Tidal Force Calculator

A tidal force is not gravity itself but the spatial gradient of gravity: the difference between what a distant mass pulls on the near side of an extended body and what it pulls on that body center. Expanding Newtonian attraction for a body of radius r at separation d and keeping the leading term gives a differential acceleration of 2GMr divided by d cubed, which multiplied by the affected mass m yields F = 2GMmr / d cubed, with G = 6.674e-11 N m2/kg2. The cube law is what matters in practice: halving the separation multiplies the tide-raising force by eight, while ordinary attraction only quadruples. For the Earth-Moon pair at 384,400 km, the lunar differential acceleration across the 6,371 km Earth radius is near 1.1e-6 m/s2, about one ten-millionth of surface gravity, enough to lift an open-ocean equilibrium bulge of only roughly 0.5 m. Because this is a first-order expansion, it is reliable only when r is far smaller than d. Near the Roche limit, about 2.44 primary radii for a fluid satellite held together by self-gravity alone, the full expression is needed. The formula also treats the body as a passive test object, ignoring rigidity, internal friction and the centrifugal terms of the rotating two-body frame. Real ocean tides depart widely from the equilibrium value because basins resonate: the Bay of Fundy exceeds 16 m of range while mid-ocean amphidromic points see almost none. The Sun raises roughly 46 percent of the lunar tide, and repeated flexing dissipates energy as heat, the mechanism that keeps Io volcanically active.

Isaac Newton gave the first physical account of tides in Book III of the Principia in 1687, showing that the Moon and Sun raise an equilibrium bulge through differential attraction. Kepler had earlier guessed at lunar attraction; Galileo dismissed it. The static picture could not predict the timing of tides at real ports, and in 1740 the Paris Academy prize on the subject was shared by Daniel Bernoulli, Leonhard Euler, Colin Maclaurin and Antoine Cavalleri, who refined equilibrium theory. Pierre-Simon Laplace replaced it in 1775 and 1776 with a dynamic theory, formulating the partial differential equations still called the Laplace tidal equations, which treat the tide as a forced wave in a rotating fluid shell. The wider consequences of tidal forces were worked out over the following two centuries. Edouard Roche derived the disruption limit that carries his name in 1848. George Howard Darwin, from about 1879, developed the theory of tidal friction and argued that dissipation steadily transfers angular momentum from the spin of the Earth to the lunar orbit. That prediction was confirmed quantitatively after the Apollo 11 crew left a laser retroreflector on the Moon in 1969, since lunar laser ranging shows the Moon receding at roughly 3.8 cm per year. In 1979 Stanley Peale, Patrick Cassen and Ray Reynolds predicted in Science that tidal heating should melt the interior of Io, and Voyager 1 photographed active volcanoes there within weeks.