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Body
Mass, size, interior stiffness (k2) and dissipation (Q).
Orbit
Distance and initial spin matter most: lock time scales as a^6.
Layers
Model checks
The science
Tidal torque
The primary raises a tidal bulge on the body. Because the body is not perfectly elastic, the bulge lags the line to the primary by a small angle δ ≈ 1/(2Q). The primary pulls on the lagging bulge and exerts a torque that changes the spin. For a body of mass m, radius R, moment of inertia C = α m R2, Love number k2 and quality factor Q at distance a from a primary of mass M:
dω/dt = −sgn(ω − n) · 3 (k2/Q) G M2 R3 / (α m a6), with mean motion n = √(G (M + m) / a3).
Locking time
The strong a6 dependence is why close moons lock quickly and far ones never do. Gladman et al. (1996) give tlock ≈ ω0 a6 α m Q / (3 G M2 k2 R3). The page integrates the full equation, including the slowly changing a, and compares.
Angular momentum
Spin plus orbital angular momentum is conserved: Cω + μ√(G(M+m)a) = const, with reduced mass μ. A body spinning faster than it orbits gives angular momentum to the orbit, so the orbit expands (the Moon recedes from Earth by about 3.8 cm per year). The page uses this to update a every step.
Honest simplifications: constant-Q, constant-torque model; circular, coplanar orbit; no obliquity, eccentricity, libration, resonances (such as Mercury's 3:2) or tides raised on the primary; k2 and Q are order-of-magnitude values and real ones are poorly known. The bulge lag drawn is exaggerated. Results are order-of-magnitude estimates.
What to try
- Choose Moon (Earth), then double the orbit radius: the lock time grows by roughly 26 = 64.
- Compare Phobos and the red-dwarf planet: close orbit means fast locking.
- Raise Q to 500 (a stiffer, less dissipative body) and watch the locking time grow.
- Watch the orbit expand while the spin slows (graph a/a0).