= Tidal force
{wiki}
Differential gravitational <acceleration>, multiplied by the affected <mass>, across a spatially extended body. In the linear approximation the <acceleration> difference is $T\Delta\boldsymbol r$, where $T$ is the <tidal tensor>. Stellar <tidal forces> limit the domain of a <circumplanetary orbit>; sufficiently strong ones can produce <tidal disruption>.
To isolate a <tidal force>, subtract the <gravitational acceleration> shared by the body as a whole. Let a reference <test particle> follow $\ddot{\mathbf R}=\mathbf g(\mathbf R)$, and let another have <position> $\mathbf R+\mathbf x$. In a <translating accelerated reference frame> with parallel, nonrotating axes,
$$
\ddot{\mathbf x}=\mathbf g(\mathbf R+\mathbf x)-\mathbf g(\mathbf R)=T(\mathbf R)\mathbf x+O\!\left(\frac{\mu|\mathbf x|^2}{R^4}\right)
$$
for a source <point mass>, when $|\mathbf x|/R\ll1$. Here $\mu=GM$ is its <standard gravitational parameter> and the last term uses <Big O notation>. This <Taylor expansion> removes the common inward pull and keeps the spatial variation. The near side has an excess pull toward the source; the far side has a deficit of inward pull, which is an outward relative <acceleration>. Both sides can stretch apart while both still accelerate toward the source.<The Feynman Lectures on Physics, Volume I by Richard Feynman and others/Chapter 7: The Theory of Gravitation>[§7–4, discussion of stronger near-side and weaker far-side attraction]{ref}
\Image[/tidal-field-difference-and-stretching.png]
{title=Subtracting common gravitational acceleration reveals radial stretching and transverse squeezing}
{description=Blue arrows show the exact <Newtonian gravitational field> at several <positions> near a reference point. Orange arrows show its linear residual, calculated with the <tidal tensor>. Arrow lengths use separate scales in the two panels; this is a comparison of direction and spatial variation, not two equal-magnitude fields.}
For a material element of <mass> $m$, the residual external <force> in this <translating accelerated reference frame> is $m[\mathbf g(\mathbf R+\mathbf x)-\mathbf g(\mathbf R)]$. Actual deformation also depends on <elasticity>, <pressure> and <self-gravity>. A <rigid body> approximation suppresses deformation but can still have internal <forces> that resist the <tidal force>. The <point-mass tidal stretching and squeezing> pattern describes freely falling, noninteracting <test particles>, rather than promising the same shape for every solid, star or ocean. A <rotating reference frame> introduces <centrifugal acceleration> and <Coriolis acceleration>, which must be included separately.
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