Initial Conditions
System Parameters
Tidal Dissipation
Visualization
Presets
Current State
Key Effects
- Tidal dissipation circularizes orbits
- Angular momentum transfer
- Spin-orbit synchronization
- Semi-major axis evolution
Watch tidal forces circularize orbits and lock rotation to revolution—explore Earth-Moon evolution over billions of years
Tidal forces between celestial bodies cause their orbits and rotation rates to evolve over time. The Moon raises tides on Earth, and these tides dissipate energy through friction. This energy loss causes the Moon to gradually recede from Earth (currently ~3.8 cm/year) and Earth's rotation to slow down. Eventually, both bodies will become tidally locked, always showing the same face to each other—though this would take tens of billions of years for Earth-Moon.
The top panel shows the orbit evolving over time. Tidal dissipation preferentially removes energy from eccentric orbits, causing them to circularize. High eccentricity means the bodies experience varying tidal forces throughout the orbit—strongest at periapse (closest approach). This asymmetric forcing dissipates more energy than a circular orbit would, driving e → 0. Most major moons in the solar system (like our Moon, or Jupiter's Galilean moons) have nearly circular orbits due to this process.
The planet (larger body) is shown rotating. Initially it may spin much faster or slower than the orbital period. Tidal bulges raised by the moon create torques that act to synchronize the spin period with the orbital period. When synchronized, the same hemisphere always faces the moon—this is called tidal locking. Our Moon is tidally locked to Earth (we always see the same face), and Earth will eventually become tidally locked to the Moon as well.
Total angular momentum of the system is conserved (ignoring external torques). As the orbit circularizes and spins synchronize, angular momentum transfers between orbital motion and rotational motion. If the planet initially spins faster than the orbital motion, angular momentum flows from spin to orbit, causing the orbit to expand (semi-major axis increases) and the planet to spin down. This is exactly what's happening with Earth-Moon: Earth's day is lengthening and the Moon is receding.
Tidal forces arise because gravitational force varies across the extended body. The near side feels stronger gravity than the far side, creating a tidal bulge. Real materials aren't perfectly elastic—as the body rotates through this bulge, internal friction dissipates energy as heat. The efficiency of this dissipation is parameterized by the quality factor Q: lower Q means more dissipation and faster tidal evolution. Earth has Q ~ 10-100, while the Moon has Q ~ 30.
Tidal evolution timescales depend sensitively on distance (∝ a⁶ for circularization) and mass ratio. Close-in systems evolve rapidly: exoplanets at a ~ 0.05 AU can tidally lock to their host stars in millions of years. Distant systems evolve glacially: the Earth-Moon system has been evolving for 4.5 billion years and is still not fully synchronized. The circularization timescale is typically shorter than the synchronization timescale, so moons often achieve circular orbits before full tidal locking.
The Moon is tidally locked to Earth (period ~ 27.3 days), but Earth's day (24 hours) hasn't caught up yet. Lunar laser ranging measurements show the Moon receding at 3.82 cm/year. Working backward, the Moon formed much closer to Earth (~15-20 Earth radii, compared to 60 today) about 4.5 billion years ago, likely from a giant impact. Early Earth had a day lasting only ~5 hours. Tidal dissipation in Earth's oceans (especially in shallow seas) dominates the energy loss.
Pluto and Charon form a double-locked system: both are tidally locked to each other with a 6.4-day period. Pluto and Charon always show the same faces to each other. This represents the endpoint of tidal evolution. Charon's mass is ~12% of Pluto's mass—unusually large for a moon, which accelerated the tidal locking process. From Pluto, Charon appears stationary in the sky (always above the same spot). From the far side of Pluto, you'd never see Charon at all.
Many exoplanets orbit extremely close to their stars (a < 0.1 AU). These "hot Jupiters" experience strong tidal forces and typically become tidally locked within millions of years. The planet's rotation synchronizes with its orbit (same hemisphere always faces the star), creating extreme day-night temperature contrasts. Some close-in planets may spiral inward and eventually be engulfed by tidal orbital decay—the reverse of Earth-Moon because the star rotates slower than the planet orbits.
Circularization: Set initial eccentricity e₀ = 0.8 and watch the orbit gradually become more circular over time. The evolution accelerates when the bodies are closer (smaller semi-major axis). Notice how the eccentricity graph shows exponential decay.
Spin Synchronization: Start with a fast spin (spin period = 1 day) while the orbital period is much longer. Watch the planet's rotation slow down as the rotation marker gradually synchronizes with the moon's orbital position. Angular momentum from the planet's spin transfers to the orbit, causing the moon to recede.
Earth-Moon System: Click the "Earth-Moon" preset. This uses realistic mass ratios and tidal parameters (though hugely sped up for visualization). Observe the Moon gradually receding and Earth's spin slowing. The Moon is already tidally locked, so only Earth's spin evolves.
Double Locking (Pluto-Charon): Click "Pluto-Charon" preset. The large mass ratio (0.12) means both bodies' spins and the orbit all synchronize to a common period. This is the endpoint of tidal evolution—a mutually locked binary system.
Quality Factor Effect: Decrease log Q (quality factor) to 1.0 or lower. This increases tidal dissipation, making the evolution much faster. Q measures how "springy" vs "viscous" the body is. Rocky bodies have Q ~ 100, icy bodies Q ~ 100-1000, gas giants Q ~ 10⁴-10⁶. Lower Q means more tidal heating and faster evolution.
Close vs Distant Orbits: Start with a = 3 (close orbit) and note how quickly the orbit evolves. Then reset and try a = 15 (distant orbit)—evolution is much slower. Tidal effects scale as (R/a)⁶ for circularization and (R/a)³ for synchronization, so distance matters enormously.
Outward Spiral: Start with the planet spinning faster than the orbital period (e.g., spin = 5 days, orbital period = 20 days at a = 5). The moon will recede (a increases) as angular momentum transfers from planet spin to orbital motion. This is Earth-Moon's current state.
Inward Spiral: If the planet spins slower than synchronous, the opposite happens: the moon spirals inward and the planet spins up. This is relevant for some exoplanets around slowly rotating stars. Eventually the moon may reach the Roche limit and be tidally disrupted.
Tidal evolution shapes planetary systems across the universe. It explains:
George Darwin (son of Charles Darwin) pioneered tidal evolution theory in the 1880s, calculating that the Moon must have been much closer to Earth in the past. He proposed the "fission hypothesis"— that the Moon split off from a rapidly rotating Earth. While we now know the Moon formed from a giant impact (not fission), Darwin correctly recognized tidal dissipation's importance and that Earth's rotation was slowing. Apollo astronauts placed retroreflectors on the Moon, enabling laser ranging measurements that confirmed the Moon's recession rate and validated tidal evolution theory quantitatively.