Editing Orbital mechanics (section)

===Perturbations===
The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors <math>x_0</math> and <math>v_0</math> at a given epoch <math>t = 0</math>.  In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation.  Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity ''would have been'' at the epoch.  In perfect two-body motion, these orbital elements would be invariant (just like the Keplerian elements would be).

However, perturbations cause the orbital elements to change over time. Hence, the position element is written as <math>x_0(t)</math> and the velocity element as <math>v_0(t)</math>, indicating that they vary with time.  The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions <math>x_0(t)</math> and <math>v_0(t)</math>.<!-- TODO: Explain it more -->

The following are some effects which make real orbits differ from the simple models based on a spherical Earth.  Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding two-body effects.
*Equatorial bulges cause [[precession]] of the node and the perigee
*[[Spherical harmonics#Visualization of the spherical harmonics|Tesseral harmonic]]s<ref>{{MathWorld |title=Tesseral Harmonic |id=TesseralHarmonic |access-date=2019-10-07}}</ref> of the gravity field introduce additional perturbations
*Lunar and solar gravity perturbations alter the orbits
*Atmospheric drag reduces the semi-major axis unless make-up thrust is used

Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behavior can become [[Chaos theory|chaotic]]. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as [[Orbital station-keeping|station-keeping]], [[ground track]] maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude.