Tidal Locking Time Calculator

Tidal Locking

Tidal locking is the long-term outcome of tidal friction: the satellite (or planet) slowly synchronizes its rotation to its orbital period, always presenting the same face to its primary. The Moon is tidally locked to Earth. Most close-in exoplanets around M-dwarfs are believed to be locked to their stars, with profound implications for atmosphere and habitability.

Standard Formula (MacDonald 1964 / Goldreich-Soter)

t_lock ≈ (ω × a⁶ × I × Q) / (3 × G × M_primary² × k₂ × R⁵)

For rigid rocky bodies, the widely used simplified estimate is:

t_lock (years) ≈ 6 × 10¹⁰ × a⁶ × R × μ × Q / (m_s × M²)

with a and R in meters, masses in kilograms, and μ the rigidity (~3 × 10¹⁰ N/m² for rock).

Where:

• a = semi-major axis
• M = mass of primary, m_s = mass of the locking body
• R = radius of the locking body
• Q ≈ 100 for rocky planets, 10⁴ – 10⁶ for gas giants
• μ = rigidity (~3 × 10¹⁰ N/m² rocky, ~4 × 10⁹ icy)

This calculator estimates the locking body’s mass from its radius assuming rocky density (5.5 g/cm³, like Earth).

Worked Example — Earth and the Moon

At the Moon’s current distance the formula gives roughly 2 × 10⁸ years, comfortably less than the Solar System’s 4.6-billion-year age, so a locked Moon is expected. In reality it locked far faster: the Moon formed perhaps 10 times closer, where the a⁶ term makes locking about a million times quicker, mere centuries. It then drifted outward already synchronized. Mercury did not lock to a 1:1 ratio but to a 3:2 spin-orbit resonance, a special case driven by orbital eccentricity.

Locking Time vs Distance

The a⁶ scaling is dramatic: doubling the orbital distance increases the locking time 64×. This is why close-in planets lock first and outer planets retain their spins for the lifetime of the system.

Earth at 1 AU around the Sun is therefore safe — the Sun will leave the main sequence long before our planet locks.

Implications for Habitable Zones

Around an M-dwarf star, the habitable zone sits inside about 0.1 AU. Locking times in that region are much shorter than the age of the universe — so M-dwarf habitable-zone worlds are likely tidally locked, with permanent day and night sides separated by a terminator. Whether such worlds can support liquid water and life is a major open question in exoplanet research.

Caveats

The standard formula is an order-of-magnitude estimate. Real tidal evolution depends on:

• Orbital eccentricity (drives spin-orbit resonances)
• Internal structure (Q varies by orders of magnitude)
• Atmospheric thermal tides
• Stellar irradiation history

Mercury’s 3:2 resonance, and the dual locking of Pluto-Charon, show that “always 1:1” is a simplification. Use the calculated time as a rough timescale, not a precise prediction.