Tidal Locking Timescale Calculator

Tidal locking (synchronous rotation) occurs when a body's rotation period equals its orbital period, so it always shows the same face to its partner. It happens because tidal forces raised by the partner create a bulge in the body. If the body rotates faster than it orbits, the bulge is slightly ahead of the partner due to internal friction (characterized by the tidal quality factor Q), creating a torque that slows the rotation. Over time, this torque brings the rotation rate into synchrony with the orbital rate.

The tidal quality factor Q measures how efficiently a body dissipates tidal energy. Low Q means high dissipation (locking is faster); high Q means low dissipation (locking is slower). For rocky bodies: Earth Q ≈ 12 at monthly periods (Moon) and Q ≈ 280 at annual periods (Sun). The Moon has Q ≈ 38. Mars has Q ≈ 80-170. For gas giants: Jupiter Q ≈ 30,000-300,000; Saturn Q ≈ 1,000-10,000. These values are poorly constrained and can change the locking timescale by orders of magnitude.

The second Love number k₂ describes how much a body deforms under an external gravitational potential. It ranges from 0 (rigid, no deformation) to 1.5 (uniform fluid sphere). Rocky planets: Earth k₂ ≈ 0.30, Moon k₂ ≈ 0.024, Mars k₂ ≈ 0.17, Mercury k₂ ≈ 0.1. Ice-rich bodies: k₂ ≈ 0.3-0.5. Gas giants: Jupiter k₂ ≈ 0.37, Saturn k₂ ≈ 0.39. A smaller k₂ means less tidal deformation and a longer locking timescale.

Yes, completely. The Moon is in 1:1 synchronous rotation: its rotation period (27.3 days) exactly equals its orbital period around Earth. This is why we always see the same side of the Moon from Earth — the near side. The Moon locked to Earth approximately 1 billion years after formation, consistent with the Peale formula using Q ≈ 38. The far side of the Moon was completely unknown until Soviet spacecraft photographed it in 1959.

Mercury is in a 3:2 spin-orbit resonance: it rotates 3 times for every 2 orbits around the Sun. This is a stable resonance maintained by Mercury's orbital eccentricity (e = 0.206). Near perihelion, the tidal torque from the Sun is much stronger, and the 3:2 resonance is the lowest-energy trapped state for a body with significant eccentricity. If Mercury's orbit were circular, it would be tidally locked in the 1:1 state like most moons.

Probably, for planets orbiting M-dwarf (red dwarf) stars. Red dwarfs are much fainter than the Sun, so the habitable zone is close in (a ≈ 0.02-0.15 AU). At such small separations, the tidal locking timescale is typically less than 1 billion years — shorter than the stellar lifetime. Proxima Centauri b, at 0.0485 AU from a 0.122 solar-mass star, is very likely tidally locked. This has important implications for habitability: the permanent day side may be too hot and the night side too cold, though atmospheric circulation could redistribute heat.

The gyration constant α (also called the moment of inertia factor) appears in I = α*m*R², the moment of inertia. It depends on how mass is distributed inside the body: α = 0.4 for a uniform sphere, α = 0.33 for Earth (denser core), α = 0.394 for the Moon. Gas giants have α ≈ 0.25 (centrally concentrated mass). Use α = 0.4 for an unknown rocky body as a reasonable default; the effect on locking timescale is proportional to α, so uncertainty here is usually minor compared to uncertainty in Q.

When a body becomes tidally locked, tidal forces also gradually circularize its orbit, damping any eccentricity toward zero. The eccentricity damping timescale is roughly t_ecc ~ (Q/k₂*n)*(a/R)^5*(m/M). It is typically comparable to or shorter than the locking timescale. Bodies in resonances like Mercury (3:2) or the Galilean moons (Laplace resonance) have their eccentricity excited by the resonance, competing with tidal damping. Io's eccentricity is maintained by its Laplace resonance with Europa and Ganymede, keeping it geologically active through intense tidal heating.

The locking timescale scales as a^6 — the sixth power of the semi-major axis. This makes it extraordinarily sensitive to distance. A moon at twice the orbital distance locks 2^6 = 64 times more slowly. At three times the distance, it takes 3^6 = 729 times longer. This steep scaling explains why close-in moons (like the Martian moon Phobos at 9,376 km) are already locked, while distant irregular satellites are not. Phobos is tidally locked to Mars; Deimos is also locked but at greater distance took longer.

By conservation of angular momentum, as the moon spirals slightly outward and locks, the planet's rotation slows slightly. The Moon has slowed Earth's rotation from about 6 hours per day 4.5 billion years ago to 24 hours today — losing about 18 hours of day length. Earth is also slowly becoming tidally locked to the Moon: Earth's rotation is currently slowing by about 2 milliseconds per century. The Moon is receding from Earth at 3.8 cm/year as Earth's rotational angular momentum transfers to the Moon's orbital angular momentum.

Yes, in principle. A sufficiently massive moon close enough to a planet can tidally lock the planet before the star can. In practice, for solar system planets, stellar tides dominate unless the planet is far from the star. For Earth, the solar tidal locking timescale (ignoring the Moon) is about 50 Gyr — longer than the stellar lifetime. The Moon's tidal effect on Earth is stronger than the Sun's at present tidal periods, so Earth will tidally lock to the Moon before locking to the Sun, if the Sun does not first expand into a red giant.