Things get a little bit more interesting when one considers non-binary systems. In the case of Earth, there are two significant sources of differential force acting, namely the moon and the sun.
To find out the relative strengths of the differential forces acting on Earth we use the approximation for the differential force strength (Eq. 3) and the values for the masses and orbital distances of the sun and moon.
Δa =~ GM(4R/r³)
rmoon=384.4*108cm
rsun=1.4960*1013cm
Mmoon=7.35*1025g
Msun=1.989*1033g
=> Δamoon =~ 4RG(1.29*10-6) m/s²
=> Δasun =~ 4RG(5.941*10-7) m/s²
Here we find that the differential force exerted by the moon on the earth is larger that that exerted by the sun, though comparable so that the sun's could hardly be considered negligible.
The ratio follows:
=> Δamoon /Δasun =~ 2.18
As the rotation of the moon is not in any way synchronized with the earth's orbit about the sun, these differential forces interact with each other in a periodic way, sometimes constructively, sometimes destructively.
These periods of constructive interaction are called "spring tides" and correspond to the times when the Sun, Earth, and moon are aligned. At these times the tidal bulges are at their largest.
The periods of destructive interaction are called "neap tides" and correspond to the times when the Sun makes a right angle with the Earth and the moon. At these times the tidal bulges are at their smallest.
