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At any given instance in time, a perturbed planetary orbit is completely determined by six osculating orbital elements. (See Section 10.2.)
Let us choose these elements to be the major radius,
; the mean anomaly at epoch,
; the eccentricity,
; the argument of the perihelion,
;
the inclination to the ecliptic,
; and the longitude of the ascending node,
. In the absence of a disturbing force,
,
,
,
,
, and
are all constants of the motion.
As is clear from the analysis of the preceding two sections, in the presence of a disturbing force of the form (I.15), the osculating elements evolve in time as follows:

These equations are known collectively as the *Gauss planetary equations*.

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Richard Fitzpatrick
2016-03-31