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Tidal forces arise from the differential gravitational attraction across an extended body. These forces circularize elliptical orbits, transfer angular momentum between bodies, and lock rotation to orbits (tidal locking). Tidal heating from friction inside moons drives volcanic activity and tectonics, as seen on Io and Europa.
Calculate the tidal force on an extended body by computing gravitational force differences across it. Show how tidal torque affects rotation. Apply to Earth-Moon system: explain why the Moon is tidally locked and why its orbit slowly recedes. Relate to other moons.
From your study of the two-body problem and gravitational potential energy, you know that gravity between two point masses produces a clean, predictable orbit. Tidal forces emerge when we drop the point-mass approximation and recognize that real bodies have finite size. The side of a moon facing its planet is closer to the planet than the far side, so it feels a stronger gravitational pull. This differential force across the body — not the absolute force — is the tidal force, and it stretches the body along the line connecting the two objects while compressing it perpendicular to that line, creating the characteristic tidal bulge.
The consequences of tidal forces depend critically on whether the body's rotation is synchronized with its orbit. Imagine a moon rotating faster than it orbits: its tidal bulge, raised by the planet's gravity, gets carried slightly ahead of the line connecting the two bodies because the moon's rotation sweeps the bulge forward. The planet's gravity then pulls back on this displaced bulge, creating a tidal torque that slows the moon's rotation. Energy is dissipated as friction inside the moon (the bulge is constantly being raised, displaced, and dragged back), and angular momentum is transferred from the moon's spin to its orbit. This process continues until the moon's rotation period exactly matches its orbital period — a state called tidal locking. Our Moon is tidally locked to Earth, which is why we always see the same face. Given enough time, the same process would lock Earth's rotation to the Moon's orbit, though this would take far longer than the age of the solar system.
Tidal locking is an endpoint, but the journey there produces remarkable effects. Tidal heating occurs when a body's orbit is eccentric — even if rotationally locked, the varying distance means the tidal bulge changes size and orientation throughout the orbit, flexing the interior and generating heat through friction. Jupiter's moon Io is the most dramatic example: gravitational interactions with Europa and Ganymede maintain Io's orbital eccentricity, and the resulting tidal flexing produces enough internal heat to drive the most volcanically active surface in the solar system. Europa's subsurface ocean is likely maintained by the same tidal heating mechanism, making tidal forces relevant not just to orbital dynamics but to questions of habitability.
The angular momentum transfer in tidal interactions also reshapes orbits over geological time. In the Earth-Moon system, tidal friction in Earth's oceans transfers angular momentum from Earth's rotation to the Moon's orbit, causing Earth's day to lengthen by about 2.3 milliseconds per century and the Moon to recede by roughly 3.8 centimeters per year. Running this process backward, the Moon was much closer to Earth in the past and Earth rotated much faster. Tidal forces also circularize orbits: an eccentric orbit dissipates more energy at closest approach (where tidal forces are strongest), and energy dissipation without angular momentum loss drives eccentricity toward zero. This explains why close-in moons and many short-period exoplanets have nearly circular orbits despite potentially forming on eccentric ones.