Tidal evolution of Kleopatra spin (dashed magenta) and moon orbits (solid green, blue), computed as the difference in the true longitudes Δλ₀, Δλ₁, Δλ₂ between dynamical models with and without tides. The value of the time-lag Δt = 47 s corresponds to Ṗ₂ in Fig. 2. The epoch when mean periods coincide was arbitrarily shifted (↔) towards 2 456 500. Moreover, the mean periods were adjusted (↕) to fit observations in 2008 and 2018.