tides and orbital resonance explained
The Moon pulls on the oceans, so why does the water bulge out on both sides of Earth rather than only toward the Moon? Because Earth is not pulled with one uniform force: the side near the Moon is pulled harder and the far side more weakly. That difference in force, the gradient of gravity, stretches Earth toward the Moon and away from it at once, raising two bulges. Over long ages the same gradient brakes a moon's spin until one face stays locked toward its planet, and resonances, in which orbital periods form whole-number ratios, hold the orbits of several moons in step. The lesson ends at the Roche limit, the distance inside which tides tear a moon apart into a ring.
Gravity falls off as the square of the distance, so the side nearest the Moon is pulled harder than the center and the far side more weakly. The three arrows at the top show that difference: beside the gold center arrow, the near arrow is longer and the far arrow shorter. Subtract the pull at the center from the pull at each point and what remains pushes the near side toward the Moon and the far side away from it. So the body below stretches in the two directions of the orange arrows, raising not one bulge but two. Bring the Moon closer with the distance slider and the difference grows steeply, as 1/r³, while the ellipse visibly elongates.
A bulge raised this way does not point exactly at the partner body. If the moon spins faster than it orbits, its rotation carries the bulge slightly ahead, and the planet's gravity pulls back on the misaligned bulge, producing a torque that slows the spin; the orange curved arrow is that brake. Press play and the spin gradually slows, the lead of the gold ellipse shrinks, and at last spin and orbit match, so the face marked with the green dot stays turned toward the planet on the left. A body that keeps one face toward its partner this way is tidally locked, which is why the Moon always shows Earth the same face.
Spin does not have to lock at exactly 1:1 with the orbit. On an elliptical orbit other whole-number ratios can hold as well, and Mercury is locked at three spins for every two orbits, or 3:2. The lower curve draws stability against the ratio as a landscape of wells; move the spin-to-orbit slider and the ball snaps into the bottom of a well near 1:1 and near 3:2. In the orbit sketch above, the marked face always points at the center for synchronous rotation, while under Mercury's lock it faces a different way on each pass. Ratios caught in between keep getting pushed away by the tides and cannot last.
Moons pull on one another too. When the orbital periods of two neighboring moons form a whole-number ratio, their conjunctions, the moments when they line up, keep recurring at the same places. The small tugs then pile up at the same spots every time and hold the orbits in step. The classic example is Jupiter's Io, Europa, and Ganymede, whose periods are locked at 1:2:4 in the Laplace resonance. The three moons in the figure really move, and when the slider sets the period ratio to a whole number, the faint dots marking conjunctions gather at a few places and the readout turns green. Move off the whole number and the conjunctions scatter around the circle.
Tides try to stretch a moon apart while the moon's own gravity holds it together. The blue bar on the left is the moon's self-gravity and the orange bar is the stretching force of the tide; as the moon approaches the planet, only the orange bar grows, as the inverse cube of the distance. The dashed gold circle marks where the two balance, the Roche limit, roughly d ≈ 2.44·R·(ρplanet/ρmoon)1/3. Slide the moon inside that circle and the stretching wins, so the moon can no longer hold together and spreads along its orbit into a ring of orange fragments. Saturn's rings are thought to have formed this way.