= Quadrupolar tidal forcing
For a point-mass companion at separation $a$, subtract the constant and linear terms of its <Newtonian gravitational potential> in a frame accelerating with the target's center. The leading residual is
$$
\Phi_{\mathrm{tide}}=-\frac{GM_2}{2a^3}\left[3(\mathbf n\cdot\mathbf r)^2-r^2\right].
$$
It is degree two in <spherical harmonics>. In a circular orbit with angular frequency $\omega_o$, its time-dependent $m=\pm2$ components force at $2\omega_o$ in a frame with no spin; there is also a static $m=0$ component. This is the leading approximation in $r/a$, and the tidal amplitude may be absorbed into a harmonic-dependent constant.
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