= 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.<Introduction to Celestial Mechanics by Richard Fitzpatrick>[Online sections Major planetary satellites in solar system and Earth–Moon–Sun system: third-body and planetary-oblateness perturbations]{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 gives
$$
a_{\rm pert}=Gm_3\left(\frac{R-r}{\lVert R-r\rVert^3}-\frac{R}{\lVert R\rVert^3}\right).
$$
Omitting the second term would mistake common acceleration of the entire system for a change in the relative orbit. For $\lVert r\rVert\ll R_0=\lVert R\rVert$, expanding yields the tidal approximation
$$
a_{\rm pert}\simeq\frac{Gm_3}{R_0^3}\bigl[3\widehat R(\widehat R\cdot r)-r\bigr].
$$
The perturbation stretches separation along $\widehat R$ and compresses it in perpendicular directions. Its characteristic ratio to a central attraction $GM/\lVert r\rVert^2$ is $(m_3/M)(\lVert r\rVert/R_0)^3$. 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>.<Introduction to Celestial Mechanics by Richard Fitzpatrick>[Earth–Moon–Sun system, relative solar attraction and expansion of the disturbing function]{ref}
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