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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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