What tides are
Gravity weakens with distance. A moon pulls the near side of a planet more strongly than its center, and the center more strongly than the far side. That difference stretches the planet along the line to the moon, raising two bulges: one facing the moon and one opposite. That is why most coasts see two high tides a day.
The formula
For a body of mass M at distance d, the tidal acceleration across a planet of radius r is approximately:
a_tidal ≈ 2 × G × M × r ÷ d³
Tidal effects fall with the cube of distance, much faster than gravity itself, which falls with the square.
Moon versus Sun
- Moon: about 1.1 × 10⁻⁶ m/s² at Earth’s surface.
- Sun: about 0.5 × 10⁻⁶ m/s², roughly 46% of the Moon’s effect.
The Sun’s gravity on Earth is about 180 times stronger than the Moon’s, yet its tidal effect is weaker because it is about 390 times farther away and the cube of distance wins.
Spring and neap tides
- Spring tides: at new and full moon, the Sun and Moon line up and their effects add, about 1.46 times the lunar tide alone.
- Neap tides: at first and last quarter, they act at right angles and partly cancel, about 0.54 times the lunar tide.
Local geography amplifies these forces enormously: tides range from a few inches in the Mediterranean to over 50 ft (16 m) in Canada’s Bay of Fundy.
Example
With the Moon’s mass (0.0123 Earth masses) at 384,400 km and Earth’s radius (6,371 km), the calculator gives about 1.10 × 10⁻⁶ m/s², about one ten-millionth of Earth’s gravity. Small as it is, acting on whole oceans it moves enormous volumes of water.
Other tidal effects
- Tidal locking: the Moon always shows the same face because Earth’s tides on it slowed its rotation.
- Tidal heating: Jupiter’s moon Io is the most volcanic body in the solar system because Jupiter’s tides flex it constantly.
- Roche limit: close enough to a planet, tides can tear a moon apart, one origin proposed for planetary rings.
- Earth’s day is lengthening: tidal friction slows Earth’s rotation by about 2 milliseconds per century and pushes the Moon about 3.8 cm (1.5 in) farther away each year.