical) osculating elements can still be defined for any point in time along the actual orbit. For each successive set of osculating elements, a mean longitude can be defined, as in the unperturbed case. But here, the changes in mean longitude over time will not only be those due to some constant rate over time; there will also be superimposed perturbations (and the rate itself is also perturbed). A set of mean elements can still be defined for such an orbit, after abstracting the perturbational variations with time. The term 'mean longitude' was already used for the unperturbed and osculating cases, and the corresponding mean longitude member in a set of mean elements, after abstraction of the periodic variations, is sometimes therefore called the 'mean mean longitude'. To arrive at a true longitude from a mean mean longitude, the perturbational terms must be applied as well as the equation of the center.

Calculation

The mean longitude $L$ can be calculated as follows: 

$L=M+\varpi =M+\mathrm{\Omega }+\omega$

where:

• $M$ is orbit's mean anomaly,
• $\varpi \equiv \mathrm{\Omega }+\omega$ is longitude of the orbit's periapsis,
• $\mathrm{\Omega }$ is the longitude of the ascending node and
• $\omega$ is the argument of periapsis.

References

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