Gravitational perturbation

A gravitational perturbation is a change to a reference motion caused by an additional gravitational force or a correction to the assumed gravitational field. In celestial mechanics, the reference is often a two-body problem with a point-mass central attractor. Another planet or the nonspherical mass distribution of the central body can change the orbital elements. This is a gravitational cause of an orbital perturbation; atmospheric drag is another kind of cause.^[ref]

The third-body acceleration must be relative to the accelerating central body. Let a satellite be at $r$ relative to that body, and let a perturbing mass $m_3$ be at $R$. Subtracting the third body's attraction on the origin givesOmitting the second term would mistake common acceleration of the entire system for a change in the relative orbit. For $\Vert r\Vert \ll R_0=\Vert R\Vert$, expanding yields the tidal approximationThe perturbation stretches separation along $\hat{R}$ and compresses it in perpendicular directions. Its characteristic ratio to a central attraction $GM/\Vert r{\Vert }^2$ is $(m_3/M)(\Vert r\Vert /R_0{)}^3$. This explains why distance as well as mass determines whether a third body strongly disturbs a satellite. Small instantaneous ratios can still produce appreciable accumulated changes over many orbital periods.^[ref]