Tidal Locking Time Calculator
Estimate how long it takes a satellite or planet to become tidally locked to its primary using the standard Q-rigidity formula.
Used in exoplanet habitability.
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)
Approximate dimensionally-cleaner form:
t_lock ≈ (ω × a⁶ × Q) / (3 × G × M² × k₂ × R⁵ / I)
For practical estimates, astronomers use:
t_lock (years) ≈ 6 × 10¹⁰ × (a / 1 AU)⁶ × (M_sun / M)² × (R_J / R)⁵ × (Q / 100)
Where:
- a = semi-major axis
- M = mass of primary
- R = radius of the locking body
- Q ≈ 100 for rocky planets, 10⁴ – 10⁶ for gas giants
- k₂ = tidal Love number (~0.3 rocky, ~0.5 gaseous)
- ω = initial spin rate
Worked Example — Earth and the Moon
Plugging in lunar parameters yields t_lock ~ 1 Gyr, which roughly matches the locked state seen today. The Moon locked relatively early in its history; 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.
| Orbit (around Sun-mass star) | Lock time (Earth-like) |
|---|---|
| 0.05 AU | ~ 5 × 10⁴ yr |
| 0.1 AU | ~ 3 × 10⁶ yr |
| 0.3 AU | ~ 2 × 10⁹ yr |
| 1.0 AU | ~ 6 × 10¹⁰ yr (longer than universe age) |
| 5.0 AU | ~ 4 × 10¹³ yr |
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.